Archivio 2021

In the first part of the seminar we introduce Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion, and we show a classical well-posedness result for this class of equations. In the second part, we give an overview on jump measures and related stochastic calculus. Then, we consider BSDEs driven by a general random measure, and we show how additional conditions have to be imposed in order to recover existence and uniqueness for the corresponding solutions.

We shall prove that in the first Heisenberg group with a sub-Finsler structure, a complete, stable, Euclidean Lipschitz surface without singular points is a vertical plane. This is joint work with Manuel Ritoré.

We shall present a characterization of simply interpolating sequences in the Dirichlet space. The same characterization is conjectured to hold in all complete Nevanlinna Pick spaces but the problem remains open despite recent progress. Finally, we are going to discuss some variants of the classical interpolation problem, such as random interpolation. This is a field where numerous questions remain open. Extended abstract: https://site.unibo.it/complex-analysis-lab/en/news-1/piniseminarabstract.pdf/@@download/file/PiniSeminarAbstract.pdf

In this seminar, I overview the research work carried out by IMATI-CNR on the extension of the Hough transform (HT) to recognize families of planar and spatial curves and surface primitives on 3D objects and point clouds. Recent developments are transforming the HT into a tool computationally affordable even outside the classical context of recognition of lines, circles and ellipses in the plane and planes and spheres in the space. In particular, I will focus on some applications such as the characterization of curves and complex patterns on artefacts and the creation of geometric models with curvilinear elements.

In the first​ part of this seminar we will introduce Coxeter groups, fully commutative elements and the Temperley-Lieb algebra, by illustrating some classical examples. In the second part, we will recall a recent construction of a diagrammatic representation of the Temperly-Lieb algebra of affine type C due to Dana Ernst, and we will show that this representation is faithfull in a new combinatorial way.

Level structures are extra data that can be added to some moduli problems in order to rigidify the situation. For example, in the case of curves, they yield smooth Galois covers of the moduli space M_g, and the problem of extending this picture to the boundary was studied by several authors, using in particular admissible covers and twisted curves. I will report on some work in progress with M. Ulirsch and D. Zakharov, in which we consider a tropical notion of level structure on a tropical curve. The moduli space of these is expected to be closely related to the boundary complex of the stack of G-admissible covers. As usual, logarithmic geometry stands in the middle and provides a convenient language to bridge the two worlds.

Reprising from my last seminar I will present some results recently obtained with G. M. Dall'Ara on the L^p boundedness of the Bergman projection of domains that can be covered by the unit ball in C^n. These results were inspired by recent works by Chen, Krantz and Yuan and by Bender, Chakrabarti, Edholm and Mainkar and these latter works were in turn motivated by the L^p regularity of the Bergman projection on generalized Hartogs triangles.

We consider convex optimisation problems defined in the variable exponent Lebesgue space L^p(·)(Ω), where the functional to minimise is the sum of a smooth and a non-smooth term. Compared to the standard Hilbert setting traditionally considered in the framework of continuous optimisation, the space L^p(·) (Ω) has only a Banach structure which does not allow for an identification with its dual space, as the Riesz representation theorem does not hold in this setting. This affects the applicability of well-known proximal (a.k.a. forward-backward) algorithms, since the gradient of the smooth component here lives in a different space than the one of the iterates. To circumvent this issue, the use of duality mappings is required; they link primal and dual spaces in a nonlinear fashion, thus allowing a sensible definition of the algorithmic iterates. However, such nonlinearity introduces further difficulties in the definition of the proximal (backward) step and, overall, in the convergence analysis of the algorithm. To overcome the non-separability of the natural induced norm on L^p(·)(Ω), we consider modular functions allowing for a an appropriate definition of proximal algorithms in this setting for which convergence properties in function values can be proved. Some numerical examples showing the flexibility of our approach in comparison with standard (Hilbert, L^p with constant p) algorithms on some exemplar inverse problems (deconvolution, denoising) are showed.

Campana proposed a series of conjectures relating algebro-geometric and complex-analytic properties of algebraic varieties and their arithmetic. The main ingredient is the definition of the class of special varieties, which conjecturally identity the class of varieties with a potential dense set of rational points (when defined over a number field) and admitting a dense entire curve (when defined over the complex numbers). In the talk we will review the main conjectures and constructions, and we will discuss some recent results that give evidence for some of these conjectures. This is joint work with E. Rousseau and J. Wang

This seminar introduces the time-frequency analysis of Gabor frames and the sampling problem for the Bargmann-Fock space, which is still an open question in the multivariate situation. It is theoretically possible to reconstruct a signal (a square-integrable function) and its Fourier transform, as integral superpositions of time-frequency shifts operators. In practical applications, Gabor theory provides a discrete version of this reconstruction formula, up to a proper choice of a window function and a discrete subset of the phase-space. More precisely, these two objects have to define a frame of L 2 . For several reasons, Gaussian windows are the best suited for the analysis of signals. It turns out that Gaussian frames are characterized by discrete subsets of the phase-space that define sampling sequences for the Bargmann-Fock space, a problem which falls within the realm of complex analysis.

