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# Archivio 2023

2023

03 ottobre

Agnese Barbensi

Seminario di algebra e geometria, interdisciplinare

Topological data analysis has been demonstrated to be a powerful tool to describe topological signatures in real-life data, and to extract complex patterns arising in natural systems. An important challenge in topological data analysis is to find robust ways of computing and analysing persistent generators, and to match significant topological signals across distinct systems. In this talk, I will present some recent work dealing with these problems. Our method is based on an interpretation of persistent homology summaries with network theoretical tools, combined with statistical and optimal transport techniques.

2023

03 ottobre

Daniele Celoria

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

After giving a general introduction to Dimofte-Gaiotto-Gukov's 3D index for cusped hyperbolic 3-manifolds, we'll dive into some of its relations with basic hypergeometric series. Then we'll describe an ongoing effort to prove how the 3D index changes under Dehn surgery. This is work in progress with Profs C. Hodgson and H. Rubinstein.

2023

29 settembre

Lorenzo Cerboni Baiardi

nell'ambito della serie: TOPICS IN MATHEMATICS 2022/2023

Seminario di sistemi dinamici

This presentation provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaotic trajectories with the related phenomena of bubbling, on–oﬀ intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly inﬂuence the eﬀects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.

2023

29 settembre

Davide Bianchi

Seminario di analisi numerica

We investigate a Tikhonov method that embeds a graph Laplacian operator in the penalty term (graphLa+). The novelty lies in building the graph Laplacian based on a first approximation of the solution derived by any other reconstruction method. Consequently, the penalty term becomes dynamic, depending on and adapting to the observed data and noise. We demonstrate that graphLa+ is a regularization method and we rigorously establish both its convergence and stability properties.
Moreover, we present some selected numerical experiments in 2D computerized tomography, where we combine the graphLa+ method with several reconstructors: Filter Back Projection (graphLa+FBP), standard Tikhonov (graphLa+Tik), Total Variation (graphLa+TV) and a trained deep neural network (graphLa+Net). The quality increase of the approximated solutions granted by the graphLa+ approach is outstanding for each given method. In particular, graphLa+Net outperforms any other method, presenting a robust and stable implementation of deep neural networks for applications involving inverse problems.

2023

27 settembre

Andre Nies

Seminario di algebra e geometria, analisi matematica, interdisciplinare, logica

The course will cover applications of logic to various mathematical areas. They will be discussed in four largely independent blocks, each consisting of 2 hours of lecturing and one hour of discussions and exercises.
1. First-order logic and group theory: We introduce the notion of a f.g. group to be QFA (quasi-finitely axiomatizable): the group can be uniquely described by a first-order sentence, among the finitely generated groups. We provide examples, and discuss recent work of the analogous notion for the class of profinite groups
2. Descriptive set theory and the isomorphism problem for Polish groups: We show that isomorphism of profinite groups is as complex as possible w.r.t. Borel reducibility, while isomorphism of oligomorphic groups is essentially countable.
3. Computabity and totally disconnected, locally compact (tdlc)groups: We initiate an algorithmic theory of tdlc groups, and discuss an example of a computably tdlc group with noncomputable scale function.
4. Continuous logic and the Connes embedding problem: We discuss an alternative method due to Goldbring and Hart, based on continuous logic, how a recent breakthrough result in quantum information theory can be used to refute a long-open conjecture on von Neumann algebras. This avoids the various equivalences to the CEP that were applied in the original refutation.

2023

26 settembre

Luigi Caputi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

Magnitude homology, as introduced by Hepworth and Willerton, is a bigraded homology theory of metric spaces that categorifies Leinster's notion of magnitude. We will show that, when restricting to graphs of bounded genus, magnitude homology is a finitely generated functor. As a consequence, we will prove that the ranks of magnitude homology, in each homological degree, grow at most polynomially in the number of vertices, and that its torsion is bounded. We will use the categorical framework of Groebner categories developed by Sam and Snowden, in the spirit of Ramos, Miyata and Proudfoot. This is joint work with C. Collari.

2023

26 settembre

Daniel Juteau

Seminario di algebra e geometria

Abstract: I will talk about joint work with Baohua Fu, Paul Levy and
Eric Sommers about the geometry of nilpotent cones of complex semisimple
Lie algebras, particularly in exceptional types (Kraft and Procesi
studied classical types in the 1980's). One very particular singularity
in E8 led us to consider a family of 4-dimensional isolated symplectic
singularities; in another paper with two more authors (Gwyn Bellamy and
Cédric Bonnafé), we show that they admit many different descriptions and
that they have trivial local fundamental group (so they are not quotient
singularities), thus answering a question that Beauville raised in 2000.

26/09/2023

28/09/2023

28/09/2023

Taro Sano

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Minicourse for PhD and master students. Title: Deformations of Fano and Calabi-Yau varieties.
Abstract: In the classification of algebraic varieties, we parametrize varieties with similarities, thus it is natural to consider families/deformations of varieties. In good cases, we can parametrize some varieties over a smooth base space and this makes the description of the moduli space of those varieties reasonable.
Fano varieties and Calabi-Yau varieties are fundamental objects in the classification. In the talks, I'll explain that, when varieties are Fano or Calabi-Yau, we have such smooth parameter spaces. In most parts, I'll concentrate on smooth Fano/Calabi-Yau varieties. If time permits, I'll also talk about more general cases.

26/09/2023

28/09/2023

28/09/2023

Yasunari Nagai

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Minicourse for master and PhD students.
Title:
Hilbert scheme of points on a surface and its degeneration.
Abstract:
Hilbert scheme of zero dimensional subschemes on a (quasi-)projectuve surface gives an interesting construction of higher dimensional smooth (quasi-)projective varieties of even dimension. For example, if the surface is a K3 surface, the Hilbert scheme gives an example of higher dimensional irreducible symplectic projective manifold. In the first part of this mini course, I explain the basic properties of the Hilbert scheme of points only assuming Hartshorne (i.e. a basic knowledge of modern algebraic geometry). In the second part, I put emphasis on the degeneration of Hilbert schemes. The motivation comes from the study of the boundary behavior of the period map of irreducibel symplectic Kähler manifolds. I also explain an explicit construction of the degeneration of Hilbert schemes.

26/09/2023

28/09/2023

28/09/2023

Daniele Faenzi

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: Moduli spaces of bundles in low genus and degeneracy loci.
Abstract: Coble hypersurfaces enjoy very special properties related to abelian varieties and moduli of semistable bundles of rank r with trivial determinant on a curve C of genus g, notably when (g,r) equal to (2,3) or (3,2).
Using orbital degeneracy loci arising from Vinberg theta-groups and Hecke cycles, we describe moduli of semistable bundles with fixed odd determinant as subvarieties of Grassmannians, again when (g,r) equals (2,3) or (3,2).
The geometry of these loci and of their singularities parallels that of Coble hypersurfaces and is related to projective models of K3 surfaces of genus 13 and 19.
Joint work with Vladimiro Benedetti, Michele Bolognesi, Laurent Manivel.

26/09/2023

28/09/2023

28/09/2023

Dario Faro

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: Gauss-Prym maps on Enriques surfaces.
Abstract:
Let C be a complex projective algebraic curve and let L and M be two line bundles on C. One can associate L and M with some natural maps between spaces of global sections of certain sheaves on C. These are called Gaussian-Wahl maps. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

26/09/2023

28/09/2023

28/09/2023

Lucas Li Bassi

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: The Fano variety of lines on a cyclic cubic fourfold.
Abstract: Among Fano varieties of K3 type (or FK3 for short) the cubic fourfold stands out for its historical significance. Indeed, long before the terminology FK3 was born there were already examples of a relation between this famous cubic hypersurface and irreducible holomorphic symplectic (or IHS for short) manifolds. One method to associate a smooth cubic fourfold with an IHS manifold involves the Fano variety of lines on it. This is, as proven by Beauville-Donagi, an IHS manifold of type K3[2]. This relation becomes even more intriguing when considering mildly singular cubic fourfolds, e.g. cubic fourfolds Y that are triple covering of P4 branched over a singular cubic threefold. In this case we have that F(Y), the Fano variety of lines on Y, is birational to an IHS manifold of type K3[2]. This fact has been used by Boissière-Camere-Sarti and by me to study some compactification of the moduli spaces of irreducible holomorphic symplectic manifolds with an order three non-symplectic automorphism. In order to achieve this result the authors do not consider the rich geometry of F(Y).
I will present recent results obtained in collaboration with Samuel Boissière and Paola Comparin that explain how the geometry of F(Y) gives us a better understanding of the deep relation between cyclic cubic fourfolds and IHS manifolds of type K3[2] with a non-symplectic automorphism of order three.

26/09/2023

28/09/2023

28/09/2023

Alex Massarenti

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: On the (uni)rationality problem for quadric bundles and hypersurfaces.
Abstract: A variety X over a field is unirational if there is a dominant rational map from a projective space to X. We will discuss the unirationality problem for quartic hypersurfaces and quadric bundles over a arbitrary field in the the perspective of the relation between unirationality and rational connectedness. We will prove unirationality of quadric bundles under certain positivity assumptions on their anti-canonical divisor. As a consequence we will get the unirationality of any smooth 4-fold quadric bundle over the projective plane, over an algebraically closed field, and with discriminant of degree at most 12.

26/09/2023

28/09/2023

28/09/2023

Benedetta Piroddi

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: Symplectic action of groups of order four on K3^[2]-type manifolds.
Abstract: When a group of order four G (either Z/4Z or (Z/2Z)^2) acts symplectically on a K3^[2]-type manifold X, then its action is always standard, meaning that we can deform the pair (X,G) to a pair (S^[2],G), where S^[2] is the Hilbert square of a K3 surface with a symplectic action of G, and the action of G on S^[2] is naturally induced.
I will describe these two actions and construct for each one the general member of a projective family that admits it, starting from a family of K3 surfaces with a mixed (symplectic and non-symplectic) action of a group of order 4.
Time permitting, I will also talk about the induced involutions on the Nikulin orbifold which is obtained by partial resolution of the quotient X/i, where i is a symplectic involution normal in G.

26/09/2023

28/09/2023

28/09/2023

Eleonora Romano

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: Recent results on Fano varieties.
Abstract: In this talk we present some recent results on complex smooth Fano varieties. To this end, we first recall an invariant introduced by Casagrande, called Lefschetz defect. We review the literature to deduce that all Fano manifolds with Lefschetz defect greater than three are well known. Then we focus on the case in which the Lefschetz defect is equal to three, by discussing a structure theorem for such varieties. As an application, we use this result to classify all Fano 4-folds with Lefschetz defect equal to three: there are 19 families, among which 14 are toric. This is a joint work with C. Casagrande and S. Secci.

26/09/2023

28/09/2023

28/09/2023

Filippo Viviani

Relazione all'interno del convegno: Algebraic Geometry Workshop in Bologna

Seminario di algebra e geometria

Title: On the classification of fine compactified Jacobians of nodal curves.
Abstract: If a smooth curve degenerates to a nodal curve, what are the possible modular degenerations of the Jacobian? I will give a complete answer to this question, using some recent results of Pagani-Tommasi.

2023

25 settembre

Andre Nies

Seminario di algebra e geometria, interdisciplinare, logica

2023

22 settembre

Gianluca Paolini

nel ciclo di seminari: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario interdisciplinare

It was asked in [J. Math. Log. , Vol. 22, No. 01, (2022)] if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups.We prove that under the assumption of weak elimination of imaginaries this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of Aut(GL(V)) for any vector space of dimension $\aleph_0$ over a finite field, in affinity with the classical description for finite dimensional spaces due to Schreier and van der Waerden.

2023

21 settembre

Marco Golla

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Libgober defined Alexander polynomials of (complex) plane projective curves and showed that it detects Zariski pairs of curves: these are curves with the same singularities but with non-homeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is work in progress with Hanine Awada.

2023

18 settembre

Andre Nies

Seminario di algebra e geometria, interdisciplinare, logica

2023

15 settembre

Joost Hooyman

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, teoria delle categorie

A well-known shortcoming of the category of smooth manifolds is its lack of arbitrary pullbacks. A pullback of manifolds, and in particular an intersection of submanifolds, exists only along maps which are transversal. This problem can be overcome by passing to the larger category of derived smooth manifolds. The construction of this category combines ideas from algebraic geometry, homotopy theory and of course differential topology.
We can describe this construction in several steps. Firstly, we consider the relation between manifolds and schemes. Here, we employ the so-called C^\infty-rings, which are algebraic objects encoding the structure of the collection of smooth functions on R^n beyond that of an R-algebra. By the general philosophy of algebraic geometry, their duals give rise to geometric objects, called C^\infty-schemes. These geometric objects are primarily studied as models for synthetic differential geometry.
Secondly, we introduce homotopy theory into the picture. This step adapts the ideas of derived algebraic geometry to the setting of C^\infty-schemes. Our approach replaces the algebraic objects involved by their simplicial counterparts. In this context, the main objective is to develop a homotopy theory of presheaves which allows us to work with sheaf axioms 'up to homotopy'.
Succinctly, a derived smooth manifold can be described as a homotopical C^\infty scheme of finite type. In my talk, I will highlight some steps of the rather intricate construction described above. Hopefully, this will give the audience a perspective from which to think further about these exciting interactions between algebraic geometry, homotopy theory and differential topology.

