Archivio 2025

For a countable homogeneous structure M, a natural question in representation theory is the classification of unitary representations of the Polish, non-archimedean group Aut(M). This question was completely answered by Tsankov for ω-categorical structures. In joint work with R. Barritault and C. Jahel, we generalize this result and go beyond ω-categoricity by addressing structures such as the integral/rational Urysohn space and the integral/rational universal diversity. We will explain how the notion of dissociation, a structural property of unitary representation for Polish non-archimedean groups, is used in our work. No background knowledge of representation theory will be assumed for this talk.

The aim of this work is to determine the optimal cyber-security investment strategy for an entity subject to cyber-attacks. Inspired by the Gordon-Loeb model, we assume that the success rate of cyber-attacks depends on the vulnerability of the security system under threat, which can be reduced investing in security measures. We introduce a dynamic version of the Gordon-Loeb setting, by exploiting Hawkes stochastic processes to model the arrival of attacks. This stochastic framework is crucial to rapidly react to the random changes which characterize cyber-risk. The problem is framed as a Markovian 2-dimensional stochastic control problem with jumps and it is addressed using dynamic programming techniques. The optimal value is characterized by a partial integro-differential equation, which is solved numerically. The corresponding optimal strategy is, hence, explicitly obtained by differentiating the optimal value function.

In this talk, we will show how representations of a (Dynkin) quiver allow to construct cluster variables for the associated Fomin-Zelevinsky cluster algebra. We will start from the basics on quiver representations, notably Gabriel's theorem establishing a bijection between positive roots and indecomposable representations. By combining this bijection with Fomin-Zelevinsky's between positive roots and non-initial cluster variables, we will obtain a map associating a non-initial cluster variable with each indecomposable representation. The starting point of additive categorification is an explicit formula for this map due to Caldero-Chapoton. It involves Euler characteristics of varieties of subrepresentations and is typical of links between cluster algebras and algebraic geometry.

Since their invention by Fomin-Zelevinsky in 2002, cluster algebras have shown up in an ever growing array of subjects in mathematics (and in physics). In this talk, we will approach their theory starting from elementary examples. More precisely, we will see how the remarkable integrality properties of the Coxeter-Conway friezes and the Somos sequence find a beautiful unification and generalization in Fomin-Zelevinsky's definition of cluster variables and their Laurent phenomenon theorem. Motivated by the periodicity of Coxeter-Conway friezes, we will conclude with a general periodicity theorem (Keller 2013), whose proof is based on the interaction between discrete dynamical systems and quiver representations through the combinatorial framework of cluster algebras.

Given an abelian category A its derived category D(A) admits a natural t-structure whose heart is A. Moreover by the Auslander’s Formula A is equivalent to the quotient category of coherent functors by the Serre subcategory of effaceable functors. Given a quasi-abelian category E its derived category D(E) admits two canonical t-structures (left and right) whose hearts L and R are derived equivalent and their intersection in D(E) is E, moreover E is a tilting torsion class (rep. cotilting torsion-free class) in the right heart R (resp. L). We generalise the Auslander’s Formula to quasi-abelian categories and we extend this picture to its higher version introducing n-quasi-abelian categories. Given X a smooth algebraic variety of dimension n the category of locally free O_x-modules of finite rank is n-quasi-abelian.

The regularity of minimizers for the classical one-phase Bernoulli functional has been extensively studied following the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers has also been investigated. This relaxed notion of minimality arises naturally in various contexts, such as variational problems with constraints, and its flexibility allows for addressing a broader range of questions. We present some results concerning the regularity of almost minimizers for a one-phase Bernoulli-type functional in Carnot groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. We outline the main challenges and techniques employed to establish local Lipschitz regularity for almost minimizers, emphasizing the role of intrinsic gradient estimates. Additionally, we discuss certain generalizations to nonlinear counterparts. Some of the results presented stem from joint works with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).