We put established Krylov subspace methods and Gauss quadrature rules to new use by generalizing the class of matrix function-based centrality measures from single-layered to multiplex networks. Our approach relies on the supra-adjacency matrix as the network representation, which has already been used to generalize eigenvector centrality to temporal and multiplex networks. We discuss the cases of unweighted and weighted as well as undirected and directed multiplex networks and present numerical studies on the convergence of the respective methods, which typically requires only few Krylov subspace iterations. The focus of the numerical experiments is put on urban public transport networks.

A projective algebraic variety is called Fano (after the Italian mathematician Gino Fano) if it has positive curvature. Fano varieties play a prominent rôle in algebraic geometry for many reasons. Recently there has been fundamental work on constructing moduli spaces of (certain) Fano varieties. The aim of my talk is to show how polytopes and combinatorics can help in proving that moduli spaces of Fano varieties are, in general, quite singular. My talk is based on joint work with Anne-Sophie Kaloghiros. The first 45 minutes of my talk do not require any knowledge in algebraic geometry.

In 1906 F. Hartogs introduced the domain which is now known as the Hartogs triangle. Such domain is a peculiar non-smooth pseudoconvex domain which often happens to be a good test domain for several conjectures. L. Edholm and J. McNeal recently introduced (2016) some generalizations of the Hartogs triangle in order to study the mapping properties of their associated Bergman projection. Since then several mathematicians studied such generalized Hartogs triangles and this investigation led to new and deep results in more general settings as well. In this expository talk I will review and describe some of these results pointing out possible directions for future research.

Optimization problems subject to PDE constraints form a mathematical tool that can be applied to a wide range of scientific processes, including fluid flow control, medical imaging, biological and chemical processes, and many others. These problems involve minimizing some function arising from a physical objective, while obeying a system of PDEs which describe the process. It is necessary to obtain accurate solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems. In this talk we consider Krylov subspace methods to solve such systems, accelerated by fast and robust preconditioning strategies. A key consideration is which time-stepping scheme to apply — much work to date has focused on the backward Euler scheme, as this method is stable and the resulting systems are amenable to existing preconditioners, however this leads to linear systems of even larger dimension than those obtained when using other (higher-order) methods. We will summarise some recent advances in addressing this challenge, including a new preconditioner for the more difficult linear systems obtained from a Crank-Nicolson discretization, and a Newton-Krylov method for nonlinear PDE-constrained optimization. At the end of the talk we plan to discuss some recent developments in the preconditioning of multiple saddle-point systems, specifically positive definite preconditioners which may be applied within MINRES, which may find considerable utility for solving optimization problems as well as other applications. This talk is based on work with Stefan Güttel (University of Manchester), Santolo Leveque (University of Edinburgh), and Andreas Potschka (TU Clausthal).

Le classi di Jordan sono state introdotte da Borho e Kraft nel loro studio delle sheet per algebre di Lie semisemplici. Sono le classi di equivalenza di elementi in un'algebra di Lie che hanno stessa decomposizione di Jordan, o, equivalentemente di elementi che hanno stabilizzatori (per l'azione aggiunta) coniugati tra loro. Sono localmente chiuse, irriducibili, lisce, e le loro chiusure danno luogo ad una stratificazione finita. La stessa costruzione può essere adattata per definire le classi di Jordan in gruppi algebrici riduttivi: la stratificazione che ne risulta compare nello studio di Lusztig dei fasci carattere. In collaborazione con Ambrosio ed Esposito abbiamo osservato che localmente le chiusure di classi di Jordan nel gruppo si comportano come chiusure di classi di Jordan in un'opportuna algebra di Lie. Un analogo di classe di Jordan per algebre di Lie Z_2-graduate è stato introdotto da Tauvel e Yu e le chiusure sono state studiate da Bulois ed Hivert: si perdono alcune delle caratteristiche dei casi precedenti ma il quadro complessivo è ancora chiaro. Motivato dallo studio della modalità per azioni di gruppi, Popov ha recentemente introdotto le classi di Jordan anche per algebre di Lie ciclicamente graduate. In collaborazione con Esposito e Santi abbiamo fornito una descrizione geometrica locale delle loro chiusure, mostrando in particolare che anche in questo caso la chiusura delle classi di Jordan è un'unione di classi. Con una serie di esempi mostreremo affinità e divergenze tra i vari contesti e le situazioni nelle quali la partizione in classi di Jordan ha un ruolo importante.