2023

14 settembre

David Rottensteiner

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

We present a pseudo-differential Weyl calculus on graded nilpotent Lie groups, especially the Heisenberg group, which extends the celebrated Weyl calculus on R^n. This Weyl calculus is a particular instance of a general symbolic calculus we develop for a large class of quantization schemes that are defined via the (operator-valued) group Fourier transform. The symbol classes we consider are the Hörmander-type classes introduced by Fischer and Ruzhansky, which on R^n coincide with the classical ones. As a by-product, we also recover the classical Kohn-Nirenberg calculus on R^n and Fischer and Ruzhansky's KN-calculus on general graded groups.
A few immediate applications of our theory are the expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\aa rding inequality for elliptic operators, and a generalized Poisson bracket on stratified groups.
We also discuss two simple algebraic criteria which determine the Weyl quantization uniquely at least on R^n and the Heisenberg group.
The talk is based on joint-work with Serena Federico and Michael Ruzhansky.

2023

14 settembre

Joost Hooyman

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, teoria delle categorie

2023

14 settembre

Andre Nies

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

2023

13 settembre

Saugata Bandyopadhyay

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

Let n ∈ N, n > 2 and let Ω ⊆ R^n be open. Let H, G : R^n → R^{n×n} be of appropriate regularity. We discuss the existence of an immersion u : Ω → R^n of appropriate regularity, satisfying
(∇u)^tH(u)(∇u) = G in Ω. (1)
We consider both local and global problems.
Equation (1) comes up in diverse contexts. When H (and hence G) is symmetric and positive definite, Equation (1) is connected to the problem of equivalence of Riemannian metrics. The symmetric case is also important in the non-linear elasticity theory because of its connection with the Cauchy-Green deformation tensor. When H (and hence G) is skew-symmetric, Equation (1) comes up in the context of the problem of equivalence of differential two-forms.
The aim of the talk is to present a survey of recent progress and advances in the context of Equation (1). We also discuss the general case when H, G are neither symmetric nor skew-symmetric.
The talk is based on joint works with Bernard Dacorogna, Vladimir Matveev and Marc Troyanov.

2023

13 settembre

Nicola Pagani

Seminario di algebra e geometria

The double ramification is a locus in the moduli spaces M_{g,n} of smooth n-pointed curves of genus g, parameterizing curves that admit a meromorphic function with prescribed orders of zeroes and poles. This locus, or rather its compactification, and its (co)-homology class, plays a role in the enumerative algebraic and symplectic geometry and in mathematical physics. We will discuss an approach to its compactification and its calculation via compactified universal Jacobians.

2023

12 settembre

2023

11 settembre

Andre Nies

Seminario di algebra e geometria, interdisciplinare, logica

2023

11 settembre

Igor Klep

Seminario di analisi matematica, interdisciplinare

An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. The main result of this talk is, for a positive integer d, the simultaneous dilation, up to a sharp factor ϑ(d), of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for ϑ(d) is derived, using probabilistic methods and an old result of Rev. Simmons on flipping biased coins.
Time permitting we will consider consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, L(x):=I-\sum A_j x_j is a linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is a spectrahedron. The set D_L of tuples X=(X_1,...,X_g) of symmetric matrices (of the same size) for which L(X):=I-\sum A_j \otimes X_j is PsD, is a free spectrahedron. A result here is: any tuple X of d-by-d symmetric matrices in a free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the spectrahedron S_L. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive.

2023

11 settembre

2023

07 settembre

Andre Nies

Seminario di algebra e geometria, interdisciplinare, logica, sistemi dinamici

2023

05 settembre

Linear matrix inequalities (LMIs) play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. These are exactly the semialgebraic matrix convex sets.
This talk will discuss analytic maps between free spectrahedra and, under certain irreducibility assumptions, classify all those that are bianalytic. The foundation of such maps turns out to be a very small distinguished class of birational maps we call convexotonic.
The results depend on new tools in noncommutative analysis, such as a Positivstellensatz for analytic functions whose real part is positive on a free spectrahedron, and fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of free spectrahedra.

2023

05 settembre

Shaun Allison

nel ciclo di seminari: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario interdisciplinare

A deep result of Gao-Jackson is that orbit equivalence relations induced by Borel actions of countable discrete abelian groups on Polish spaces are hyperfinite. Hjorth asked if indeed any orbit equivalence relation induced by a Borel action of an abelian Polish group on a Polish space, which is also essentially countable, must be essentially hyperfinite. We show that any countable Borel equivalence relation (CBER) which is treeable must be classifiable by an abelian Polish group (such as $\ell_1$). As the free part of the Bernoulli shift action of $F_2$ is a treeable CBER, and not hyperfinite, this answers Hjorth's question in the negative.
On the other hand, for certain abelian Polish groups such as $\matbb{R}^\omega$, Hjorth's question has a positive answer. Indeed, we show that any orbit equivalence relation induced by a Borel action of a countable product of locally compact abelian Polish groups which is also potentially $\BPi^0_3$ must be Borel-reducible to $E_0^\omega$. By a dichotomy result of Hjorth-Kechris, this implies that essentially countable such orbit equivalence relations are hyperfinite. This uses a result of Cotton that locally compact abelian Polish groups yield essentially hyperfinite orbit equivalence relations, as well as the Hjorth analysis of Polish group actions.

30/08/2023

01/09/2023

01/09/2023

Caterina Mazzetti

A sub-Riemannian model of the functional architecture of M1 for arm movement direction

Seminario di analisi matematica

30/08/2023

01/09/2023

01/09/2023

Mattia Galeotti

Differential operators heterogenous in orientation and scale in the cortical $V_1$ cortex

Seminario di analisi matematica

30/08/2023

01/09/2023

01/09/2023

Vasiliki Liontou

Gabor Frames and Contact structures: Signal encoding and decoding in the primary visual cortex

Seminario di analisi matematica, interdisciplinare

30/08/2023

01/09/2023

01/09/2023

Ilya Shirokov (PDMI RAS)*; Dmitry Alekseevsky (IITP RAS)

Geometry of saccades and saccadic cycles

Seminario di analisi matematica

30/08/2023

01/09/2023

01/09/2023

Dario Prandi (CNRS)*; Cyprien Tamekue (Université Paris-Saclay); Yacine Chitour

MacKay-type visual illusions via neural fields

2023

19 luglio

Roberto Ognibene

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk, I will discuss the behavior of the spectrum of the Laplacian on bounded domains, subject to varying mixed boundary conditions. More precisely, let us assume the boundary of the domain to be split into two parts, on which homogeneous Neumann and Dirichlet boundary conditions are respectively prescribed; let us then assume that, alternately, one of these regions “disappears” and the other one tends to cover the whole boundary. In this framework, I will first describe under which conditions the eigenvalues of the mixed problem converge to the ones of the limit problem (where a single kind of boundary condition is imposed); then, I will sharply quantify the rate of this convergence by providing an explicit first-order asymptotic expansion of the “perturbed” eigenvalues. These results have been obtained in collaboration with L. Abatangelo, V. Felli and B. Noris.

2023

17 luglio

Georgios Katsimpas

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica, probabilità

The theory of free probability was developed by Voiculescu in the 1980's as an extension of classical probability to the non-commutative setting and aims to study algebras of operators from a probabilistic point of view. The connection of free probability with random matrix theory, as well as the development of notions of entropy in the non-commutative framework, lead to significant breakthroughs regarding the structure of certain von Neumann algebras. More recently, in 2013 Voiculescu laid the foundations of bi-free probability theory, which extends the free setting and involves the simultaneous study of left and right actions of algebras on reduced free product spaces.
In this talk, we will give an overview of the contributions of free probability theory to the field of operator algebras and will discuss the development of notions of entropy within the context of bi-free probability theory.

2023

13 luglio

Giusi Vaira

Seminario di analisi matematica

In this talk we discuss about the existence of positive non-radial solutions for a system of Schrödinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity. Moreover, we also consider the critical growth case.

2023

10 luglio

While density compensation has been introduced artificially in iterative SENSE reconstructions as a preconditioner for accelerating the convergence of the conjugate gradient descent algorithm, it turns out that this density compensation appears naturally if the problem is reformulated with appropriate Euclidean products in the image space and the data space. This is also true for compressed sensing reconstruction, which can be seen as solving the same least-squares problem as iterative SENSE, but with L1 regularization. Modifying the Euclidean products also changes the adjoint operator of the linear model, which has important consequences for implementations. The purpose of this talk is to illustrate how conjugate gradient descent can be modified to take different inner products into account.

2023

10 luglio

PhD thesis defense

2023

08 luglio

Finite automata are a model of computation in finite memory in theoretical computer science. An automaton gets a string of symbols as an input, and either accepts or rejects it. A deterministic two-way finite automaton (2DFA) has a finite set of states and works by moving over the string back and forth, changing its state. The question studied in this talk is how long could be the shortest string accepted by an $n$-state 2DFA, as a function of $n$. The previously known lower bound is of the order $\Omega(1.626^n)$, and the upper bound is $\binom{2n}{n+1}=O(\frac{1}{\sqrt{n}}4^n)$.
In my talk, I construct a family of $n$-state automata with shortest accepted strings of length $\frac{3}{4} \cdot 2^n - 1$. Also I study a special case of direction-determinate automata (those that always remember in the current state whether the last move was to the left or to the right), and determine the maximum length of the shortest accepted string precisely as $\binom{n}{\lfloor\frac{n}{2}\rfloor}-1 = \Theta(\frac{1}{\sqrt{n}} 2^n)$.

2023

06 luglio

Louis Dupaigne

Seminario di analisi matematica

Due to its conformal invariance, the sharp Sobolev inequality takes equivalent forms on the three standard model spaces i.e. the Euclidean space, the round sphere and the hyperbolic space. By analogy, we introduce three weighted manifolds named after Caffarelli, Kohn and Nirenberg (CKN) for the following reason: the sharp Caffarelli-Kohn-Nirenberg inequality in the standard Euclidean space can be reformulated as a (sharp) Sobolev inequality written on the CKN Euclidean space. It is equivalent to similar (but new) Sobolev inequalities on the CKN sphere and the CKN hyperbolic space. In addition, the Felli-Schneider condition, that is, the region of parameters for which symmetry breaking occurs in the study of extremals, turns out to have a purely geometric interpretation as an (integrated) curvature-dimension condition. To prove these results, we shall use Bakry's generalization of the notion of scalar curvature, (a weighted version of) Otto's calculus, the reformulation of all the inequalities (and many more) as entropy-entropy production inequalities along appropriate gradient flows in Wasserstein space, and eventually elliptic PDE methods as our best tool for building rigorous and concise proofs. This is joint work with Ivan Gentil (Lyon 1) and Simon Zugmeyer (Paris 5).

2023

06 luglio

The analysis of linear, time-invariant systems by superposition of modes is a longstanding idea, tracing back to the early works by Daniel Bernoulli on the vibrating string and later formalised by Fourier. Time-invariance and linearity allow to describe systems in terms of eigenfunctions and frequencies, called the modes of the system. Such modal shapes and frequencies may be determined experimentally or from a suitable mathematical model. In the latter case, the model is a system of partial differential equations (PDEs), depending on material and geometric properties, type of excitation, and initial and boundary conditions. Modal equations result after an appropriate projection is applied to the system of PDEs, yielding an eigenvalue problem from which the modal frequencies and shapes are determined. The resulting modal equations depend exclusively on time, and output may be extracted as a suitable combination of the time-dependent modal coordinates. Usually, one is interested in computing a physical output at one or more points of the system via a weighted sum resulting from an inverse projection.
Besides being a practical analysis tool, this approach lends itself naturally to the simulation of mechanical vibrations and, thus, to sound synthesis via physics-based modelling. Modal synthesis began in earnest in the 1990s, when frameworks such as Mosaic and Modalys emerged. The early success of modal synthesis was partly due to the ease of implementation, and efficiency, of the modal structure: the orthogonality of the modes yields a bank of parallel damped oscillators. Including complicated loss profiles (necessary for realistic sound synthesis) is also trivial and inexpensive within the modal framework, as is the fine-tuning of the system's resonances. In direct numerical simulation, such as finite differences, distributed nonlinearities can be resolved locally and, in some cases, efficiently via linearly implicit schemes. For the modal approach, the presence of nonlinearities, either lumped or distributed, may become problematic since a coupling occurs between the modes of the associated linear system.
In this seminar, an extension of the modal approach, including nonlinearly coupled subsystems, is presented. The enabling idea is the quadratisation of the potential energy in a fashion analogous to that proposed within the SAV (scalar auxiliary variable) approach. It is then possible to derive discrete-time equations whose update remains explicit while guaranteeing pseudo-energy conservation (necessary for the system's stability). Musical examples, including a real-time music plugin developed within this framework, will be offered.

