Archivio 2023

In this talk we will give a brief introduction to the theory of bicategories, in order to present a bicategorical gadget, called Mackey 2-functor, axiomatizing an ubiquitous phenomena in finite equivariant mathematics, especially in representation theory of finite groups and equivariant topology. The key point of this theory so far is the existence of a universal bicategory (the Mackey 2-motives) encoding the properties of such 2-functors, and allowing furthers constructions.

The theory of machine learning has been stimulated, in recent years, by a series of empirical observations that challenged the standard knowledge inherited by the classical statistical theory. In the first part of the talk, I will review some results on simple mean-field models, that allowed statisticians and physicists to understand some of these unexpected behaviors, e.g., the double-descent phenomenon or the effectiveness of ensembling. The models rely on some simplifying assumptions, one of them related to some kind of "Gaussianity of the dataset". In the second part of the talk, I will present therefore two recent works in which we characterized regression and classification tasks on fat-tailed datasets. We showed how Gaussian universality can break down and how non-Gaussianity can affect the generalization performances, for example, the generalization rates, the existence of an MLE in a classification task, and the robustness of a Huber estimator.

In this talk, after reviewing the main concepts related to forcing and giving some examples of frequently used forcing notions (i.e. partially ordered sets deployed for this technique), we discuss some "concrete" mathematical statements that can be shown to be undecidable. In particular, we will show that the existence of a Suslin Line is independent of ZFC.

In this talk we shall discuss condensed points in the Polish space of left-orderings of a fixed left-orderable groups. We will describe new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. We discuss how these methods apply to a large class of left-orderable groups, and they shed light on spaces of left-orderings with low Borel complexity. This is joint work with Adam Clay.

In this talk, we introduce some set-theoretic tools to prove consistency results. More precisely, the presentation will cover Goedel's Constructible Universe as well as Cohen's method of forcing. No previous knowledge on this subject will be assumed.

Diophantine approximation in number theory is to approximate a given irrational number with rational numbers. From a geometric perspective, it quantifies the rate at which a given geodesic flow approaches the cusp on the fundamental domain of the modular group within hyperbolic space. In this talk, we will discuss various Diophantine approximations on the real line and also on the complex plane. In the context of the complex plane, we approximate a complex number using elements from a specified imaginary quadratic field. Lastly, we will consider Diophantine approximation on circles and spheres and study intriguing examples.

In this talk we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. More precisely, we develop an existence theory in the cases of linear, superlinear and singular perturbations; in the particular case of linear perturbations, we also investigate an associated mixed Sobolev inequality, detecting the optimal constant, which we show that is never achieved. The results discussed in this seminar are obtained in collaboration with S. Dipierro, E. Valdinoci and E. Vecchi.

With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.

While the dynamical behaviour of the iteration of holomorphic functions in one variable is well known, the situation is drastically different in several variables. This should not be a surprise. After all, even from the geometrical point of view the two situations are drastically different: in several variables there is no theorem similar to the Riemann uniformization theorem, and even simple domains as the ball and the polydisk are not biholomorphically equivalent; a holomorphic function of several variables is not determined if known on a set with an accumulation point; there are open domains which are not the maximal natural domain of any holomorphic function (Hartogs' phenomenon). Thus, understanding the dynamical behaviour of the iterations of holomorphic maps, even of automorphisms of C^2, is quite difficult. There are some classes of functions, which can be thought of as being of dimension 1.5, for which it is easier to find results, using theorems of the 1-dimensional theory together with some tools of geometrical flavour. Among these, are the Hénon maps: F(z,w)=(f(z)-\delta w , z) where f is a one-dimensional entire function, and \delta is a complex number. If f is a polynomial, they are a valid playground to understand the behaviour of all polynomial automorphisms of C^2. If f is trascendental, they are not enough to grasp all the possible dynamical behaviours of automorphisms of C^2, but nevertheless they are a starting point. In the first part of the seminar I will present the state of the art of holomorphic dynamic in C^2, while in the second part I will talk about recent results on trascendental Hénon maps, in collaboration with Anna Miriam Benini, Veronica Beltrami and Michela Zedda.