The tensor rank decomposition or canonical polyadic decomposition (CPD) is a generalization of a low-rank matrix factorization from matrices to higher-order tensors. In many applications, multi-dimensional data can be meaningfully approximated by a low-rank CPD. In this talk, I will describe a Riemannian optimization method for approximating a tensor by a low-rank CPD. This is a type of optimization method in which the domain is a smooth manifold, i.e. a curved geometric object. The presented method achieved up to two orders of magnitude improvements in execution time for challenging small-scale dense tensors when compared to state-of-the-art nonlinear least squares solvers.

In the first part, we present classical results about hyperplane arrangements and matroids: starting from the definitions, we present the construction by De Concini and Procesi of wonderful models. We discuss the cohomology of the wonderful model and of the complement of the arrangement. We also sketch the proof by Huh, Adiprasito, Katz of log-concavity of coefficients of the characteristic polynomial. In the second part we introduce subspace arrangements and polymatroids. We provide a generalization of the Goresky, MacPherson formula and we discuss the Hodge package (i.e. Poincaré duality, Hard Lefschetz and Hodge Riemann bilinear relations) for the Chow ring of a polymatroid. This is a joint work with Gian Marco Pezzoli.

We show a link between weighted Hilbert-Schmidt norms of Wick operators on Bargmann space and $L^2$-norm of Wick symbols with respect to a class of measures on the complex phase space. As an application, we derive the flow of discrete NLS equations by the mean field asymptotics of a many body quantum model for $N$ interacting particles as $N$ becomes large.

There is a fruitful analogy between Lie group actions and the dynamics of SL(2,R) on the moduli space of abelian differentials. Along the lines of this dictionary, the horocycle flow corresponds to a unipotent flow on a homogeneous space, for which Ratner’s theory is available. I will report on recent progress regarding the dynamics of the horocycle flow in the moduli space of abelian differentials.

Abstract: Compact and Normal Operators are two of the best understood classes of operators in terms of existence of invariant subspaces. Nevertheless, it is still open the Halmos Problem: Does every compact perturbation of a normal operator have a non-trivial invariant subspace? In this talk, we will motivate the interest of this particular problem and will focus on the 'baby' case: rank-one perturbations of diagonal operators. We will also introduce some basic aspects from Local Spectral Theory and will show the relation of this theory with invariant subspaces, that will play a prominent role on > the study of these particular operators.

Very few techniques are available for performing calculations in enumerative geometry. One of these is localisation. We will present different examples, flavours and refinements of the localisation formula, including applications to Donaldson-Thomas invariants.

The Invariant Subspace Problem asks if every bounded, linear operator on a complex Hilbert space has a non-trivial invariant subspace. In spite of its simple statement, this is one of the most famous unsolved problems on Operator Theory. We will present and skecth some of the most important techniques that have been developed to try to answer this question. In particular, we will discuss methods to provide non-trivial invariant subspaces for some classes of operators (Compact and Normal Operators) and methods to reduce the problem to the study of lattices of non-trivial invariant subspaces for concrete operators defined on spaces of analytic functions.

We introduce some types of fractional derivatives in one real variable, and try to solve some analogs of ordinary differential equations, in the form \[ D^\alpha u(t) = Au(t) + f(t), \quad t \in [0, T], \] where $A$ is a square $N \times N-$matrix, $f : [0, T] \to \C^N$, and prescribed proper initial conditions are given. Finally, we try to explain how to extend these elementary results to the important case that $A$ is a linear, not necessarily continuous, operator in a Banach space $X$.

Dimostriamo formule di media, di superficie e di volume, per soluzioni classiche di equazioni paraboliche in forma di divergenza sotto ipotesi naturali sulla regolarità dei coefficienti. La dimostrazione si basa sulle proprietà usuali della soluzione fondamentale delle equazioni paraboliche, su un teorema di divergenza generalizzato e su un preciso risultato dovuto a Dubovickii sulla regolarità degli insiemi di livello delle funzioni C^1. Discuteremo infine la generalizzazione di queste formule di media al contesto degli operatori subellittici nei gruppi di Carnot. I risultati di questo seminario sono stati ottenuti in collaborazione con Diego Pallara ed Emanuele Malagoli.