2023

06 luglio

Gianluca Pacienza

Seminario di algebra e geometria

Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this talk is to prove the Morrison-Kawamata cone conjecture for such manifolds when the degree of the cover is prime using the analogous result (established by Amerik-Verbitsky) for their universal cover. We also verify the conjecture for the known examples having non-prime degree.

2023

03 luglio

03/07/2023

05/07/2023

05/07/2023

Luca Capogna

Relazione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023 - GHAIA

Seminario di analisi matematica

TBA

03/07/2023

05/07/2023

05/07/2023

Matteo Bonforte

Relazione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di analisi matematica

TBA

03/07/2023

05/07/2023

05/07/2023

Jorge Antezana

Relazione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Pablo Berná

Relazione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Guy David

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Fernando Quirós

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Yannick Sire

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Manuel Ritoré

Relazione come Organizzatore e Chair di una sessione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

03/07/2023

05/07/2023

05/07/2023

Julián Pozuelo

Relazione breve all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di algebra e geometria, analisi matematica

03/07/2023

05/07/2023

05/07/2023

Davide Barbieri

Relazione come Organizzatore e Chiar di una sessione all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Gianmarco Giovannardi

Relazione Breve all'interno del convegno: Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications 2023

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Nicolò Forcillo

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Juan Manfredi

Seminario di analisi matematica

03/07/2023

05/07/2023

05/07/2023

Virginia Bolelli

Seminario di analisi matematica

2023

29 giugno

We give an effective characterization of semi-abelian surfaces extending to the logarithmic setting the famous theorem of Enriques about abelian surfaces. In addition we will give a characterization of semi-abelian varieties in any dimension, strengthening a result of Kawamata. This is a joint work with M. Mendes Lopes and R. Pardini

2023

28 giugno

Shigeyuki Kondo

nel ciclo di seminari: PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS

Seminario di algebra e geometria

Part 2: Quadratic line complexes and Kummer surfaces.
A quadratic line complex is a nonsingular 3-fold which is the intersection of the Grassmannian G(1,3) (= lines in P^3) and
a quadric in P^5. It naturally gives us a Kummer quartic surface S with 16 nodes, a curve C of genus 2, and an abelian surface A.
Then A is isomorphic to the Jacobian of C and S is the quotient of A by its inversion. We give a sketch of this classical theory
and extend the theory to the case of characteristic 2.
Main references are Griffiths, Harris, Principles of Algebraic Geometry, the last chapter and T. Katsura, S. Kondo, arXiv:2301.01450.

2023

28 giugno

We introduce the definition of tensorized block rational Krylov subspaces and their relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in [2]. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or Tensor Train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis and some strategies for poles selection based on the techniques developed in [1].
References
[1] A. A. Casulli and L. Robol. “An effcient block rational Krylov solver for Sylvester equations with adaptive pole selection”.
In: arXiv preprint arXiv:2301.08103 (2023).
[2] D. Kressner and C. Tobler. “Krylov subspace methods for linear systems with tensor product structure”.
In: SIAM Journal on Matrix Analysis and Applications 31.4 (2010), pp. 1688–1714

2023

28 giugno

Genki Ouchi

nel ciclo di seminari: PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS

Seminario di algebra e geometria

Mukai found the relation between polarized symplectic
automorphism groups and certain subgroups of the Mathieu group.
After the discovery of Mathieu moonshine, Huybrechts established the
relation between autoequivalence groups of derived categories of K3
surfaces and certain subgroups of the Conway group. Although we know
explicit examples of polarized K3 surfaces with maximal symplectic
automorphism groups, it is difficult to find explicit examples of finite
autoequivalences of derived categories of K3 surfaces not conjugate to
automorphisms of K3 surfaces.
In this talk, I would like to study how to construct finite
autoequivalences of derived categories of K3 surfaces and discuss their
difficulties.
First, we recall automorphism groups of compact Riemann surfaces from
point of view of algebraic geometry, topology and derived categories.
Second, I would like to discuss autoequivalence groups of derived
categories of K3 surfaces as an analogue of the case of compact Riemann
surfaces.

2023

27 giugno

Alessandro Codenotti

Seminario di algebra e geometria

The Menger universal curve M is a well known compact connected metric fractal, usually described as a decreasing intersection of cubical complexes. By adapting techniques developed for manifolds by Gutman-Tsankov-Zucker, we prove that the group G of self-homeomorphisms of M has a nonmetrizable universal minimal flow M(G). This is a compact space on which G acts minimally, and such that for any other compact space Y on which G acts minimally, there is a continuous equivariant surjection from M(G) onto Y. Much work has been done in recent years to study M(G) for a variety of Polish groups G, as this provides a natural dividing line in the class of Polish groups: groups G for which M(G) is nonmetrizable are considered to have wild dynamics, while groups G for which M(G) is metrizable are considered to have tame dynamics. This is joint work with Gianluca Basso and Andrea Vaccaro.

2023

23 giugno

Ana Cecilia Goncalves

Seminario di didattica della matematica

Games, in particular board games, have the potential to cause high states of satisfaction, an essential characteristic for lasting learning. In this work, it is carried out a study with 15 students of the 6th grade of Basic Education, where the tabletop game Absolute Blast! is used to address concepts related to algebraic operations on the set of integers.
From the implementation of the game in the classroom context, a question arises 'in what way Absolute Blast! helps students to learn addition with integer numbers?'. To answer this question, a pretest was applied, two fifty-minute sessions were performed with the game Absolute Blast! and, at the end, a post-test was also applied. A quantitative analysis of the results was carried with the Wilcoxon test for non-parametric paired data. As a main result, it was concluded that Absolute Blast! had a positive impact regarding addition with integer numbers.

2023

22 giugno

Francesca Anceschi

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

The aim of this talk is to discuss the most recent developments of the De Giorgi-Nash-Moser weak regularity theory for kinetic operators. The analysis will be presented in the model case of the Fokker-Planck equation with measurable coefficients in divergence form and the focus will be on exploring qualitative and quantitative methods to prove an invariant Harnack inequality for non-negative solutions. Finally, some applications of this inequality to various kinetic models will be discussed.

2023

22 giugno

Shigeyuki Kondo

nel ciclo di seminari: PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS

Seminario di algebra e geometria

A Kummer surface was first found by A. Fresnel (1822) for special case and by E. Kummer (1864) for a general one.
Later F. Klein (1870) discovered a relation between quadratic line complexes and Kummer surfaces, studied their automorphisms
and raised a question on the automorphism group. In this lecture we discuss two topics:
Part 1: The Leech lattice and the automorphism group of a generic Kummer surface.
By applying the theory of Leech lattice, we present a generator of the automorphism group of a generic Kummer surface
associated with a curve of genus 2. This gives an answer of Klein’s question.
Main reference is S. Kondo, K3 surfaces, EMS 2020, the last chapter.

2023

22 giugno

Genki Ouchi

nel ciclo di seminari: PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS

Seminario di algebra e geometria

2023

21 giugno

Jarosław Buczyński

Seminario di algebra e geometria

In order to study projections of smooth curves and the singularities of the image curve, we introduce multifiltrations obtained by combining flags of osculating spaces. In our research we classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree d rational curves in P^n when d-n <4 and d<2n. During the talk I will focus on an illustrative example. The set of linear spaces giving a fixed singularity type is related to a specific Schubert cycle (which depends on the singularity).
Based on a joint work with Nathan Ilten and Emanuele Ventura, https://arxiv.org/abs/1905.11860, International Mathematical Research Notices.

2023

20 giugno

Alessandra Bernardi (Università degli Studi di Trento)

nel ciclo di seminari: APPLIED ALGEBRAIC GEOMETRY

Seminario di algebra e geometria

The ground state of a Hamiltonian is a most suitable state to play with if one is interested in seeing quantum mechanics effects. These kinds of states naturally appear in a special "region" of the tensor space, namely on/very close to Tensor Network varieties. I would like to illustrate these connections and, in dependence on the time, to show one of the few examples where all the construction is exactly possibile, namely the so-called AKLT model.

2023

20 giugno

Donatella Iacono

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

In this talk we first introduce differential graded Lie algebra (DGLA) and their role in deformation theory. Then, we apply this techniques to analyse some deformation problems, such as deformations of pair (variety,sheaf) or deformations of locally free sheaves and subspace of sections.

2023

19 giugno

In this talk, I will present some techniques to deal with quaternionic slice regular functions in order to obtain global topological properties. These techniques come from abstract algebra and complex analysis. In particular, I will discuss the nature of quaternionic exponential and that of k-th roots as covering maps. I will try to present the topic with several explicit examples. The results are obtained in some works in collaboration with Chiara de Fabritiis and with Samuele Mongodi.

2023

19 giugno

Genki Ouchi

nel ciclo di seminari: PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS

Seminario di algebra e geometria

2023

16 giugno

2023

16 giugno

2023

16 giugno

José A. Carrillo

Seminario di analisi matematica

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem.

2023

16 giugno

Sebastien Bertrand

Seminario di fisica matematica

Superintegrable Hamiltonian systems possess remarkable mathematical properties. Among others, for maximal superintegrability, if the trajectory is finite, then the motion will be closed and periodic. The Hamiltonian formalism allows us to completely characterize the dynamics of a physical system with one conservation law, the Hamiltonian, using a symplectic geometry. By imposing more integrals of motion than the number of dimensions, we are looking to classify “Natural” Hamiltonians possessing a magnetic field leading to superintegrability and how to linearize them. We will discuss applications of superintegrability, and recent results, after a more historic approach.

2023

16 giugno

David Joao Brandligt de Jesus (prossimo assegnista di ricerca di Dipartimento dal 1° luglio 2023)

Seminario di analisi matematica

We will discuss the regularity of viscosity solutions for fully nonlinear Hessian equations with coefficients in some Muckenhoupt class. We prove Holder and higher regularity under mild assumptions on the coefficients. We use approximation techniques in the spirit of Caffarelli's seminal approach. A special attention has to be paid to the degeneracy of the coefficients and we assume some natural structural assumptions which entail applications to curvature equations on conic manifolds.

2023

15 giugno

Berardo Ruffini

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In the talk I will introduce some variational models where an aggregating term, like the perimeter or a Dirichlet-type energy, is in competition with a repulsive one. Examples of such models arise naturally in different fields of physics. It is the case of the Gamow [liquid drop] model and the Hartree energies in quantum mechanics, or the Rayleigh liquid charged drop model in electrowetting theories.
I will give an overview of the recent strategies to get well-or-ill posedness of these energies. Then I will focus on a particular case -the reduced Hartree energy of the atom of Helium in a confined potential field- and show a strategy to characterize minimizers for such an energy.
The talk is mostly motivated by an ongoing project with Dario Mazzoleni (Pavia) and Cyrill B. Muratov (Pisa).

2023

15 giugno

In the first part of this talk I will try to give a (very partial) overview on some of the phenomena of
interest in the area of brain diseases, on the kind of contributions mathematical modelling could give in
this respect and on which mathematical instruments can be used when it comes to models.
In the second part I will present a mathematical model for the onset and progression of Alzheimer’s
disease. The synergistic interplay of proteins Amyloid-beta and tau is a subject of considerable interest
when it comes to the study of Alzheimer’s disease. The model I will describe is based on transport
and reaction-diffusion equations for the two proteins. In the model neurons are treated as nodes of a
graph (the connectome) and structured by their degree of malfunctioning. Three different mechanisms are
assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble
Amyloid-beta, ii) effects of misfolded tau protein and iii) neuron-to- neuron prion-like transmission of
the disease. These processes are modelled by a system of Smoluchowski equations for the Amyloid-beta
concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation
for the distribution function of the degree of malfunctioning of neurons. I will explain the structure of the
model, give a hint of the main analytical results obtained and I will show the output of some numerical
simulations

2023

13 giugno

Marco Manetti

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Let A\subset L be a flat inclusion of Lie algebroids, i.e., a Lie pair, on a smooth separated scheme over a field of characteristic 0. For every locally free A-module M we define its semiregularity maps and prove that, under some additional assumptions, they annihilate obstructions to deformations of M. In case A=0 and L=tangent bundle, this construction gives to the usual Buchweitz-Flenner's semiregularity maps for coherent sheaves.

2023

09 giugno

Lorenza Foschini

Seminario interdisciplinare

Incontro su Renato Caccioppoli con l'autrice del libro "L'attrito della vita. Indagine su Renato Caccioppoli matematico napoletano", la quale dialogherà con Pietro Cimmino, Salvatore Coen e Francesco Serra Cassano (Università di Trento) sulla figura di Renato Caccioppoli.