With every proper metric space are associated C*-algebras of special geometric importance (Roe algebras), originally defined for index-theoretic considerations. It is a simple observation that metric spaces that are coarsely equivalent give rise to isomorphic C*-algebras, and recently we could prove that the converse also holds. Namely, a C*-rigidity Theorem shows that spaces with isomorphic Roe algebras must be coarsely equivalent. This establishes a strong connection between the categories of coarse spaces and C*-algebras. In this introductory talk I aim to properly introduce the C*-rigidity question and illustrate the roadmap of the proof of rigidity.

Understanding quantum magnetism in two-dimensional systems represents a lively branch in modern condensed-matter physics. In the presence of competing super-exchange couplings, magnetic order is frustrated and can be suppressed down to zero temperature. Still, capturing the correct nature of the exact ground state is a highly complicated task, since energy gaps in the spectrum may be very small and states with different physical properties may have competing energies. Here, we introduce a variational Ansatz for two-dimensional frustrated magnets by leveraging the power of representation learning. The key idea is to use a particular deep neural network with real-valued parameters, a so-called Transformer, to map physical spin configurations into a high-dimensional feature space. Within this abstract space, the determination of the ground-state properties is simplified and requires only a shallow output layer with complex-valued parameters. We illustrate the efficacy of this variational Ansatz by studying the ground-state phase diagram of the Shastry-Sutherland model, which captures the low-temperature behavior of SrCu2(BO3)2 with its intriguing properties. With highly accurate numerical simulations, we provide strong evidence for the stabilization of a spin-liquid between the plaquette and antiferromagnetic phases. In addition, a direct calculation of the triplet excitation at the Γ point provides compelling evidence for a gapless spin liquid. Our findings underscore the potential of Neural-Network Quantum States as a valuable tool for probing uncharted phases of matter, and open up new possibilities for establishing the properties of many-body systems.

TBA

We present a duality between compact \( T_1 \)-spaces and a class of distributive lattices (subfit, compact, and complete), which captures key aspects of both Stone duality and \(\Omega\)-point duality in particular instances. We then show how this duality extends to a contravariant adjunction between \( T_1 \)-spaces and bounded distributive lattices. This adjunction gives rise to a canonical compactification---the \emph{Wallman compactification}---for \( T_1 \) spaces, such that any \textit{strongly continuous} map from a \( T_1 \) space \( X \) into a compact \( T_1 \) space factors uniquely through the Wallman compactification of \( X \). This is joint work with Matteo Viale and Mai Gehrke.

We are interested in Gizatullin’s problem which consists of the following question: Given a smooth quartic surface S ⊂ P3, which automorphisms of S are induced by Cremona transformations of P3? Cremona transformations of P3 can be written as a composition of a finite sequence of elementary maps. This is an algorithmic process called the Sarkisov Program. In this talk, we will solve Gizatullin’s problem when S ⊂ P3 has Picard number two by using the Sarkisov program. The results that will be presented are in collaboration with Ana Quedo, and with Carolina Araujo and Sokratis Zikas.

Feature learning - or the capacity of neural networks to adapt to the data during training - is often quoted as one of the fundamental reasons behind their unreasonable effectiveness. Yet, making mathematical sense of this seemingly clear intuition is still a largely open question. In this talk, I will discuss a simple setting where we can precisely characterise how features are learned by a two-layer neural network during the very first few steps of training, and how these features are essential for the network to efficiently generalise under limited availability of data.

This is joint work with Andreas Stavrou. For a compact orientable surface S of genus g with one boundary component and for an odd prime number p, we study the homology of the unordered configuration spaces C(S) := coprod_{n>=0} C_n(S) with coefficients in F_p. We describe H_*(C(S); F_p) as a bigraded module over the Pontryagin ring H_*(C(D); F_p), where D is a disc, and give a splitting as direct sum of certain well-behaved quotients of this ring. We also consider the action of the mapping class group Mod(S) on the homology, and identify the kernel of the action with the subgroup of Mod(S) generated by separating Dehn twists and p-th powers of Dehn twists. We compare with some partial results about the Mod(S)-action on ordered configuration spaces, obtained in joint work with Jeremy Miller, Jennifer Wilson, and Andreas Stavrou.