The Hardy space and signal processing

The three-dimensional reconstruction of an object is an interesting topic with many applications in different fields and has attracted several researchers. The applications range goes from the biomedical 3D reconstruction of human tissues to the approximation of the surface of astronomical objects, from archeology for the digitization of artistic works to the recent development of 3D printing. The first being interested in this problem were some opticians in the Fifties-Sixties. Afterwards, B.K.P. Horn first formulated the Shape-from-Shading (SfS) problem for a single gray-level image of the object. The goal was to get the 3D surface represented in the input image solving a partial differential equation or a variational problem. This problem gave rise to an expansion in the field of mathematics and some researchers tried to prove the well-posedness in the framework of weak solutions. The first works of Lions, Rouy and Tourin in the early 90s inserted the SfS problem in the context of the viscosity solutions frameworks, hence in a much more theoretical area. In this seminar I will start dealing with the orthographic SfS problem with Lambertian reflectance model, the classical and simplest setup for this ill-posed problem that can be modeled by first order Hamilton-Jacobi equations. During the seminar I will briefly introduce some notions of Hamilton-Jacobi equations, viscosity solutions and other ingredients necessary to understand the problem in a general setting. I will continue exploring some non-Lambertian reflectance models and we will see how it is possible to derive a well-posed problem adding information in a natural way. Finally, I will talk about the more recent Shape-from-Polarization problem and the advantages of it with respect to the SfS.

We study the size of the Dobinski set, a subset of the unit interval defined in terms of dyadic approximation. This can be done by using two distinct approaches: first, we give an analogous of Jarnik’s Theorem in Diophantine approximation that fits our dyadic setting, and we use such a result ( together with some reference to quasi-independence) to prove that the Dobinski set has logarithmic Hausdorff dimension equal to 1, with associated logarithmic Hausdorff measure equal to infinity. We will also see how the same result can be proven by using technical estimates on the Hausdorff dimension for willow Cantor sets constructed inside the Dobinski set. This is a joint work with José Luís Fernández and Maria Jose González.

In this talk, we introduce higher-order expansions in the mixing local limit theorems for Birkhoff sums. We will discuss the general results under technical assumptions, and illustrate them by different examples (e.g. subshifts of finite type, Young towers, Sinai billiards, and random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting. This is joint work with Françoise Pène.

In this talk, we shall exhibit a mini-max characterization of the second eigenvalue of the p-Laplacian operator on p-quasi-open sets, using a construction based on minimizing movements on non-linear gradient flows. The following outline shall be presented: the notion of non-linear eigenvalues and their properties, the statement of the characterization, the notion of Quasi-open sets, and a sketch of the proof of the theorem. This is based on a joint work with Nicola Fusco and Yi Zhang.

This talk will present some of the results obtained in the paper "The c-map as a functor on certain variations of hodge structure" by Mantegazza and Saha. In particular we will see how projective special Kähler manifolds can be interpreted in terms of variations of polarised Hodge structure, and how the latter can be used to give a description of the c-map that is manifestly functorial.

The talk will introduce the main concepts and tools of Optimal Transport between probability measures and its recent extensions to the unbalanced case, involving entropic regularizations. A few applications will also be discussed.

Double rotations are the simplest subclass of interval translation mappings. A double rotation is of finite type if its attractor is an interval and of infinite type if it is a Cantor set. It is easy to see that the restriction of a double rotation of finite type to its attractor is simply a rotation. It is known due to Suzuki – Ito – Aihara and Bruin – Clark that double rotations of infinite type are defined by a subset of zero measure in the parameter set. We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization, we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic. In my talk I plan to outline this proof that is based on the recent result by Fougeron for simplicial systems. I also hope to discuss briefly some challenging open questions and further research plans related to double rotations. The talk is based on joint work with Charles Fougeron, Pascal Hubert and Sasha Skripchenko.

We prove that skew systems with a sufficiently expanding base have “approximate” statistical properties similar to random ergodic Markov chains. For example, they exhibit approximate exponential decay of correlations, meaning that the exponential rate is observed modulo a controlled error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. The error in the approximation is shown to go to zero when the expansion of the base tends to infinity.

I will start the talk by recalling the notion of gradient flow in its easiest occurrence: the evolution equation x'(t)=-grad F(x(t)) in the Euclidean space. In particular, the focus will be on the implicit Euler scheme as a sequence of iterated minimization problems. I will then move to a more involved setting, where the point x is replaced by a probability density ρ evolving in the space of probabilities endowed with the so-called Wasserstein distance, induced by optimal transport. For suitable choices of the functional F one can recover linear diffusion PDEs (heat and Fokker-Planck equations) as well as non-linear ones (porous medium, fast diffusion, models for crowd motion). The iterated minimization scheme is called in this case JKO scheme (from Jordan-Kinderlehrer-Otto). After explaining why this scheme heuristically provide the desired equation at the limit, I will show how its optimality conditions can be exploited to prove estimates on its solutions, in particular BV, Sobolev and Lipschitz bounds. Lipschitz estimates can also be interpreted as bounds on the maximal displacement of each particle in the optimal transport map, and have a numerical interest, which I will discuss in two examples, where a potential drift is coupled either with linear diffusion or with a pressure effect due to density constrained in crowd motion.