2023

08 giugno

Vincenzo Vespri

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Simone Ciani

Anisotropic p-Laplacean Equations - the pursuit of a comprehensive theory of regularity

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Anna Chiara Zagati

On the sharp Hardy inequality in fractional Sobolev spaces

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Alessandro Goffi

L^p contraction estimates for parabolic equations via the nonlinear adjoint method

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Gianmarco Giovannardi

SCHAUDER ESTIMATES UP TO THE BOUNDARY ON H-TYPE GROUPS: AN APPROACH VIA THE DOUBLE LAYER POTENTIAL

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Giorgio Stefani

A distributional approach to fractional Sobolev and BV spaces

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Francesca Bianchi

GEOMETRICAL ESTIMATES FOR THE FIRST EIGENVALUE OF LINEAR OPERATORS: THE FRACTIONAL DIRICHLET–LAPLACIAN AND THE DIRICHLET–BILAPLACIAN

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Alessandra De Luca

Nonlocal capillarity problems with anisotropic kernels

Seminario di analisi matematica

08/06/2023

09/06/2023

09/06/2023

Francescantonio Oliva

Dirichlet problems involving the 1-Laplacian and general nonlinearities

Seminario di analisi matematica

2023

07 giugno

Ermanno Lanconelli

Seminario di analisi matematica

2023

07 giugno

Giacomo De Palma (Alma Mater Studiorum – Università di Bologna)

nel ciclo di seminari: APPLIED ALGEBRAIC GEOMETRY

Seminario di algebra e geometria

I will present entanglement as a resource theory where the free operations are local operations and classical communication.
I will not closely follow any particular reference, but all the topics I will present can be found in:
- M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010, sec. 12.5;
- I. Bengtsson, K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition, Cambridge University Press, 2017, chpts. 16-17;
- E. Chitambar, G. Gour, Quantum resource theories, Rev. Mod. Phys. 91 (2019), 025001.

2023

06 giugno

Luca Casarin

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

TBA

2023

06 giugno

In this talk I try to provide an idea of how the mathematical structures of quantum mechanics allow to construct novel techniques in machine learning and artificial intelligence. After an introduction to the fundamentals, I overview some recent results.

2023

01 giugno

Marco Mughetti

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk we discuss the problem of the real analytic regularity for the solutions of sums of squares of vector fields. While the problem of the C^\infty hypoellipticity has been settled from the very beginning by Hörmander, the problem of the analytic hypoellipticity is still open and seems much more involved.
Treves conjecture states that a “sum of squares”-type operator is analytic hypoelliptic if and only if all the Poisson strata of its characteristic set are symplectic.
We show that this conjecture, as stated, does not hold. However, we briefly discuss some model examples which would suggest that the analytic regularity still depends on a suitable stratification of the characteristic variety of the operator.

2023

30 maggio

Martino Lupini

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

TBA

2023-05-30

Leonardo Banchi

Relazione all'interno del convegno: Workshop Frontiers of Machine Learning: Hard-Sciences for Machine learning

Seminario di fisica matematica, interdisciplinare

In recent years there have been an increasing number of results where quantum physics has been combined with machine learning for different reasons. On the one hand, quantum computers promise to significantly speed up some of the computational techniques used in machine learning and, on the other hand, “classical” machine learning methods can help us with the verification and classification of complex quantum systems. Moreover, the rich mathematical structure of quantum mechanics can help define new models and learning paradigms. In this talk, we will introduce quantum machine learning in all of these flavors, and then discuss how to bound the accuracy and generalization errors via entropic quantities. These bounds establish a link between the compression of information into quantum states and the ability to learn, and allow us to understand how difficult it is, namely how many samples are needed in the worst case scenario, to learn a quantum classification problem from examples. Different applications will be considered, such as the classification of complex phases of matter, entanglement classification, and the optimization of quantum embeddings of classical data.

2023

29 maggio

Data assimilation tries to predict the most likely state of a dynamical system by combining information from observations and prior models, often represented by a discretized partial differential equation.
The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches.
In particular, three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var), and, if time allows, the Kalman filter.
The data assimilation problem usually results in a very large, yet structured, nonlinear optimization problem. The dimension of the latter represents a quite challenging aspect of the entire solution procedure. Indeed, the inclusion of both the time and space dimensions leads to extremely large optimazion problems which need to be carefully handled by designing smart numerical schemes able to fully exploit the structure of the problem at hand.
Therefore, the second part of this talk aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches mentioned above.

2023

29 maggio

Grazia Rago

nell'ambito della serie: TOPOLOGIA E GEOMETRIA DELLE VARIETÀ

Seminario di algebra e geometria

2023

27 maggio

Andrea Cianchi (Università di Firenze)

Seminario interdisciplinare

In certe circostanze, le soluzioni di problemi variazionali e delle equazioni di
Eulero a loro associate godono di speciali proprietà di simmetria. Esse possono essere
conseguenza del fatto che le soluzioni in questione rappresentano le estremali di certe
quantità geometriche o fisiche fondamentali, o ereditano simmetrie dell’operatore
differenziale, del dominio e del dato al bordo, o ancora sono soggette a ulteriori condizioni
al contorno oltre a quelle naturali. Problemi al contorno di quest’ultimo tipo sono chiamati
sovradeterminati in letteratura. Dopo una breve panoramica su risultati classici di
simmetria del tipo sopra accennato, saranno discussi problemi sovradeterminati per
equazioni di tipo ellittico. In particolare, verranno analizzate soluzioni di equazioni
ellittiche anisotrope, dette di tipo Finsler o Wulff, che presentano simmetrie non standard
dettate dall’anisotropia dell’operatore.

2023

27 maggio

Antonella Grassi (Università di Bologna)

Seminario interdisciplinare

In questi giorni ricorre anche il centenario di Eugenio Calabi. In linguaggio
non tecnico descriverò qualche aspetto dell’influenza dell’analista Calabi sulla ricerca in
geometra algebrica birazionale e fisica teorica negli ultimi decenni.

2023

26 maggio

Nicholas Braun Rodrigues

Seminario di analisi matematica

In this talk I will present a recent result obtained together with Paulo D. Cordaro and Gerson Petronilho, where we give a partial answer to a conjecture made by F. Treves in 1981, concerning the hypoellipticity of corank one locally integrable structures. Besides hypoellipticity, we also consider the same regularity problem for Gevrey classes, and regular Denjoy-Carleman, which includes quasi-analytic classes.

2023

25 maggio

Marcello Malagutti

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In questo seminario trattiamo lo studio di una condizione necessaria e sufficiente per l'esistenza di soluzioni con una determinata regolarita` per un sistema di equazioni lineari a coefficienti di regolarita` data.
Iniziamo esponendo un metodo risolutivo per la determinazione di soluzioni continue di equazioni lineari a coefficienti continui dovuto a C. Fefferman. Passiamo poi a presentare il risultato di C. Fefferman e G. Luli di soluzioni di classe C^m nel caso di coefficienti polinomiali. Consideriamo infine un nostro risultato per determinare una condizione necessaria e sufficiente per l'esistenza di soluzioni semialgebriche continue nel caso di un sistema di equazioni lineari con coefficienti semialgebrici continui.

2023

25 maggio

Grazia Rago

nell'ambito della serie: TOPOLOGIA E GEOMETRIA DELLE VARIETÀ

Seminario di algebra e geometria

2023

24 maggio

Paola Boito, Dipartimento di Matematica, Università di Pisa

nell'ambito della serie: SEMINARI MAT/08 TEAM

Seminario di analisi numerica

The widespread interest in quantum computation has motivated (among others) applications to network analysis, where quantum advantage may turn out to be especially beneficial for the treatment of large-scale problems. In the past years, several authors have proposed the use of quantum walks - as opposed to classical random walks - in the definition and analysis of centrality measures for graphs, which in turn are the basis for ranking algorithms. Here we focus on unitary continuous-time quantum walks (CTQW) applied to directed graphs and propose new quantum algorithms for hub and authority ranking of the nodes. In particular we explore - the choice of Hamiltonian operator that defines the time evolution of a CTQW, - the choice of the initial state of the system (which turns out to have a non-negligible effect on the final ranking), and perform numerical comparisons with well-known classical ranking algorithms such as HITS and PageRank.

2023

24 maggio

Gianna M. Del Corso, Dipartimento di Informatica, Università di Pisa

Seminario di analisi numerica

The hitting time of a random walk on a graph is the expected number of steps required to reach a marked node starting from a given node or a given distribution. Hitting time finds a crucial application in search problems, where it tells us how many steps are needed to detect a marked node. In a quantum framework, random walks are replaced by quantum walks, which exhibit peculiar properties. In particular, quantum walks typically tend to diffuse faster on a graph than classical random walks. One way in which this remark can be made more precise is through the definition of a quantum notion of hitting time. We focus in this talk on quantum hitting time for discrete-time quantum walks. Usually, quantum hitting time is defined in terms of the stationary distribution associated with the given graph, while the classical hitting time can be defined in terms of a generic distribution. We generalize the notion of quantum hitting time in terms of a generic distribution emphasizing analogies and differences with the case where the stationary distribution is used. We provide conditions for the quadratic speedup of quantum hitting time over the classical counterpart and we report the results of numerical experiments on several examples of graphs both directed and undirected and for several different distributions. (joint work with Paola Boito).

2023

23 maggio

Chiara Pagani

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

TBA

22/05/2023

26/05/2023

26/05/2023

Markus Reineke

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

We will first work through the construction of moduli spaces of quiver representations as GIT quotients, and collect basic geometric properties. We will then work out classes of examples where these spaces can be described explicitly. We will describe geometric techniques for studying moduli spaces, for example coordinates, vector bundles, torus localization, Hilbert scheme.

22/05/2023

26/05/2023

26/05/2023

Fernando Rodriguez Villegas

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

In these lectures I will present the calculation of the title in the case of the star-shaped quivers related to character varieties based on my joint work with E. Letellier and T. Hausel. The starting point will be a formula of Hua for a general quiver. The basic tool used is the combinatorics of symmetric functions and generating functions, which I will discuss from scratch.

22/05/2023

26/05/2023

26/05/2023

Jerzy Weyman

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

Semi-invariants of quivers and their applications.

22/05/2023

26/05/2023

26/05/2023

Grzegorz Bobinski

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

TBA

22/05/2023

26/05/2023

26/05/2023

Giovanni Cerulli-Irelli

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

In 2021 Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti (IMRN 2023, arXiv 2206.10281) we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term there is a very beautiful (as yet conjectural) formula for the coefficients.

22/05/2023

26/05/2023

26/05/2023

Søren Gammelgaard

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

TBA

22/05/2023

26/05/2023

26/05/2023

Martina Lanini

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

Symmetric quivers and symmetric varieties
In this talk I will report on ongoing joint work with Ryan
Kinser and Jenna Rajchgot on varieties of symmetric quiver
representations. These varieties are acted upon by a reductive group via
change of basis, and it is natural to ask for a parametrisation of the
orbits, for the closure inclusion relation among them, for information
about the singularities arising in orbit closures. Since the Eigthies,
same (and further) questions about representation varieties for type A
quivers have been attached by relating such varieties to Schubert
varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I
will explain that in the symmetric setting it is possible to interpret
the above questions in terms of certain symmetric varieties. More
precisely, we show that singularities of an orbit closure of a symmetric
quiver representation variety are smoothly equivalent to singularities
of an appropriate Borel orbit closure in a symmetric variety.

22/05/2023

26/05/2023

26/05/2023

Steve Oudot

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

This talk will be an introduction to the field of topological data analysis, emphasizing the role played in it by quiver representation theory. Specifically, I will describe how the supports of the indecomposables can be used as descriptors for data, with stability guarantees under suitable choices of metrics on the representation categories of the quivers under consideration. I will also explain how one proceeds when those quivers are of wild representation type.

22/05/2023

26/05/2023

26/05/2023

Csaba Szántó

Relazione all'interno del convegno: Quiver Representations, Quiver Varieties and Combinatorics

TBA

2023

18 maggio

2023

18 maggio

Makhlouf Derridj

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

My aim is to give , in this talk , some topics on the question of regularity of Analytic-Gevrey vectors of partial differential operators (p.d.o.) with analytic-Gevrey coefficients . Since the results obtained in the sixties on elliptic p.d.o's , which are both hypoelliptic (Cˆ{\infty} setting) , analytic-Gevrey hypoelliptic (analytic-Gevrey setting) and satisfy the so-called Kotake-Narasimhan property , a lot of works and articles were devoted to these problems in case of non elliptic p.d.o's under suitable hypotheses (for example on the degeneracy of ellipticity).
I will consider the third problem on analytic-Gevrey vectors in the three cases of global (on compact manifolds ), local ( near a point in the base-space ), microlocal (near a point in the cotangent space ) , situations , and say few words on the main two methods used in order to obtain positive (or negative) results . Finally I will focus on some new microlocal results on degenerate elliptic (also called sub-elliptic ) p.d.o's of second order , obtained in a common work with Gregorio Chinni .

18/05/2023

19/05/2023

19/05/2023

Giulio Galise

On the strong maximum principle for nonlocal degenerate operators

Seminario di analisi matematica

This talk is devoted to the validity and the failure of the strong maximum principle for equations involving the k-th fractional truncated Laplacian or the k-th fractional eigenvalue, which are fully nonlinear integral operators whose nonlocality is somehow k-dimensional. We give in particular geometric characterizations of the sets of minima for nonnegative supersolutions. Based on joint works with I. Birindelli (Sapienza University), H. Ishii (Tsuda University) and D. Schiera (University of Lisbon).