The Asymptotic Plateau Problem in the hyperbolic space is the problem of existence of minimal surfaces with a prescribed Jordan curve as a boundary “at infinity”. Since the work of Anderson in the 1980s, it is known to have a solution, which is in general not unique. In this talk, I will present an example of a Jordan curve bounding uncountably many minimal discs. I will also present some criteria for uniqueness. This is joint work with Zheng Huang and Ben Lowe.

Moving from the abstract definition of monads, we introduce a version of the call-by-value computational λ-calculus based on Wadler’s variant. We call the calculus computational core and study its reduction, prove it confluent, and study its operational properties on two crucial properties: returning a value and having a normal form. The cornerstone of our analysis is factorization results. In the second part, we study a Curry-style type assignment system for the computational core. We introduce an intersection type system inspired by Barendregt, Coppo, and Dezani system for ordinary untyped λ-calculus, establishing type invariance under conversion. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterizes convergent terms via their types. For greater accessibility, the presentation will begin with a brief introduction to lambda calculus, monads, and intersection types.

Thanks to a result by Fournier-Facio and Wade, we now know that for nonelementary hyperbolic groups, there exists an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. However, their construction is not very explicit. Therefore, it makes sense to restrict ourselves to specific Aut-invariant subspaces of quasimorphisms, derived, for example, from geometric or combinatorial constructions. During this talk, our focus is on de Rham quasimorphisms, which are defined for fundamental groups of hyperbolic surfaces. In this context, the action of the automorphism group can be translated into two different frameworks: bounded cohomology and the set of continuous functions on the circle. We delve into how these interpretations provide a more geometric perspective, facilitating a deeper understanding of the action of the automorphism group. By the end of the talk, after translating the results back into the original context, we will be able to describe all the finite-dimensional invariant subspaces of the action of the automorphism group on the space of de Rham quasimorphisms.

In 1982, Gromov introduced a homotopy invariant of manifolds called simplicial volume. Although being purely homotopic in nature, this invariant is sensitive to the geometric structures that a (closed) manifold can carry. The vanishing of the simplicial volume in the case of closed manifolds is implied by the amenability of the fundamental group. For open manifolds the situation is different, since an open manifold with amenable fundamental group can have either vanishing or infinite simplicial volume. A finiteness criterion for simplicial volume of open tame manifolds (i.e., homeomorphic to the interior of a compact manifold with boundary) was given by Clara Loeh in her PhD thesis. It implies that, if the "missing" boundary has amenable fundamental group, then the simplicial volume of the interior is finite. We generalize this result to a wider class of manifolds via the fundamental group at infinity, a topological invariant which detects the topology at infinity of an open manifold. In particular, we prove the amenability of the fundamental group at infinity implies the finiteness of the finite simplicial volume.

Le algebre di vertice e le pseudoalgebre di Lie hanno definizioni notoriamente ostiche e dure da digerire. Proverò ad illustrarne il significato motivandone l’origine.

Delle tre geometrie a curvatura costante K, quella iperbolica (corrispondente a K = -1) è di gran lunga la più ricca. In questo seminario parleremo di varietà iperboliche, cioè varietà Riemanniane di curvatura costante -1 compatte (o più generalmente complete e di volume finito). Vedremo che tali oggetti esistono in ogni dimensione, e cercheremo di capirne la topologia - in particolare esaminando quali fra queste varietà possono avere una struttura di fibrato (da un lavoro in collaborazione con Italiano e Migliorini).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

PhD thesis defense

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 $\binom{n}{\lfloor\frac{n}{2}\rfloor}-1 = \Theta(\frac{1}{\sqrt{n}} 2^n)$.

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.

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

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.

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.

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.

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).

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

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.

TBA

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.

TBA

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.

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.

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.

TBA

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 .

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.

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.

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).

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.

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.

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.

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.

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.

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).

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.

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.

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.

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).

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.

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.

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.

TBA

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).