I numeri reali si differenziano per le loro caratteristiche diofantee, che riflettono diverse proprietà geometrico-dinamiche delle orbite sotto l'azione di PSL(2,Z). La teoria analitica dei numeri studia le proprietà metriche, come misura o dimensione di Hausdorff, dell'insieme dei numeri reali di un certo tipo diofanteo. In particolare, l'insieme dei numeri male approssimabili ha misura zero e dimensione 1 nella retta. Inoltre esiste un' esaustione naturale in sottoinsiemi Bad(c), la cui dimensione converge ad 1 quando c va a zero. D. Hensley ha ottenuto l'asintotico al primo ordine in c della dimensione di Bad(c), attraverso un'analoga stima per l'insieme dei numeri reali la cui frazione continua ammette coefficienti parziali uniformemente limitati. La dimostrazione di questo risultato illustra come la frazione continua permette di introdurre strumenti dinamici, quali l'operatore di trasferimento ed il formalismo termodinamico, che forniscono informazioni metriche molto più precise che un'analisi puramente analitico-geometrica. Tale approccio si estende a diversi contesti geometrico-dinamici. In particolare vedremo una versione generalizzata dell'asintotico di Hensely, in cui la nozione di numeri male approssimabili è riferita non più all'azione di PSL(2,Z) ma a quella di un lattice non-uniforme in PSL(2,R).

After a very brief review of basics on elliptic curves and their families, we shall consider "sections" of such  families, and especially their "torsion values".  For instance, what can be said of the complex numbers b for which (2, \sqrt{2(2-b)}) is torsion on the Legendre curve y^2=x(x-1)(x-b)? In particular, we shall recall results of "Manin-Mumford type" and focus to illustrate some applications to elliptical billiards. Finally, if time allows we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.)

In this talk, I will discuss the development of semi-classical analysis for sub-elliptic operators such as sub-Laplacians. For an elliptic operator, this is well understood as the tools and methods to study e.g. quantum ergodicity or Schrödinger equations have become well established over the past fifty years. They rely on the pseudo-differential theory, and in the elliptic case the space of principal symbols is commutative. The aim of this talk is to present my approach to define similar tools for sub-Laplacians, leading to more non-commutative concepts.

Some new ideas in Calculus of Variation and Geometric Measure Theory allowed, in the last decade, to revisit from a precise mathematical point of view some physical models. Instances of such models are the Lord Rayleigh model of charged liquid drops in electrowetting, the liquid drop model by Gamow to describe nuclear fissions, the Hartree equations in atomic physics. In the seminar I will give a brief overview on such results. Later I will focus on recent results about some of those topics. The topic of the talk is partially based upon works in collaboration with M. Goldman, D. Mazzoleni, C.B. Muratov and M. Novaga.

I will introduce the nonlocal curvature flows, discussing existence, uniqueness and stability of solutions. In the particular case of the fractional mean curvature flow, I will also describe the long time behaviour of graphical solutions and some issues related to the formation of singularities.

In this talk I will consider the following question: if one picks a piecewise affine map of the circle “at random”, what dynamical behaviour are we likely to observe? The case of standard circle diffeomorphisms has been studied by Herman in the 80s; in this emblematic case the problem can be reduced to theorems close to KAM theory. For the piecewise affine case, we put forward a geometric approach, inspired by methods from both Teichmüller theory and hyperbolic geometry.

Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past) or by one-dimensional space (which is the case, for example, for one-dimensional Markov fields), where conditioning is two-sided (on the right and on the left). I will discuss some examples, in particular generalising this distinction to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided) produce the same objects and when they are different. We show moreover the role one-dimensional entropic repulsion plays in this setting. Based on joint work with R. Bissacot, E. Endo and A. Le Ny, and S. Shlosman

There is currently a great deal of interest in the scientific community in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease. I will present a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations for the two proteins. In the model neurons are treated as a continuous medium and structured by their degree of mal- functioning. 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. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. I will explain in detail the structure of the model and give a hint of the main results obtained and the techniques used for the purpose. Eventually I will also show the output of some numerical simulations, of some significance even if performed in an over-simplified 2D geometry.