18/05/2023

19/05/2023

19/05/2023

Francesca Tripaldi

Sobolev-Gaffney type inequalities on differential forms in the subRiemannian setting

Seminario di analisi matematica

In this talk, I will show what problems arise when trying to obtain Gaffney-type inequalities in subRiemannian geometry, since one cannot simply apply the classical Riemannian tools to this particular setting. I will then present some of the tools that are currently available to tackle this problem, and how they can be applied to obtain the desired results in the case of contact manifolds.

18/05/2023

19/05/2023

19/05/2023

Carlo Mercuri

On some p-Laplacian problems involving critical nonlinearities

Seminario di analisi matematica

I will discuss a class of quasilinear elliptic equations involving the p-Laplace operator and nonlinearities of Sobolev-critical growth, focusing on existence, non-existence, and compactness issues related to their variational formulation.

18/05/2023

19/05/2023

19/05/2023

Annunziata Loiudice

Critical subelliptic equations with Hardy potential and related Brezis-Nirenberg type problems

Seminario di analisi matematica

We study existence and qualitative properties of solutions to subelliptic problems with Hardy potential and critical nonlinearities on stratified groups. We investigate both the semilinear and the quasilinear case. First, we determine the existence, Lorentz regularity and asymptotic behavior of entire solutions. By convenient transformations, we are naturally lead to study the equation satisfied by the extremal functions to some weighted Sobolev-type inequalities on groups, whose analytic expression is not known. As a byproduct, we derive existence results for the associated Brezis-Nirenberg type problem, depending on the involved parameters. We also obtain non-existence Pohozaev-type results.

18/05/2023

19/05/2023

19/05/2023

Stefano Biagi

A Brezis-Nirenberg type result for mixed local and nonlocal operators

Seminario di analisi matematica

In this seminar we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a critical problem driven by a mixed local and nonlocal linear operator. More precisely, given a bounded open set in R^n (with n ≥ 4), we consider a perturbed critical problem and we develop an existence theory, both in the case of linear (that is, p = 1) and superlinear (that is, p > 1) perturbations. In the particular case p = 1, we also investigate the mixed Sobolev inequality associated with (P), detecting the optimal constant, which we show that is never achieved. The results discussed in this talk are obtained in collaboration with S. Dipierro, E. Valdinoci and E. Vecchi.

18/05/2023

19/05/2023

19/05/2023

Dimiter Vassilev

The fractional Yamabe equation on homogeneous groups

Seminario di analisi matematica

The general themes of the talk are Dirichlet forms, fractional operators and associated Sobolev type spaces on groups of homogeneous type. Our results lead to explicit integral formulas of the infinitesimal generators, which are the studied fractional operators, and embedding theorems between the relevant spaces. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type conditions. Finally, we will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.

18/05/2023

19/05/2023

19/05/2023

Federica Sani

Extremal functions for Adams inequalities with Navier boundary conditions

Seminario di analisi matematica

We consider the problem of existence of extremal functions for second order Adams' inequalities with Navier boundary conditions on balls in R^n in any dimension n\geq 4. We also discuss some sharp weighted versions of Adams' inequality on the same spaces. The weights that we consider determine a supercritical exponential growth, except in the origin, and the corresponding inequalities hold for spherically symmetric functions only.

18/05/2023

19/05/2023

19/05/2023

Carlo Orrieri

Wasserstein stability of porous medium equation on Riemannian manifolds

Seminario di analisi matematica

Given a complete, connected Riemannian manifold with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium equation with respect to the 2-Wasserstein distance. We produce stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg.

18/05/2023

19/05/2023

19/05/2023

Giusi Vaira

Clustering phenomena in low dimensions for a boundary Yamabe problem

Seminario di analisi matematica

We consider the classical geometric problem of prescribing scalar and boundary mean curvature via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that this is not true anymore in low dimensions (that is n=4, 5, 6, 7). In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and minimizes the norm of the trace-free second fundamental form of the boundary.

2023

17 maggio

Eurica Henriques

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this seminar we revisit several previous works concerning anisotropic differential equations,
i.e. evolution equations in which the diffusion takes a different form within different space
directions.
Recently, in a joint collaboration with S. Ciani, we went back to these anisotropic PDEs,
focusing on the singular ones, and started to work on some of their open problems.
We’ll briefly discuss this ongoing project pointing out some of the difficulties one needs to address
and overcome.

2023

17 maggio

Daniela Giorgi

Seminario di algebra e geometria, analisi numerica

In this talk, I will describe recent works in the fields of 3D design and computational fabrication, from small objects up to the architectural scale.
I will describe how solving real-world problems requires blending notions from geometry, shape analysis, optimization, simulation and deep learning.
Finally, I will talk about challenges and open research directions in computational geometry and shape analysis, also in other application fields.

2023

16 maggio

Annalisa Grossi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Hyperkähler manifolds are one of the building blocks of compact complex Kähler manifolds with trivial first Chern class. Huybrechts proved that if X and Y are birational HK then they have to be deformation equivalent, but this implication is far from being an equivalence. It is then natural to ask which additional assumptions are needed to get that two HK manifolds of the same deformation type are birational. In this talk I will give a gentle introduction on HK manifolds and lattice theory for HK manifolds, then I will provide a lattice-theoretic criterion to determine when two HK manifolds of OG10 type are birational (joint work with C.Felisetti and F.Giovenzana), and I will show an application to the Li-Pertusi-Zhao variety. If time permits I will discuss another related topic about understanding sufficient numerical conditions to determine the deformation type of a given HK manifolds of a fixed dimension. This is a first result about a joint work in progress with P.Beri.

2023

12 maggio

Eva Löcherbach

nel ciclo di seminari: MCKEAN-VLASOV MODELS FOR SYSTEMS OF SPIKING NEURONS

Seminario interdisciplinare

Lecture V: Some outlooks : Models including learning (plasticity) and models with random synaptic weights.

2023

11 maggio

Nicola Abatangelo

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

We will present results concerning existence and uniqueness of solutions to a nonlinear equation driven by an operator arising from the superposition of a Laplacian and a signed power of a fractional Laplacian. Of particular interest are the boundary conditions naturally associated to the equation. We will also go over a couple of strongly related problems. This is a joint work with M. Cozzi (UniMi).

2023

11 maggio

We will report on recent results concerning “unusual” integro-differential operators given by the sum of a local and a nonlocal one. There are two new features: either the operator is not positive definite or
the spatial domain where the two operators act is not the same.

2023

11 maggio

Paolo Cavicchioli

nell'ambito della serie: TOPOLOGIA E GEOMETRIA DELLE VARIETÀ

Seminario di algebra e geometria

2023

10 maggio

Eva Löcherbach

nel ciclo di seminari: MCKEAN-VLASOV MODELS FOR SYSTEMS OF SPIKING NEURONS

Seminario interdisciplinare

Lecture IV: Longtime behaviour of the limit process, metastability.

2023

09 maggio

Paolo Cavicchioli

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

This dissertation will deal with the equivalence of links in 3-manifolds of Heegaard genus two. We construct an algorithm (implemented in C++) which, starting from a description of such a manifold introduced by Casali and Grasselli that uses 6-tuples of integers and determines a Heegaard decomposition of the manifold, allows to find the words in B_2,2n, the braid group on 2n strands of a surface of genus two, that realize the plat-equivalence for links in that manifold. In this way we extend the result obtained by Cattabriga and Gabrovšek for 3-manifolds of Heegaard genus one to the case of genus two. We describe explicitly the words for a notable family of 3-manifolds.

2023

08 maggio

Eva Löcherbach

nel ciclo di seminari: MCKEAN-VLASOV MODELS FOR SYSTEMS OF SPIKING NEURONS

Seminario interdisciplinare

Lecture III: Mean field limits in the case when the limiting equation is a McKean-Vlasov equation driven by Poisson random measure. Well-posedness of the limit equation and strong convergence results.

2023

05 maggio

Il problema e' di sapere quali gruppi di Lie nilpotenti
nascono naturalmente nell'analisi al bordo di domini complessi alla
Nagel--Stein--et al, e la risposta e' tanti.

2023

05 maggio

Eva Löcherbach

nel ciclo di seminari: MCKEAN-VLASOV MODELS FOR SYSTEMS OF SPIKING NEURONS

Seminario interdisciplinare

Lecture II: Proof of the convergence via coupling arguments. Extension to the spatially structured case and convergence to the so-called neural field equation.

2023

04 maggio

Eleonora Cinti

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk, I will present a recent result which establishes optimal regularity for isoperimetric sets with densities, under mild H\¨older regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach the optimal regularity class $C^{1, \frac{α}{2−α}}$ in any dimension. This is a joint work with L. Beck and C. Seis.

2023

04 maggio

Eva Löcherbach

nel ciclo di seminari: MCKEAN-VLASOV MODELS FOR SYSTEMS OF SPIKING NEURONS

Seminario interdisciplinare

Lecture I: Systems of interacting neurons described by Hawkes processes.
- Representation by means of differential equations driven by Poisson random measure;
- A limiting ODE describing the evolution of the mean firing rate;
- Emergence of oscillatory behavior.

2023

04 maggio

Alessia Cattabriga

nell'ambito della serie: TOPOLOGIA E GEOMETRIA DELLE VARIETÀ

Seminario di algebra e geometria

2023

02 maggio

Rosanna Grassi

Seminario interdisciplinare

Assortativity is a global indicator that provides meaningful insights about the network structure. In the classical definition, the assortativity is a global measure based on the Pearson correlation between the degrees of nodes. This definition can be extended into two different directions. On the one side, one can consider other quantitative attributes of the nodes different from the degree; on the other side, one can move from the adjacency of the nodes – which is the basis of Newman’s degree-degree assortativity – and propose more general ways to connect them. We provide a generalized concept of the assortativity measure for directed and weighted networks, moving beyond the adjacency relations in both directions. The proposed concept is formulated on a node attribute that is not necessarily the degree or strength, and nodes are connected through walks or paths. In this way, we totally extend the assortativity definitions provided in the literature until now.
We provide an empirical application of these measures for the paradigmatic case of the trade network. Interestingly, this interpretation of the higher-order assortativity measure allows stating a natural bridge between complex networks and stochastic processes. In so doing, we are able to move from the information content of the higher order assortativity of a network to the dynamical properties of the underlying Markov chain. Specifically, the temporal dimension of the network and the regularities captured by the autocorrelation – which are hidden in the network structure – become clear in moving to the Markov chain theory.

2023

02 maggio

Lorenzo Vecchi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Matroids encode in a combinatorial way the notion of linear independence and can be seen as a generalization of matrices, graphs and hyperplane arrangements.
The main protagonist of this talk is an invariant called the Chow ring of a matroid, whose definition is given in analogy with the one arising from Algebraic Geometry. Long-standing combinatorial conjectures were solved by the introduction of this and other related geometric tools, which in turn have remarkable combinatorial features; for example, their Hilbert series seem to be real-rooted.
After a friendly introduction to Matroid Theory, the plan of the talk is to answer the following questions.
1) How can we study the Hilbert series without actually building the whole graded vector space?While trying to answer this question, different algebraic and combinatorial objects will arise along the way, like the Kazhdan-Lusztig-Stanley polynomials. Help will come both from Poset Theory and Polytope Theory.
2) After obtaining these combinatorial answers, which tools can be lifted back to the higher categorical level we started from?In particular, we are concerned with questions regarding properties of some functors in a new category of matroids. Time permitting, we will also transform all these invariant into graded representations of the group of symmetries of the matroid.
This is based on a joint work with Luis Ferroni, Jacob Matherne, and Matthew Stevens and an ongoing project with Ben Elias, Dane Miyata, and Nicholas Proudfoot.

2023

28 aprile

Alessandro Monguzzi (U. di Bergamo)

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica

It is known that no general statements can be made about the rate of convergence in Birkhoff’s ergodic theorem. However, it is possible to obtain some results in specific settings. In this talk I will consider weighted sums and discuss their speed of convergence to the mean value. I will provide estimates for such speed of convergence in terms of Diophantine properties of the involved parameters.
This talk is based on an ongoing investigation with N. Chalmoukis, L. Colzani and B. Gariboldi.

2023

28 aprile

Stefania Bellavia

Seminario di analisi numerica

We present and discuss gradient-based optimization methods for solving minimization problems arising in machine learning applications. The analysed methods employ first-order models for the objective function and stochastic gradient approximations. We will focus on the stochastic gradient and some of its widely used variants. The application of such methods to the training of classifiers and neural networks will be considered and numerical results will be also shown.

2023

27 aprile

Alessandra Pluda

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature,
and can be interpreted as the gradient flow of the length.
In this talk I will consider the same flow for networks (finite unions of sufficiently smooth curves whose
endpoints meet at junctions).
I will explain how to define the flow in a classical PDE framework, and then I will list some examples of singularity formation,
both at finite and infinite time, and explain the resolution of such singularities obtained by geometric microlocal analysis
techniques.
I will describe a stability result based on Lojasiewicz–Simon gradient inequalities and give a rough estimate
on the basin of attraction of critical points. Furthermore, I will motivate the coarsening-type behavior
clearly visible in numerical simulations.
This seminar is mainly based on recent papers in collaboration with Jorge Lira (Uni-
versidade Federal do Ceará), Rafe Mazzeo (Stanford University), Mariel Saez (P. Universidad
Catolica de Chile) and Marco Pozzetta (Università di Napoli Federico II).