We will provide sharp weak type estimates for the Hardy-Littlewood maximal operator in the context of Gromov hyperbolic metric measure spaces, which satisfy a locally doubling condition and the measures of balls grows exponentially for large radii. This result generalizes previous results on symmetric spaces of non compact type and rank 1, Damek-Ricci spaces, and Riemannian manifolds of pinched negative curvature. This is a joint work with S. Meda and F. Santagati.

We discuss a class of pro-groups whose derived limits are relevant to the additivity of strong homology. These are indexed by the set of functions {}^\kappa \omega. The case of width \omega, that is, the one where \kappa = \omega, is linked to the additivity of strong homology on the class of locally compact separable metric spaces. While an equivalence was proved for \lim^1 between certain narrow and wider system, the analogous equivalence for higher limits has long been an open question. Here we present a negative answer to that question and, time permitting, some ideas toward a consistency result on simultaneous vanishing for all cardinals and all \lim^n. This is joint work with Jeffrey Bergfalk. 

Given a measured groupoid G, together with Filippo Sarti, we defined a cohomology theory which generalizes the measurable bounded cohomology of a locally compact group. In the particular case of a groupoid associated to a measure preserving action, our cohomology boils down to the usual bounded cohomology of the group with twisted coefficients. We will discuss the possible applications of this result to orbit equivalence. 

We give an introduction to Topological Data Analysis (TDA), focusing on Persistent Homology. This is a technique to extract information about the shape of data and summarize it as a collection of intervals, also called a barcode. If the data is given in the form of a nice enough function on a smooth manifold, there are connections to Morse theory, where the topology of the underlying manifold is related to the critical points of the function. We then move the focus to Morse-Smale vector fields, which are a class of vector fields with good structural properties, and present new approaches to extend the aforementioned methods to these objects. We do so by applying different algebraic methods such as parametrized chain complexes and spectral sequences.

The aim of Electrical Impedance Tomography (EIT) is to determine the electrical conductivity distribution inside a domain by applying currents and measuring voltages on its boundary. Mathematically, the EIT reconstruction task can be formulated as a non-linear inverse problem. The Bayesian inverse problems framework has been applied expensively to solutions of the EIT inverse problem, in particular in the cases when the unknown conductivity is believed to be blocky. In this talk, we demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data. Numerical tests on synthetic and real data illustrate the computational efficiency of the proposed algorithm.

We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The Receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the Receiver through a device designed by the Sender that generates the observation process. The commitment of the Sender is enforced. We develop this approach in the case where all dynamics are linear and the preferences of the Receiver are linear-quadratic. We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the Receiver’s value function. An extension to the case of persuasion of a mean field of interacting Receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve. This is a joint work with: R. Aïd (Paris Dauphine), O. Bonesini (LSE) and G. Callegaro (Padova).

An elementary Mori contraction from a smooth variety X is a morphism with connected fibres onto a normal variety which contracts a single extremal ray of K_X-negative curves. Thanks to a result by P. Ionescu and J. Wisniewsi, we know that the length of such a contraction (i.e. the minimal degree -K_X can have on contracted rational curves) is bounded from above. In a paper which dates back to 2013, A. Höring and C. Novelli studied elementary Mori contractions of maximal length, that is, elementary Mori contractions for which the upper bound is met. Their main result exhibits the structure of a projective bundle for the locus of positive-dimensional fibres up to a birational modification. In my talk, I will move to the submaximal case, in other words the case where the length equals its upper bound minus one, and focus on the divisorial case.