I will discuss a theme which, at various levels of complexity, is pervasive in algebra and geometry: The attempt to parameterize algebraic or geometric objects naturally leads to the problem of constructing quotients, almost always by the action of a group. I will start by discussing elementary problems such as parameterizing finite subsets of points in the plane or conjugacy classes of matrices, showing how they lead to the so called geometric invariant theory, which also has a more differential geometric counterpart, called symplectic reduction. In the second half of the talk I will discuss the parameterization of representations of the fundamental group of a surface, and the closely related notion of moduli spaces of vector or Higgs bundles on a Riemann surface, still from the point of view of quotient constructions.

We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then — by a classical theorem by Aaronson — there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum adding a suitable number of terms depending on the point of evaluation. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. Joint work with Claudio Bonanno.

We will discuss the maximal subspace of strong continuity of a semigroup of composition operators acting on the space of analytic functions of bounded mean oscillation in the unit disc. The minimality of this space is related to a well known theorem of Sarason about the space of analytic functions of vanishing mean oscillation. In the case of elliptic semigroups we give a complete characterization in terms of the Koenigs function of the semigroups that can replace rotations in Sarason's Theorem. This improves previous results of Blasco et al. Similar results are also obtained for the Bloch space. This is a joint work with V. Daskalogiannis.

The Kotelnikov theorem recovers a Paley-Wiener function from its restriction onto an arithmetic progression. A Paley-Wiener function can also be recovered from its restriction onto a realization of the sine-process with one particle removed. If no particles are removed, then the possibility of such interpolation for the sine-process is due to Ghosh, for general determinantal point processes governed by orthogonal projections, to Qiu, Shamov and the speaker. If two particles are removed, then there exists a nonzero Paley-Wiener function vanishing at all the remaining particles. How explicitly to interpolate a function belonging to Hilbert space that admits a reproducing kernel, given the restriction of our function onto a realization of the determinantal pont process governed by the kernel? For the sine-process, the Ginibre process, the determinantal point process with the Bessel kernel and the determinantal point process with the Airy kernel, A.A. Borichev, A.V. Klimenko and the speaker proved that if the function decays as a sufficiently high negative power of the distance to the origin, then the answer is given by the Lagrange interpolation formula.

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, In the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago. We end up this talk addressing an apparently different question: What kind of physics might be non-computational? Using the former universality result, we can establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. But, in view of the increase of dimension yielded by our proof, the question is: can this be done in dimension 3? We will prove the existence of Turing complete fluid flows on a 3-dimensional geometric domain. Our novel strategy uses the computational power of symbolic dynamics and the contact mirror again. This talk is based on joint work with Robert Cardona and Fran Presas (arXiv:1911.01963 and arXiv:2012.12828).

We consider compact elliptic four-folds whose fiber ceases to be flat over Riemann surfaces of genus g in the base. We show that those contributions generically lead to non-trivial threeform cohomology proportional to g and the number of non-flat fiber components. These non-flat components can be viewed as compactifications of non-flat three-folds where they correspond to superconformal matter theories. Moreover we show, that one can perform conifold transitions that remove those non-flat fibers, corresponding to non-flat fibers in codimension three and second to birational base changes. The former phase is interpreted as a non-perturbative gauge invariant fourpoint coupling and the second one is closer to a classical 4D Coulomb branch.

When one considers manifolds with boundary, billiard dynamics are the natural analogue of standard geodesic dynamics. Namely, instead of having geodesics escape at the boundary, we force them back into the manifold using the reflection law. In other dynamical settings, similar constructions are possible: In 2006, B. Khesin and S. Tabachnikov initiated the study of billiards in the semiriemannian setting, studying the integrability of various tables. In recent years we have also seen the appearance of several billiard setups of symplectic nature. In this talk I will discuss recent work with L. Dahinden in which we look at billiards in subriemannian geometry. I will sketch how the reflection law arises naturally both from the control-theoretical and symplectic perspectives, how the reflection is problematic at tangency points between the distribution and the boundary of the table, and I will introduce some concrete examples. My ultimate goal will be to pose several intriguing open questions.

The Siegel domain plays in several complex variables the role of the upper half plane in the one-dimensional theory. One of its advantages over the unit ball is that the boundary behavior of the Bergman metric is rather transparent in Siegel coordinates; another one is that its boundary has a (noncommutative) group structure and (noncommutative) Fourier analysis can be used; another one is that the derivative with respect to the "vertical" direction encodes much information concerning classical holomorphic spaces. In these seminars we will give an overview of the rich toolbox available when studying function theory over such domain. The first one will be devoted to the Bergman metric and the automorphism group.