2023

27 aprile

In the first part of this talk we introduce integer programming models and the two ingredients for their practical solution, namely, the simplex and the branch-and-bound algorithms.
Next, we show how to formulate the Traveling Salesman problem, probably the most famous problem in combinatorial optimization, as an integer program having an exponential number of inequalities. Despite the huge size of the resulting formulation, we present a solution approach where only a small subset of inequalities has to be explicitly considered for computing an optimal solution.

2023

27 aprile

We present a manifestly nonnegative integral expansion for the
cohomology class of the permutahedral variety in terms of Schubert
classes. This is thus a contribution to the Schubert calculus of the
flag variety, which we will recall. In addition, we introduce
algebraic and combinatorial objects of independent interest. In
particular we present a new basis of the polynomial ring that refines
Schubert polynomials. We then use it to give a combinatorial
interpretation of the coefficients in the above expansion in terms of
a novel parking procedure. This is joint work with Vasu Tewari.

2023

27 aprile

Martin Stoll

Seminario di analisi numerica

2023

27 aprile

2023

21 aprile

Alessandro Oneto (Università degli Studi di Trento)

nel ciclo di seminari: APPLIED ALGEBRAIC GEOMETRY

Seminario di algebra e geometria

The degree of entanglement of a pure quantum state can be measured by its distance or angle to the closest product state. This is called Geometric Measure of Entanglement (GME) and can be extended to mixed states. I will try to introduce this notion, by providing examples from the literature.

2023

21 aprile

Zeev Wiesman

Seminario di analisi numerica, interdisciplinare

TBA

2023

20 aprile

Sven Jarohs

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk I present some recent results of a joint work with Paolo Salani and Tadeusz Kulczycki concerning the fractional Bernoulli problem, where I will focus on the so called spectral half Laplacian. After a short introduction, I will explain how to construct a solution to this problem using the Beurling method on the extended problem. Moreover, we discuss geometric properties of the solution and, if there is time, give some remarks on the problem with the usual half Laplacian.

2023

20 aprile

Marco Moraschini

nell'ambito della serie: TOPOLOGIA E GEOMETRIA DELLE VARIETÀ

Seminario di algebra e geometria

2023

20 aprile

This is the second seminar of a cicle devoted to learn properties of brain neural network from neurophysiological data. This seminar is quite technical and open only to a restricted specialistic pubblic.

2023

20 aprile

Kieran Gregory O'Grady

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Prima parte: Introduzione ai moduli di fasci stabili su varieta' complesse proiettive lisce, in particolare superfici K3.
Seconda parte: Esporro' la costruzione di componenti irriducibili di spazi di moduli di fasci (semi)stabili su varieta' HK polarizzate di tipo K3^{[n]}
che sono deformazioni di spazi di moduli di dimensione arbitrariamente alta di fasci (semi)stabili su superfici K3. La costruzione e' stata motivata da esempi di Enrico Fatighenti.

2023

19 aprile

This is the therd seminar of a cicle devoted to learn properties of brain neural network from neurophysiological data. It will be quite technical and open to a restricted specialistic pubblic.

2023

18 aprile

S. Zucker

nell'ambito della serie: NEUROMATEMATICA

nel ciclo di seminari: NEUROMATEMATICA

Seminario di analisi matematica, interdisciplinare

This is the second seminar of a cicle of seminar, devoted to learn properties of brain neural network from neurophysiological data. This second seminar is more technical than the first one, and open to a restricted specialistic pubblic.

2023

17 aprile

Martin Stoll

Seminario di analisi numerica

In this talk we briefly review some basic PDE models that are used to model phase separation in materials science. They have since become important tools in image processing and over the last years semi-supervised learning strategies could be implemented with these PDEs at the core. The main ingredient is the graph Laplacian that stems from a graph representation of the data. This matrix is large and typically dense. We illustrate some of its crucial features and show how to efficiently work with the graph Laplacian. In particular, we need some of its eigenvectors and for this the Lanczos process needs to be implemented efficiently. Here, we suggest the use of the NFFT method for evaluating the matrix vector products without even fully constructing the matrix. We illustrate the performance on several examples.

2023

14 aprile

S. Zucker

nell'ambito della serie: NEUROMATEMATICA

Seminario di analisi matematica, interdisciplinare

How might one infer circuit properties from neurophysiological data.
We address this challenge for the mouse visual system with a novel neural manifold obtained using unsupervised machine learning algorithms. Each point on our manifold is a neuron; nearby neurons respond similarly in time to similar parts of a stimulus ensemble. This ensemble includes drifting gratings and flows, i.e. patterns resembling what a mouse would “see” while running through fields. Our manifold differs from the standard practice in computational neuroscience, of embedding trials in neural coordinates. Importantly, for our manifolds topology matters: from spectral theory we infer that, if the circuit consists of separate components, the manifold is discontinuous (illustrated with retinal data). If there is significant overlap between circuits, the manifold is nearly-continuous (cortical data). To approach real circuits, local neighborhoods on the manifold are identified with actual circuit components. For the retinal data we show these components correspond to distinct ganglion cell types by their mosaic-like receptive field organization, while for cortical data, neighborhoods organize neurons by type (excitatory/inhibitory) and anatomical layer. The manifold topology for deep CNN's will also be developed.
Joint research with Luciano Dyballa (Yale), Marija Rudzite (Duke), Michael Styrker (UCSF) and Greg Field (UCLA).

2023

13 aprile

Vittorio Martino

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

Starting from motivations coming from Physics, we will introduce the Dirac-Einstein equations on a spin manifold. After reviewing some known background material, we will show some recent results, in particular: the compactness of the variational solutions, the classification of the Palais-Smale sequences for the related conformal problem, and finally, an existence result of Aubin type.

2023

12 aprile

Daniele Faenzi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Una foliazione di codimension 1 su una varietà proiettiva X puo' essere vista come un sotto fascio (saturato) F del fibrato tangente TX, stabile per il bracket di Lie.
Una volta fissato il determinante di F, tali foliazioni formano un sottoinsieme localmente chiuso dello spazio delle 1-forme sur X a valori in un fibrato in rette L.
Studieremo lo spazio di queste foliazioni quando X è una varietà razionale omogenea di numero di Picard 1, per le scelte più semplici possibili di L, in particolare quando X è una grassmanniana o più generalmente una varietà cominuscola.
Lavoro in collaborazione con V. Benedetti e A. Muniz.

2023

06 aprile

Giovanna Citti

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

We prove a parabolic version of the standard Poincaré inequality, and we show that the elliptic version of the Moser argument can be applied even in the parabolic and Kolmogorov setting to deduce the Hölder regularity of the solutions. The price to pay is the lack of uniformity, in the constants. The proof is elementary and unifies in a natural way several results in the literature on Kolmogorov equations, subelliptic ones and some of their variations. This result is a joint work with M. Manfredini and Y. Sire.

2023

04 aprile

Celeste Damiani

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

TBA

2023

30 marzo

Annalisa Baldi

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

Abstract:
Recently, in a joint work with Bruno Franchi and Pierre Pansu, we have proved some new interior Poincaré and Sobolev inequalities for the Rumin complex in the Heisenberg group in the endpoint situation p = Q (the homogeneous group dimension) or p = Q/2, depending on the degree of the forms. We refer to these inequalities as (\infty, Q)-Poincaré or Sobolev inequalities (or (\infty, Q/2)-Poincaré or Sobolev inequalities respectively).
These results complement and complete the program on Poincaré-Sobolev inequalities that we developed in a series of previous papers for $1\le p< Q$ (or p<Q/2). In this talk I'll present a further improvement of the global (\infty, Q)-Poincaré inequality, still obtained in collaboration with Franchi and Pansu, showing that it possible to upgrade bounded primitives to bounded and continuous primitives in the case (\infty, Q) (or (\infty, Q/2), depending on the degree of the forms). The argument we use relies on our previous results and duality (i.e. Hahn-Banach) and generalizes to differential forms a Bourgain-Brezis's duality argument for a Poincaré inequality for periodic functions (Bourgain-Brezis, JAMS 2003).

2023

30 marzo

Christophe Hohlweg

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

In this talk, I will survey the notion of Garside families in relation with the word problem for Artin-Tits groups. In particular, Dehornoy, Dyer and I proved that there exist a finite Garside family in any Artin-Tits group G. Then I will discuss how this family is (suprisingly) related to a well-known finite hyperplane sub-arrangements, the Shi arrangement, of the reflection arrangement of the Coxeter group associated to G.

2023

29 marzo

Luca Ratti

Seminario di analisi numerica

The task of inverse problems is to determine an unknown quantity from measurements obtained through a forward operator, possibly corrupted by noise. Such problems are usually unstable: small perturbations of the observed measurements may cause large deviations in the reconstructed solutions. Variational regularization is a well-established technique to tackle ill-posedness, and it requires solving a minimization problem in which a mismatch functional is endowed with a suitable regularization term. The choice of such a functional is a crucial task, and it usually relies on theoretical suggestions as well as a priori information on the desired solution.
In recent years, statistical learning has outlined a novel and successful paradigm for this purpose. Supposing that the exact solution and the measurements are distributed according to a joint probability distribution, which is partially known thanks to a suitable training sample, we can take advantage of this statistical model to design data-driven regularization operators.
In this talk, I will consider linear inverse problems (associated with relevant applications, e.g., in signal processing and in medical imaging), and aim at learning the optimal regularization operator, first restricted to the family of generalized Tikhonov regularizers. I will discuss some theoretical properties of the optimal operator and show error bounds for its approximation as the size of the sample grows, both with a supervised-learning strategy and with an unsupervised-learning one. Finally, I will discuss the extension to different families of regularization functionals, with a particular interest in sparsity-promotion.
This is based on joint work with G. S. Alberti, E. De Vito, M. Santacesaria (University of Genoa), and M. Lassas (University of Helsinki)

2023

28 marzo

Silvia Romanelli

Seminario di analisi matematica

We discuss certain mixed Cauchy-Dirichlet problems for degenerate parabolic equations in the unbounded interval R^+, which are generalizations of the Black-Scholes equation and of the Cox-Ilgersson-Ross equation. We present existence and regularity results for the (C0) semigroups
associated to them in suitable spaces of continuous functions on (0, ∞) highlighting,
in particular, analogies and differences between the corresponding generators.

2023

28 marzo

Arvid Perego

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

There are very few known deformation classes of irreducible
symplectic manifolds, i. e. compact, connected Kahler manifolds that
are simply connected and carry a holomorphic symplectic form spanning
the space of closed holomorphic 2-forms. Among the tools that are used
to study the birational geometry of these manifolds, the monodromy
group is of the utmost important, and has been calculated for all the
known deformations classes by Markman, Mongardi, Rapagnetta and
Onorati. As soon as we allows singularities, we get into the world of
irreducible symplectic varieties, for which we know many more
deformations classes, and the monodromy group is still defined and
plays a major role in the study of their birational geometry. In a
joint work with Onorati and Rapagnetta we calculate the monodromy
group of the examples of irreducible symplectic varieties given by
moduli spaces of semistable sheaves on K3 surfaces.

2023

24 marzo

Attraverso una serie di esempi tratti dalle sue opere, si mostrano alcune caratteristiche essenziali del metodo scientifico di Archimede e le si confrontano con quelle di alcuni scienziati della prima età moderna.

2023

23 marzo

Gonçalo dos Reis

Seminario di analisi numerica, probabilità

In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain “iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology.

2023

23 marzo

Emanuele Macrì

Seminario di algebra e geometria

In this lecture, we survey results on the Kuznetsov component of the derived category of a cubic fourfold, and its links with Hyper-Kähler geometry.

2023

21 marzo

Andrea Maffia

nell'ambito della serie: TOPICS IN MATHEMATICS 2022/2023

Seminario di didattica della matematica

International research on tertiary mathematics education has produced till the present days important solid findings which have profoundly influenced research in mathematics education even at other grade levels. In this seminar, after a brief introduction to what research into mathematics education is, results from classical research studies about the teaching/learning of mathematics at the undergraduate level will be presented. Furthermore, classical results will be related to recent research about modern practices including digital resources.

2023

21 marzo

Davide Cesare Veniani

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Non-degeneracy of Enriques surfaces
Enriques' original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron.
The question whether all Enriques surfaces arise through Enriques' construction has remained open for more than a century.
In two joint works with G. Martin and G. Mezzedimi, we have now settled this question in all characteristics by studying particular configurations of genus one fibrations, and two invariants called maximal and minimal non-degeneracy. The proof involves so-called `triangle graphs' and the distinction between special and non-special 3-sequences of half-fibers.
In this talk, I will present the classification of Enriques surfaces of low non-degeneracy and explain how this classification solves this long-standing problem.