Given a telecommunication network represented by a directed graph, our problem is to route one single stream of packets on the IP network along a min-cost path with a constraint on the maximum delay that any packet may incur. From a mathematical point of view, this problem, known as Delay Constrained Routing (DCR), can be formulated as a Mixed-Integer Second-Order Cone Program (MISOCP), where one needs to simultaneously (and "optimally") compute paths and reserve resources along the paths of the network. The DCR problem presents an interesting mixture of combinatorial and continuous structures and naturally lends itself to decomposition methods. We will discuss formulations, algorithms and computational results on real/realistic network instances.

In the first part of the seminar, I will discuss the concept of Birational Geometry, introduce some birational iInvariants and the moduli spaces of varieties of general type, along with a few examples that I consider significant. In the second part, I will present some recent results, obtained in collaboration with S. Coughlan, Y. Hu, and T. Zhang, on certain moduli spaces of three-dimensional varieties of general type.

Peridynamics is a nonlocal version of continuum mechanics theory able to incorporate singularities since it does not take into account spatial partial derivatives. As a consequence, it assumes long-range interactions among material particles and is able to describe the formation and the evolution of fractures. The discretization of such nonlocal model requires the use of raffinate numerical tools for approximating the solutions to the model. Due to the presence of a convolution product in the definition of the nonlocal operator, we propose a spectral collocation method based on the implementation of Fourier and Chebyshev polynomials to discretize the model. The choice can benefit of the FFT algorithm and allow us to deal efficiently with the imposition of non-periodic boundary conditions by a volume penalization technique. We prove the convergence of such methods in the framework of fractional Sobolev space and discuss numerically the stability of the scheme. We also investigate the qualitative aspects of the convolution kernel and of the nonlocality parameters by solving an inverse peridynamic problem by using a Physics-Informed Neural Network activated by suitable Radial Basis functions. Additionally, we propose a virtual element approach to obtain the solution of a nonlocal diffusion problem. The main feature of the proposed technique is that we are able to construct a nonlocal counterpart for the divergence operator in order to obtain a weak formulation of the peridynamic model and exploit the analogies with the known results in the context of Galerkin approximation. We prove the convergence of the proposed method and provide several simulations to validate our results. References: [1] Lopez, L., Pellegrino, S. F. (2021). A spectral method with volume penalization for a nonlinear peridynamic model International Journal for Numerical Methods in Engineering 122(3): 707–725. https://doi.org/10.1002/nme.6555 [2] Lopez, L., Pellegrino, S. F. (2022). A space-time discretization of a nonlinear peridynamic model on a 2D lamina Computers and Mathematics with Applications 116: 161–175. https://doi.org/10.1016/j.camwa.2021.07.0041 [3] Lopez, L., Pellegrino, S. F. (2022). A non-periodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models International Journal for Numerical Methods in Engineering 123(20): 4859–4876. https://doi.org/10.1002/nme.7058 [4] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for an inverse problem in peridynamic models Engineering with Computers. https://doi.org/10.1007/s00366-024-01957-5 [5] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for learning the horizon size in bond-based peridynamic models Computer Methods in Applied Mechanics and Engineering. https://doi.org/10.1016/j.cma.2024.117727

We discuss the strong unique continuation property for linear differential operators of the form sum of squares of vector fields satisfying the Hörmander condition. We provide some negative and some positive results.

I will discuss the existence of a solution to the logarithmic Schrödinger equation in a bounded convex domain, with the property that log u is concave. Since the reaction is sign-changing and non-monotone, the classical techniques to attack the problem fail. We instead rely on a continuity argument for the approximating Lane-Emden problems based on the heuristic argument. We will discuss the optimality of the result, exhibiting for any α > 0 a solution such that u^a is not concave. This is a joint work with M. Squassina and M. Gallo.