Lo scopo del seminario è quello di esporre ad un vasto pubblico la recente dimostrazione della log-concavità di certi polinomi. Il polinomio cromatico di un grafo conta il numero di colorazioni possibili di un grafo. Negli anni '70 è stato congetturato che i suoi coefficienti $\omega_i$ formino una sequenza log-concava, cioè \[ \omega_i^2\geq \omega_{i-1}\omega_{i+1}.\] L'enunciato della congettura si può dare più in generale per i coefficienti del polinomio caratteristico di un matroide. Queste congetture sono state dimostrate rispettivamente nel 2012 e nel 2018. Le tecniche usate sono sorprendenti e proverò a darne un'idea: costruirò, tramite blow up, una varietà proiettiva e ne studierò l'anello di Chow (che coincide con la coomologia). Infine dal teorema di Hodge-Riemann segue banalmente la disuguaglianza cercata. Nel caso di matroidi la corrispettiva varietà non esiste, ma si può comunque definire un anello con le proprietà desiderate e dimostrare la log-concavità.

Gromov hyperbolic spaces are a generalization of negative curved spaces in the general context of metric spaces. We present different (but equivalent) definitions and main results (Shadowing lemma, Gromov compactification and homeomorphic extension of quasi-isometries). Finally, we apply the general theory to complex domains endowed with the Kobayashi distance.

Aguiar and Ardila defined a Hopf monoidal structure on the collection of generalized permutahedra of all dimensions; and from it, constructed a Hopf algebra on the same polytopes, which is isomorphic to the Hopf algebra of symmetric functions, Sym. This endows each element of Sym with a formal sum of permutahedra, so that we can think of symmetric functions as members of McMullen’s polytope algebra. In this talk, we give geometric models for the Schur and power sum symmetric functions, when regarded as elements of the aforesaid polytope algebra. This is accomplished through a combinatorial rule for the former ones and in the way of an explicit description for the latter ones. We also characterize when the resulting geometric objects correspond to polytopes with missing faces. (Joint work with Carolina Benedetti and Mario Sanchez)

Matroids are combinatorial structures that admit several "cryptomorphic" definitions. It is possible to define matroids as a certain kind of polytopes with vertices with 0/1-coordinates. Also, for every lattice polytope, the function counting the number of integer points inside of every integral dilation is known to be a polynomial, named after Eugene Ehrhart. In this seminar we will talk about the Ehrhart polynomials of matroid polytopes. We will discuss some open problems and recent results on the area, using just minimal prerequisites.

The symbolic action of a substitution on the binary expansion generates naturally an interval map. In case of erasing substitutions, the map is typically Baire-1 and not Darboux. As a model case, we consider what is arguably the simplest erasing substitution, mapping 0s to the empty word and 1s to 0 or 1 depending on the parity. The corresponding interval map is shown to have fractal properties (bounds are given on the Hausdorff dimension of the fibers) and to display rich dynamical behavior, including Devaney chaos, uniform distributional chaos of type 1, infinite topological entropy as well as the presence of cycles attracting in a finite time every rational.

We give a introduction, assuming no prerequisite, to the notion of pseudoconvexity in several complex variables.

In this talk we present some recent results concerning the regularity of the unique weak solution vanishing at infinity of the prescribed mean curvature equation in the Lorentz-Minkowski space for spacelike hypersurfaces, when the mean curvature belongs to $L^p(R^N)$, with $p>N$. This equation is also known as the ``Born-Infeld'' equation, as it comes from the nonlinear model of electromagnetism introduced by M. Born and L. Infeld, but it also plays a crucial role in Relativity. In the first part of the talk we will show a new gradient estimate for smooth solutions of the prescribed mean curvature equation and prove that, under our assumptions, the unique minimizer of the Born-Infeld energy, which is a priori only Lipschitz continuous, is actually a strictly spacelike weak solution of class $W^{2,p}$. In the second part the we will discuss some other related results concerning the existence of spacelike radial graphs of prescribed mean curvature and some open problems. These results are collected in a series of joint works with Prof. D. Bonheure (Université Libre de Bruxelles).

In this talk I will present some results about the GIT stability of Halphen pencils of index two under the action of SL(3). These are pencils of plane curves of degree six having nine (possibly infinitely near) base points of multiplicity two. Inspired by the work of Miranda on pencils of plane cubics, I will explain how we can explore the geometry of the associated rational elliptic surfaces.

We will present a series of results regarding the behaviour of solutions to boundary value problems driven by non-integer powers of the Laplacian operator. Special attention will be paid to the failure of maximum principles and its consequences.

A characterization of the Carleson measures for the Dirichlet space is given.

Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some recent work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective.