2023

17 marzo

After having recalled the content of the first part, I will present a framework developed in the context of logic that allows one to measure the complexity of classification problems in mathematics. I will then explain how the "definable" algebraic invariants developed with Bergfalk and Panagiotopoulos can be used to obtain information on the complexity of classification problems mentioned in the title.

2023

17 marzo

Valentina Amitrano

nel ciclo di seminari: APPLIED ALGEBRAIC GEOMETRY

Seminario di algebra e geometria

In this seminar I would like to introduce the formalism commonly used to describe quantum computing: from the vectorial definition of multi-qubit states, to the description of their dynamics through the application of quantum gates as unitary matrix multiplication. The main goal is to compare this formalism with a purely tensorial one that is not used by the physics community but which could be useful in the quantum gate decomposition procedure.

2023

14 marzo

Gabriele Viaggi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

An open subset of real projective space is said to be properly convex if it is contained in an affine chart where it is convex and bounded. Together with their natural Hilbert metric and group of projective symmetries, properly convex sets are a rich source of geometry, dynamics, and group theory. Of particular interest are those that are divisible, that is, they admit a compact quotient by a discrete group of symmetries. In general, it is a challenging problem to construct such examples. After reviewing the global classification scheme, I will describe a new class of examples of divisible convex sets with special geometric properties (non-symmetric and non-strictly convex) that completes a missing part of the picture. This is joint work with Pierre-Louis Blayac.

2023

09 marzo

Bruno Franchi

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

Abstract (joint research with
P. Ciatti and Y. Sire): In this talk we derive several properties of the fundamental solution for the heat equation associated with the Rumin's complex on
Heisenberg groups. As an application, we use the heat kernel for Rumin's differential forms to construct a Calder\'on reproducing formula for Rumin’s
differential forms.
Titolo: Sul nucleo del calore per il complesso di Rumin e la reproducing formula di Calder\’on
( ricerca in collaborazione con P. Ciatti e Y. Sire).
Riassunto: In questo seminario otteniamo varie proprie\`a della soluzione fondamentale dell’equazione del calore associata
al complesso di Rumin nei gruppi di Heisenberg. Come applicazione, usiamo il nucleo del calore
per le forme differenziali di Rumin per costruire la \it{reproducing formula} di Calder\’on per le forme differenziali di Rumin.

2023

07 marzo

Alessio Savini

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Since the formulation of Dupont's conjecture, it has been evident the importance to understand the boundedness of characteristic classes appearing in the cohomology ring of a semisimple Lie group. This problem is deeply related to Monod's conjecture, which relates the continuous bounded cohomology of a semisimple Lie group with its continuous variant.
An important step towards a possible proof of those conjectures was the isometric realization of the continuous bounded cohomology of a semisimple Lie group G as the cohomology of the complex of essentially bounded functions on the Furstenberg-Poisson boundary (and more generally for any regular amenable G-space). Surprisingly, Monod has recently proved that the complex of measurable unbounded functions on the same boundary does not compute the continuous cohomology of G unless the rank of the group is not one, but an additional term appears. Nevertheless, there is a way to characterize explicitly the defect in terms of the invariant cohomology of a maximal split torus.
In this seminar we will exhibit two main examples of such phenomenon: the product of isometry groups of real hyperbolic spaces and the group SL3.
The first part of the seminar will be devoted to an overview about the state of art. Then we will move to examples and we will give a characterization of Monod's Kernel in low degree. Finally we will show that Monod's conjecture is true in those cases.
In the second part of the seminar we will discuss in details the main results and the techniques we used, such as the explicit computation on Bloch-Monod spectral sequence. If time allows we will show how we can implement all this stuff using a software like Sagemath.

2023

03 marzo

Nikolaos Chalmoukis, Università di Milano-Bicocca

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica

In this talk we will discuss some problems about interpolation in Hilbert spaces of analytic functions and their multiplier algebras. In particular we give a positive answer to a question raised by J. E. McCarthy and J. Agler about strong separation and simply interpolating sequences in reproducing kernel Hilbert spaces with the complete Nevanlinna Pick property. Furthermore, we use a result of A. F. Beardon from Fuchsian groups to show that, in the same class of spaces, there exist sequences with infinite associated measure which are simply interpolating.

2023

28 febbraio

Daniele Valeri

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Nel seminario rispolvereremo alcune nozioni base di geometria di Poisson e di algebre di vertice di Poisson e il loro legame con le gerarchie integrabili di ODE/PDE. Parleremo in seguito di triple integrabili per le algebre di Lie semplici e della loro classificazione. La classificazione è usata per dimostrare che (quasi) tutte le W-algebre classiche affini W(g,f), una vasta famiglia di algebre di vertice di Poisson associate a un'algebra di Lie semplice g e un suo elemento nilpotente f, possiedono gerarchie integrabili di PDE bi-Hamiltoniane. Queste gerarchie generalizzano le gerarchie di Drinfeld-Sokolov che sono ottenute quando f è la somma dei vettori radice corrispondenti alle radici semplici.

2023

25 febbraio

2023

24 febbraio

Giuseppe Lamberti (Université de Bordeaux)

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica

The study of interpolating sequences for analytic functions in one or more complex variables is one of the main research areas in complex analysis. It has plenty of applications in fields such as signal theory, control theory, operator theory, etc. For many spaces, like Hardy spaces, these sequences are well understood while for others, like Dirichlet spaces, there exists a characterization which is not very easy to verify. In other circumstances, a characterisation does even not exist. In this scenario it is useful to consider a random setting, which can help us to understand when interpolation is “generic”. In particular in this talk we are going to start introducing deterministic interpolation in the Nevanlinna class, to then consider a random setting, more specifically a radial model (where points’ radii are fixed, while the arguments are uniformly distributed). In this case we will give sufficient condition for a sequence to be or to not be interpolating.

2023

22 febbraio

Ermanno Lanconelli

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

Let D be a bounded open set of R^n with \sigma(\partial D)< \infty and let x_0 be a point of D. Assume that u(x_0) equals the average of u on \partial D for every harmonic function u in D continuous up to the boundary. In this case one says that D is a harmonic pseudosphere centered at x_0.
In general, harmonic pseudospheres are not spheres as a two-dimensional example due to Keldysch and Lavrentiev (1937) shows. As a consequence, the following problem naturally arose: when a pseudosphere is a sphere? Or, roughly speaking: is it possible to characterize the Euclidean spheres via the Gauss mean value property for harmonic function? The answer is yes.
The most general result in this direction was obtained by Lewis and Vogel in 2002: they proved that a harmonic pseudosphere \partial D is a sphere if D is Dirichlet regular and the surface measure on \partial D has at most an Euclidean growth.
Preiss and Toro, in 2007, proved the stability of Lewis and Vogel's result. Namely: a bounded domain D satisfying the Lewis and Vogel’s regularity assumptions, has the boundary geometrically close to a sphere centered at x_0 if the Poisson kernel of D with pole at x_0 is close to a constant.
In collaboration with Giovanni Cupini we proved that the previous rigidity and stability results hold true if the domain D has the boundary with finite area and only satisfies the following property: the boundary of D is Lyapunov-Dini regular in at least one point of \partial D closest to x_0.
Our approach to the rigidity ad stability properties of the Surface Mean Value Theorem for harmonic functions is quite elementary in spirit: it does not uses the profound harmonic analysis and free boundary techniques instead used by Lewis and Vogel and by Preiss and Toro, but it relies on careful estimates of the Poisson kernel of the biggest ball centered at x_0 and contained in D.

2023

21 febbraio

Tommaso Cremaschi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

In this talk I will give some purely topological construction for hyperbolic 3-manifolds with infinitely generated fundamental group, this will let us construct many infinite-type hyperbolic 3-manifolds. Then, I will say something about the set of hyperbolic structures that they can admit.

2023

17 febbraio

Nathan Wagner (Brown University)

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica

The Bergman projection is a fundamental operator in complex analysis in one and several complex variables. Consequently, its regularity properties on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results in this direction on strongly pseudoconvex domains with near minimal boundary smoothness. In particular, weighted L^p estimates are obtained on strongly pseudoconvex domains of class C^4, where the weight belongs to a suitable generalization of the Bekolle-Bonami class. For such domains, precise estimates on the Bergman kernel function are unavailable. Consequently, we use a kernel free, operator-theoretic technique that goes back to Kerzman, Stein, and Ligocka, and was subsequently refined by Lanzani and Stein to prove (unweighted) L^p regularity. We will also discuss the fundamental obstruction to improving this result to domains of class C^2 (minimal smoothness). This talk is based on joint work with Brett Wick.

2023

16 febbraio

Annalaura Rebucci

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

We study a class of second order strongly degenerate kinetic operators L in the framework of special relativity. More precisely, the operator L we consider here is a possible suitable relativistic generalization of the kinetic Fokker-Planck operator. We first describe L as a Hormander operator which is invariant with respect to Lorentz trans- formations. We then prove a Lorentz-invariant Harnack type inequality, and we derive accurate asymptotic lower bounds for positive solutions to L f = 0. As a consequence, we obtain lower bounds for the density of the relativistic stochastic process associated to L .
This is a joint work with Francesca Anceschi (Università Politecnica delle Marche) and Sergio Polidoro (Università degli Studi di Modena e Reggio Emilia).

2023

16 febbraio

2023

15 febbraio

Vladimir Druskin

Seminario di analisi numerica

Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.

2023

14 febbraio

Gian Marco Todesco

nel ciclo di seminari: LABORATORIO PLS "METODI MATEMATICI PER LE ANIMAZIONI"

Seminario interdisciplinare

2023

14 febbraio

Jeffrey Bergfalk

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

The rapid development, in the few short years since their introduction, of the closely related condensedand pyknotic mathematics frameworks of Clausen and Scholze, and Barwick and Haine, respectively, forms the immediate background for this talk. In the standard heuristic, these are frameworks for ``doing algebra with objects carrying a topology''; more concretely (but still imprecisely), they are frameworks for coordinated approaches to a wide variety of mathematical objects via their (pre)sheaf representations over the category of profinite sets (i.e., of totally disconnected compact Hausdorff spaces). In our talk's first half, we will introduce and survey these frameworks. In its second half, we will discuss some of the questions in infinitary combinatorics and set theory which these frameworks pose (with a particular focus on the ``condensed image'' of the Whitehead Problem), as well as some answers. This work is joint with Chris Lambie-Hanson.

2023

10 febbraio

We will present a new Reproducing Kernel Hilbert Space of functions on the real line, having the complete Pick Property. We then characterize its Carleson measures, multipliers and interpolating sequences. Work in collaboration with Nikolaos Chalmoukis.

2023

10 febbraio

Gian Marco Todesco

nel ciclo di seminari: LABORATORIO PLS "METODI MATEMATICI PER LE ANIMAZIONI"

Seminario interdisciplinare

2023

07 febbraio

Gian Marco Todesco

nel ciclo di seminari: LABORATORIO PLS "METODI MATEMATICI PER LE ANIMAZIONI"

Seminario interdisciplinare

2023

07 febbraio

Olivier Debarre

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Given an algebraic variety $X$ in a projective space, one can try to describe the families of curves (lines, conics...), surfaces (planes, quadrics...), and other subvarieties contained in $X$. These families are themselves algebraic varieties that can help understand the geometry of $X$. I will review the classical case when $X$ is a cubic hypersurface and then continue with the case where $X$ is a so-called Gushel-Mukai variety. This is work in collaboration with Alexander Kuznetsov.

2023

06 febbraio

Latva-Äijö Salla-M

Seminario di analisi numerica

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials.
A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional.
Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

2023

06 febbraio

Marta Lazzaretti

Seminario di analisi numerica

We consider structured optimization problems defined in terms of the sum of a smooth and convex function and a proper, lower semicontinuous (l.s.c.), convex (typically nonsmooth) function in reflexive variable exponent Lebesgue spaces $L^p(cdot)$. Due to their intrinsic space-variant properties, such spaces can be naturally used as solution spaces and combined with space-variant functionals for the solution of ill-posed inverse problems. For this purpose, we propose a new proximal gradient algorithm in L^p(), where the proximal step, rather than depending on the natural (non-separable) L^p()-norm, is defined in terms of its modular function, which, thanks to its separability, allows for the efficient computation of algorithmic iterates. To highlight the effectiveness of the modeling, we report some numerical tests for the CT imaging application.

2023

03 febbraio

Nikolaos Chalmoukis, Università di Milano-Bicocca

nell'ambito della serie: COMPLEX ANALYSIS LAB

Seminario di analisi matematica

A classical theorem of Fatou [1] says that a function in the Hardy space of the disc has non tangential limits at almost every boundary point. Conversely for every compact set of Lebesgue measure zero in the boundary of the unit disc there exists a function in the Hardy space which fails to have non tangential limits exactly on this set. So we say that the exceptional sets are exactly the sets of Lebesgue measure zero.
In the unit ball of C^n there are two (different) spaces which play the role of the Hardy space in the unit disc. One is the Hardy space of the ball and the other one is the Drury Arveson space. For the first one, exceptional sets in the sense of Fatou have been characterized by Korányi [2].
In this talk we will give a characterization of the exceptional sets of the Drury Arveson space by introducing a new potential theory on the boundary of the unit ball in C^n or equivalently in the Heisenberg group.
[1] P. Fatou, (1906) Séries trigonométriques et séries de Taylor, Acta Mathematica, Acta Math. 30, 335–400.
[2] A. Korányi, (1969). Harmonic Functions on Hermitian Hyperbolic Space. Transactions of the American Mathematical Society, 135, 507–516.