We investigate the optimal self-protection problem, from the point of view of an insurance buyer, when the loss process is described by a Cox-shot-noise process and a Hawkes process with an exponential memory kernel. The insurance buyer chooses both the percentage of insured losses and the prevention effort. The latter term refers to actions aimed at reducing the frequency of the claim arrivals. The problem consists in maximizing the expected exponential utility of terminal wealth, in presence of a terminal reimbursement. We show that this problem can be formulated in terms of a suitable backward stochastic differential equation (BSDE), for which we prove existence and uniqueness of the solution. We extend in several directions the results obtained by Bensalem, Santibanez and Kazi-Tani [Finance Stoch. 2023] and compare our results with those presented therein. The talk is based on a joint paper with M. Brachetta, G. Callegaro and C. Sgarra.

Transmission problems describe physical phenomena where the behavior of a system changes across a fixed interface, resulting in different PDEs on each side. These PDEs are coupled through a transmission condition, typically of the Neumann type. In this talk, we will discuss some recent results concerning transmission problems governed by fully nonlinear operators which degenerate near the transmission interface. We obtain Holder differentiability of solutions up to the interface, with optimal exponent, which depends pointwise on the rate of degeneracy. Research activity supported by PRIN 2022 7HX33Z - CUP J53D23003610006, Pattern formation in nonlinear phenomena

The roto-translation group SE(2) has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined in this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this talk, I will enumerate a computational framework for optimal transportation over Lie groups, with a special focus on SE(2). I will describe several theoretical aspects such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. Finally, I will illustrate a Sinkhorn-like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report advancements with the experiments on 1) 2D shape/ image barycenters, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE(2). We observe that our framework of lifting images to SE(2) and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image and leads to meaningful interpolations compared to their counterparts on R^2. *Joint work with Daan Bon, Gijs Bellaard, Olga Mula, and Remco Duits from CASA – TU/e. Preprint: https://arxiv.org/abs/2402.15322 (to appear in SIAM Journal in Imaging Sciences 2025)

The classical Plateau problem asks which surface in three-dimensional space spans the least area among all the surfaces with boundary given by an assigned curve S. This problem has many variants and generalizations, along with (partial) answers, and has inspired numerous new ideas and techniques. In this talk, we will briefly introduce the problem in both its classical and modern contexts, and then we will focus on a specific vectorial (planar) type of the Plateau problem. - Given a curve S in the plane, we can ask which diffeomorphism T of the disk D maps the boundary of D to S and spans the least area, computed as the integral of the Jacobian of T, among competitors with the same boundary condition. For simply connected curves, the answer is provided by the Riemann map, and the minimal area achieved is the Lebesgue measure of the region enclosed by S. For more complex curves, possibly self-intersecting, new analysis is required. I will present a recent result in this sense, obtained in collaboration with Prof. Riccardo Scala from the University of Siena, where the value of the minimum area is computed with an explicit formula that depends on the topology of S.

Given a (nice) group G, we are interested in how fast the length of a group element can grow when we apply powers of a given outer automorphism of G. If the group G is free or the fundamental group of a closed surface, classical train-track techniques give a complete and precise picture. This can be extended to automorphisms of all negatively curved (a.k.a. Gromov hyperbolic) groups G, using Rips-Sela theory and the canonical JSJ decomposition. Very little seems to be known beyond this setting. We study this problem for a broad class of non-positively curved groups: "special" groups in the Haglund-Wise sense. In this setting, we prove that: (1) the top exponential growth rate of any automorphism is an algebraic integer; (2) if the automorphism is untwisted, then it admits only finitely many growth rates, and each of these is polynomial-times-exponential.