McMullen used the fundamental operation of Minkowski sum to construct the polytope algebra of real vector space. In this talk, I will consider the subalgebra generated by deformations of a fixed zonotope and endow it with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. In the particular case of Coxeter arrangements of type A and B, we find striking relations between the corresponding module structure and certain statistics on permutations and signed permutations, respectively. I will explain how these statistics give information on families of polytopes that generate all (type B) generalized permutahedra as signed Minkowski sums.

I will present some decay of correlations results on skew products which are locally accessible. The results rely on the study of a twisted transfer operator and could be generalized to many other situations. I will also present numerical counterparts to such results.

The Carleson measures for the Dirichlet space: their use and their characterizations. Codice Zoom: 837 9129 8028

The goal of this very informal, online workshop is bringing together some researchers who use, or who are interested in, Wolff's inequality and potentials in different sub-areas of mathematics, to exchange their viewpoints on the topic. id Zoom: 842 2115 6698 Nicola Arcozzi (UniBo): Wolff's inequality and dyadicization 2.00pm Cristiana De Filippis (U. Torino): Calderón-Zygmund estimates and non-uniformly elliptic operators 2.45PM Pavel Mozolyako (St. Petersburg State University): A probabilistic proof of Wolff's inequality 3.30pm Nikolaos Chalmoukis (UniBo): TBA 4.15pm

Numerical first-order methods are the most suitable choice for solving large-scale nonlinear optimization problems which model many real life applications. Among these approaches, gradient methods have widely proved their effectiveness in solving challenging unconstrained and constrained problems arising in machine learning, compressive sensing, image processing and other areas. These methods became extremely popular since the work by Barzilai and Borwein  (BB) (1988), which showed how a suitable choice of the steplength can significantly accelerate the classical Steepest Descent method. It is well-known that the performance of gradient methods based on the BB steplength does not depend on the decrease of the objective function at each iteration but relies on the relationship between the steplengths used and the eigenvalues of the average Hessian matrix; hence BB based methods are also denoted as Spectral Gradient methods. The first part of this seminar will be devoted to a review of spectral gradient methods for unconstrained optimization while the second part will focus on recent advances on the extension of these methods to the solution of large nonlinear systems of equations, the so-called Spectral Residual methods. These methods are derivative-free, low-cost per iteration and are particularly suitable when the Jacobian matrix of the residual function is not available analytically or its computation is not relatively easy. In this framework, numerical experience will be presented on sequences of nonlinear systems arising from rolling contact models which play a central role in many important applications, such as rolling bearings and wheel-rail interaction.

The Dirichlet space is the only conformally invariant, holomorphic Hilbert space on the unit disc. In this elementary introduction (no prerequisites) we see it from the viewpoint of hyperbolic geometry, Reproducing Kernel Hilbert Spaces, fractional derivatives, and dyadic analysis. (In presenza: Seminario VIII piano)

Positroid varieties are intersections of cyclically rotated Schubert varieties in the Grassmannian, introduced in my work with Knutson and Speyer. I will discuss some aspects of these very nice spaces, including a recent computation of the cohomology of open positroid varieties in joint work with Galashin.

Kogan gave a combinatorial rule (rediscovered by Lenart and by Assaf) for computing the product of a Schur polynomial S_lambda(x_1..x_k) by a Schubert polynomial S_pi, subject to the condition that pi's last descent is at or before k. Yong and I gave a streamlined proof, using Lascoux' transition formula for Schubert polynomials. As the Lee-Lam-Shimozono bumpless pipe dream formula for Schuberts is tightly compatible with transition (an observation of Weigand's), using them we can give an even tighter formula for Kogan's coefficients.

In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C^1-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. The main ingredients of the proof are: (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of 4-periodic orbits; and (3) the integral-geometry approach initiated by the author for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on joint work with Andrey E. Mironov (Novosibirsk).

Inspired by the problem of modeling the so called anthropogenic forcing in climatology, e.g. the effects of the emissions of greenhouse gases in the atmosphere, we introduce a novel type of random perturbation for the classical Lorenz flow and prove its stochastic stability. The perturbation acts on the system in an impulsive way, hence is not of diffusive type. Namely, given a cross-section M for the unperturbed flow, each time the trajectory of the system crosses M the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the unperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes. Joint work with Sandro Vaienti.

We prove for the N-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level h>0 of the motion can also be chosen arbitrarily. Our approach is based on the construction of a global viscosity solutions for the Hamilton-Jacobi equation H(x,du(x))=h. Our hyperbolic motion is in fact a calibrating curve of the viscosity solution. The presented results can also be viewed as a new application of Marchal’s theorem, whose main use in recent literature has been to prove the existence of periodic orbits. Joint work with Ezequiel Maderna.