2023

02 febbraio

Christian Seis

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap. This improves on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the absence of a spectral gap, as occurs when the vanishing profile fails to be isolated (and may belong to a continuum of such profiles). Joint work with Beomjun Choi and Robert J. McCann.

2023

02 febbraio

Christian Seis

Seminario di analisi matematica

2023

02 febbraio

Roberto Conti

Seminario di algebra e geometria, analisi matematica, fisica matematica

Finite multiplets of isometries with orthogonal ranges summing up to 1 were systematically used by Doplicher and Roberts in the early 70's in their study of the superselection structure of Quantum Field Theory. Nowadays, the C*-algebra generated by any such multiplet is known as Cuntz algebra O_n, where n is the cardinality of the given multiplet. Since their introduction, Cuntz algebras have been the subject of endless investigations, from many different points of view. They are perhaps the most studied/used class of C*-algebras ever. Notably, the study of their automorphisms presents many challenging facets, where operator algebras meet Lie theory, dynamical systems and combinatorics. We will present an overview of recent results, along with some open questions.

2023

31 gennaio

Filippo Sarti

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Measurable cocycles arise in different fields of mathematics, and cocycle rigidity is a self standing research topic that goes back to Zimmer's work.
In the first part of this talk I will give a gentle introduction to cocycles, describing some explicit examples coming out in different situations such as orbit equivalences and measure equivalences.
In the the second part I will describe a technique to investigate rigidity that involves bounded cohomology.
A comparison with groupoids will be carried on and, time permitting, in the last part I will show how the above machinery might be described in the category of measured groupoids.
This is the starting point of a joint work in progress with A. Savini about a theory of bounded cohomology for groupoids.

2023

27 gennaio

Let A be a discrete, unbounded, infinite set in R. Can
we find a "large" measurable set E in R which does not contain
any affine copy x + tA of A (for any in R, t > 0)?
If a(n) is a real, nonnegative sequence that does not increase ex-
ponentially, then, for any 0 <= p < 1, we construct a Lebesgue
measurable set which has measure at least p in any unit interval
and which contains no affine copy of the given sequence. We gen-
eralize this to higher dimensions and also for some "non-linear"
copies of the sequence. Our method is probabilistic.
Joint work with M. Kolountzakis (Univ. of Crete).
Current address: Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Greece

2023

26 gennaio

Bianca Stroffolini

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

2023

26 gennaio

Marcello D'Abbicco

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk, we show recent Lp-Lq estimates, for general (p,q), obtained in collaboration with M.R. Ebert, for the solution to the Cauchy problem for the visco-elastic damped wave equation. These results complete the picture that was started by Y. Shibata (MMAS, 2000), who obtained the endpoint estimates (p,q)=(1,1) and (p,q)=(1,∞). We show that optimal Lp-Lq estimates in the general case cannot be derived by a simple interpolation due to the “noneffective” nature of the damping term. We show the analogous but different result which holds for other strongly damped p-evolution models, as the plate equation. We mention a straightforward application of the obtained estimates to derive global existence of small data solutions for problems with critical or supercritical power nonlinearities.

2023

25 gennaio

X-ray tomography (CT Scan) is a widely used method to inspect an object without damaging its structure (Non Destructive Testing). It allows the conformity of the object to be checked with respect to the intended dimensions, material composition, homogeneity, etc.
At the French Atomic Energy Commission (CEA), we are developing such a technique for different objectives and objects: verification of the conformity of nuclear waste drums (safety objectives), nuclear fuel (performance objectives) and metal additive manufacturing (cost objectives).
Nuclear waste drums are very large objects (more than one cubic metre and two tonnes), nuclear fuel is very dense and metallic additive manufacturing is an intermediate case.
For these three objects, the scanners are specific and rely on linear accelerators (high energy and dose rate) and thick scintillators. These components bring an intrinsic blur (Point Spread Function) which degrades the scanner results.
In order to correct this degradation and to improve the control capabilities, different PSF deconvolution methods are currently under study and will be presented. They can be applied on radiographs (projections with known Poissonian noise but low gradient) before the tomographic reconstruction process or directly on CT images (non-Poissonian noise and artefacts but with a high gradient).
The two corrections can lead to different performances. Finally, if they effectively reduce the blur in the final CT images, they must also deal with the noise corruption that is always present.

2023

24 gennaio

Salvatore Floccari

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

I will present a construction which associates to any sixfold K of generalized Kummer type a hyper-Kähler manifold Y deformation equivalent to a Hilbert scheme of lenght 3 subscheme on a K3 surface, relating the most well studied deformation types of hyper-Kähler manifolds in dimension 6. Our construction is reminiscent of the classical construction of Kummer K3 surfaces, in the sense that Y is obtained as resolution of the quotient of K by a group of symplectic automorphisms. As a consequence we are able to show that any projective sixfold K as above determines a well-defined K3 surface. We use this construction to prove that the Kuga-Satake correspondence is algebraic for infinitely many new families of K3 surfaces of general Picard rank 16.

2023

23 gennaio

Filippo Zanetti

Seminario di analisi numerica

When an iterative method is applied to solve the linear equation system in interior point methods (IPMs), the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box with a standard termination criterion which asks for a sufficient reduction of the residual in the linear system. Such an approach often leads to an unnecessary 'oversolving' of linear equations. In this paper, an IPM that relies on an inner termination criterion not based on the residual of the linear system is introduced and analyzed. Moreover, new indicators for the early termination of the inner iterations are derived from a deep understanding of IPM needs. The new technique has been adapted to the Conjugate Gradient (CG) and to the Minimum Residual method (MINRES) applied in the IPM context. The new criterion has been tested on a set of quadratic optimization problems including compressed sensing, image processing and instances with partial differential equation constraints, and it has been compared to standard residual tests with variable tolerance. Evidence gathered from these computational experiments shows that the new technique delivers significant improvements in terms of inner (linear) iterations and those translate into significant savings of the IPM solution time.

2023

23 gennaio

Gondzio Jacek

nel ciclo di seminari: "MODERN TECHNIQUES OF LARGE SCALE OPTIMIZATION FOR DATA SCIENCE"

Seminario di analisi numerica

Lectures 1 and 2 (2h)
Interior Point Methods (IPMs) for LP
- Motivation, logarithmic barrier function, central path, neighbourhoods,
- path-following method, convergence proof, complexity of the algorithm,
- practical implementation issues.
Lectures 3 and 4 (2h)
Interior Point Methods for QP, (convex) NLP, SOCP and SDP
- Quadratic Programming (QP) problems, primal-dual pair of QPs,
- Nonlinear (convex) inequality constraints,
- Second-Order Cone Programming,
- Semidefinite Programming,
- Newton method, logarithmic barrier function, self-concordant barriers.
Lectures 5 and 6 (2h)
- Sparse Approximations with IPMs:
modern applications of optimization which require a selection
of a 'sparse' solution originating from computational statistics,
signal or image processing, compressed sensing, machine learning,
and discrete optimal transport, to mention just a few.
- Alternating Direction Method of Multipliers (ADMM).

2023

20 gennaio

Chiara Razzetta

Seminario di analisi numerica

Medical ultrasound is the most widely used non-invasive real-time imaging system: it exploits the ability of human tissues to reect ultrasound signals. In detail, reected ultrasound can be processed to obtain images and quantitative measurements of the physical properties of tissues. If we refer to all the imaging modalities that a common ultrasound apparatus can handle, their properties change in accordance with numerous structural and non-structural factors of the machine. More in detail, the Point Spread Function (PSF) turns out to be space variant and dependent on the acquisition geometry and probe that is used.
In addition, for each mode and each probe there are many parameters that govern the goodness of the image (transmitted waveform, transmission frequency, number of active probe elements and so on). These parameters are historically chosen according to experience and must be selected each time the machine specifications change. Is it then possible to develop one or more methods to make an automatic estimate of all the parameters involved for any given mode? And, in particular, can we hope to optimize the parameters so that the PSF is as uniform as possible? We have tried to answer these questions by formulating a model and an associated optimization problem.

2023

20 gennaio

I will recall the notion of continuous trace C*-algebra, and present classical classification results for such C*-algebras and their automorphisms in terms of the Cech cohomology of the spectrum due to Dixmier--Douady and Phillips--Raeburn. I will then explain how the framework of Borel-definable homological algebra recently developed with Jeffrey Bergfalk and Aristotelis Panagiotopoulos allows one to refine the analysis and obtain more precise classification results.

2023

19 gennaio

Fast accurate and robust localization of barcodes in images is essential to Datalogic core business. Hereto, this seminar will present a proprietary barcode-specific low-resolution feature extraction technique enabling fast barcode localization. The seminar will focus both the analytical derivation as well as the numerical and hardware-specific optimization steps required to make the solution deployable on embedded devices with real time constraints.

2023

19 gennaio

Francesca Colasuonno

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

Seminario di analisi matematica

In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation
-\Delta u + u = a(x)|u|^{p-2}u
set in an annulus of R^N. Here p>2 is allowed to be supercritical in the sense of Sobolev embeddings, and a(x) is a positive weight with additional symmetry and monotonicity properties, which are shared by the solution that we construct. For this problem, we find a new type of positive, axially symmetric solutions. Moreover, in the case where the weight a(x) is constant, we detect a condition, depending only on the exponent p and on the inner radius of the annulus, that ensures that the solution is nonradial. In this setting, the major difficulty to overcome is the lack of compactness in a nonradial framework. The proofs rely on a combination of variational methods and dynamical system techniques.
This is joint work with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), and Tobias Weth (Goethe-Universitat Frankfurt).

2023

19 gennaio

1. Spazi di moduli di fasci (semi)stabili su superfici K3:
deformazioni, fasci sferici, fasci su K3 ellittiche,
decomposizione in camere di stabilita' del cono ampio.
2. Fasci su varieta' hyperkähler
(HK) di dimensione maggiore (di 2):
fibrati (proiettivamente) iperolomorfi, fasci modulari,
decomposizione in camere di stabilita' del cono ampio,
fasci su HK Lagrangiane, risultati di esistenza e unicita'
di fibrati vettoriali stabili su varieta' HK di tipo K3^{{n]}
e di tipo Kum_2.

2023

18 gennaio

Micaela Verucchi e Giorgia Franchini

Seminario di analisi numerica

The Indy Autonomous Challenge Powered by Cisco (IAC) is the first autonomous racecar competition at the Indianapolis Motor Speedway and took place the 23rd of October. On that day, 9 teams from 21 universities competed to win the $1 million grand prize. The rules of the IAC required each team to compete in a fastest lap competition that included an obstacle avoidance component.
TII Euroracing took part in the challenge and represented our University in this international and innovative event. They placed second @IMS, and third both @CES22 and @TMS.
In this presentation, we will present their experience at the competition and will explain how to make a Dallara AV-21, based on the Indy Light chassis, drive by itself at over 270 kph and overtake another autonomous car at over 230 kph.

2023

17 gennaio

Riccardo Moschetti

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

Hyperelliptic Odd Coverings" are a class of odd coverings of C -> P^1, where C is a hyperelliptic curve. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of P^1. I will talk about some ways for studying this type of coverings: by fixing an effective theta characteristic, they correspond to the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and then studied from the point of view of deformations. When C is general in H_g, the number of possible Hyperelliptic Odd Coverings C -> P^1 of minimum degree is finite. The main result will be how to compute this number. This is a work in collaboration with Gian Pietro Pirola.
In the first part of the talk, I will introduce the Hyperelliptic Odd Coverings from a geometric point of view, contextualizing them in the panorama of other enumerative works (in collaboration with Farkas, Naranjo, Pirola, Lian). In the second part of the talk he will dedicate myself to deepening some demonstrations and some open problems subject to future analysis.

2023

13 gennaio

In this seminar, I will overview some applications of logic in mathematics, focusing on a framework recently developed with Bergfalk and Panagiotopoulos. In this context, we apply techniques developed for the study of "definability" notions in mathematical logic to algebraic invariants in homological algebra, topology, and functional analysis.
In questo seminario presenterò una panoramica di applicazioni di logica in matematica, concentrandomi su un ambito recentemente sviluppato in collaborazione con Bergfalk e Panagiotopoulos, nel quale tecniche svulippute nel contesto dello studio di nozioni di "definibilità" in logica matematica vengono applicate allo studio di invarianti algebrici in algebra omologica, topologia, e analisi funzionale.

2023

10 gennaio

Simone Billi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Seminario di algebra e geometria

2023

04 gennaio