Accurately estimating landslides’ failure surface depth is essential for hazard prediction. However, most of the classical methods rely on overly simplistic assumptions [1]. In this work, we will present the landslide thickness estimation problem as an inverse problem Aw = b, obtained from discretization of the thickness equation [2]: ∂(hf vx)/∂x + ∂(hf vy)/∂y = − ∂ζ/∂t , (1) where the forward operator A contains information on the surface velocity (v_x, v_y), the right-hand side b corresponds to the surface elevation change ∂ζ/∂t, and w is the thickness hf . By employing a regularization approach, the inverse problem is reformulated as an optimization problem. In real-world scenarios, often no information on neither the noise type nor the noise level affecting data is available. In this context, the correct choice of the regularization parameter becomes a pressing issue. We propose a method to determine this parameter in a fully automatic way for the thickness inversion problem. Results obtained on both synthetic data generated by landslide simulation software and data measured from real-world landslides will be shown. [1] Jaboyedoff M., Carrea D., Derron M.H., Oppikofer T., Penna I.M., Rudaz B. (2020): A review of methods used to estimate initial landslide failure surface depths and volumes. Engineering Geology, 267, 105478 [2] Booth A. M. ; Lamb M. P. ; Avouac J.P. ; Delacourt C. (2013): Landslide velocity, thickness, and rheology from remote sensing: La Clapière landslide, France. Geophysical Research Letters, Vol. 40, 4299 - 4304.

The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a Davis' manifolds, obtained from the so-called reflection group trick, which is a powerful method for constructing aspherical manifolds. I will describe an approach based on the study of triangulations of spheres and simplicial maps between them. This approach also presents connections with the theory of graph minors. No knowledge about simplicial volume or Davis' reflection group trick is expected from the audience.

In this talk we present an explicit area formula to compute the spherical Hausdorff measure of an intrinsic regular graph in an arbitrary homogeneous group. We assume the intrinsic graph to be intrinsically differentiable at any point with continuous intrinsic differential. The key aspect of the result lies in the introduction of a suitable notion of intrinsic Jacobian and in the computation of an explicit expression for this object. Eventually, we present recent results about the symmetries of some homogeneous distances for which the area formula takes a simplified expression. This is a joint work with V. Magnani (Unipi). Attività di ricerca supportata dal progetto INDAM-GNAMPA-2024: "Free boundary problems in noncommutative structures and degenerate operators" CUP E53C23001670001

We introduce the concept of Ramsey pairs, and show how they can be prove several infinitary results in combinatorics.

Group operations on a fixed countably infinite universe form a Polish space G. Thus we can view group properties as isomorphism-invariant subsets of G, and it makes sense to ask: what properties are generic (in the sense of Baire category)? In my talk, I will address this question and if time permits, I may also say a few words about generic properties of compact groups.

Inspired by a recent characterisation of coherent topoi as a class of Kan injectives, we provide a tentative definition of fragment of geometric logic. We treat them as mathematical objects, and study them from the point of view of Lindstrom-type theorems.

In this seminar, I will talk about Objective Function Free Optimization (OFFO) in the context of pruning the parameter of a given model. OFFO algorithms are methods where the objective function is never computed; instead, they rely only on derivative information, thus on the gradient in the first-order case. I will give an overview of the main OFFO methods, focusing on adaptive algorithms such as Adagrad, Adam, RMSprop, ADADELTA, which are gradient methods that share the common characteristic of depending only on current and past gradient information to adaptively determine the step size at each iteration. Next, I will briefly discuss the most popular pruning approaches. As the name implies, pruning a model, typically a neural networks, refers to the process of reducing its size and complexity, typically by removing certain parameters that are considered unnecessary for its performance. Pruning emerges as an alternative compression technique for neural networks to matrix and tensor factorization or quantization. Mainly, I will focus on pruning-aware methods that uses specific rules to classify parameters as relevant or irrelevant at each iteration, enhancing convergence to a solution of the problem at hand, which is robust to pruning irrelevant parameters after training.Finally, I will introduce a novel deterministic algorithm which is both adaptive and pruning-aware, based on a modification Adagrad scheme that converges to a solution robust to pruning with complexity of $\log(k) \backslash k$. I will illustrate some preliminary results on different applications.