Rafael Greenblatt
Title: Solvable and non-solvable models in 2D lattice statistical mechanics
Abstract: Some models in 2D statistical mechanics (which are defined by a limiting procedure from families of random variables indexed by points in a discrete two-dimensional lattice) are known to exhibit emergent behavior such as the existence of a critical point where the asymptotic correlation functions are conformally covariant, and correspond to those of a conformal quantum field theory (CFT). So far, the most complete results of this type are known only for a few models with very special properties which allow them to be exactly solved, but the phenomenon is expected to be much more general.

I will discuss my contribution to this problem (some partial results for nonsovable variants of the Ising model obtained via the so-called Constructive Renormalization Group approach) and the prospects for the future.

Niels Kowalzig
Title: Centres, traces, and cyclic cohomology
Abstract: While centres of monoidal categories are the key to Gerstenhaber algebras structures, bimodule category centres allow for cyclic cohomology to be defined and hence are the key to BV algebra structures. In this talk, we will discuss the biclosedness of the monoidalcategories of modules over a (left) Hopf algebroid, along with a categorical equivalence between bimodule category centres and anti Yetter-Drinfel’d contramodules. This is directly connected to the existence of a trace functor on the monoidal category in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.

Guido Lido
Title: Isogeny graphs with level structure
Abstract: An object of central importance in isogeny-based cryptography (a family of post-quantum cryptographic algorithms) is the graph of ℓ-isogenies between supersingular elliptic curves, namely the graph whose vertices are supersingular elliptic curves over a fixed finite field and whose edges are ℓ-isogenies for ℓ a prime different from the characteristic of the field. It is immediate to prove that from each vertex there are exactly ℓ+1 outgoing edges (i.e. the graph is (ℓ+1)-regular), while it is less obvious that, as proven by Eichler, such a graph is connected and that it has the Ramanujan property.

In our talk we look at a generalization of these graphs, namely isogeny graphs whose vertices are supersingular elliptic curves with level structure: we see that new interesting phenomena arise, since the graph can be k-multipartite, and up to this, we extend Eichler’s result on the Ramanujan property using modular forms. These results have a cryptographic application: in a work with Basso, Codogni, Connolly, De Feo, Fouotsa, Morrison, Panny, Patranabis and Wesolowski they are used to define a protocol that generates supersingular elliptic curves with unknown endomorphisms.
Joint work with Giulio Codogni

Cristian Mendico
Title: Aubry-Mather theory for sub-Riemannian control systems
Abstract: The long-time average behavior of the value function in the classical calculus of variation is known to be connected with the existence of solutions of the so-called critical equations, that is, a stationary Hamilton-Jacobi equation which includes a sort of nonlinear eigenvalue called the critical constant (or effective Hamiltonian).
In this talks, we will address a similar issues for the dynamic programming equation of an optimal control problem, namely a control problem of sub-Riemannian type, for which coercivity of the Hamiltonian is non longer true. We introduce the Aubry set for sub-Riemannian control systems and we show that any fixed point of the Lax-Oleinik semigroup is horizontally differentiable on such a set. Furthermore, we obtain a variational representation of the critical constant by using an adapted notion of closed measures, introduced by A. Fathi and A. Siconolfi (2004). In conclusion, defining a new class of probability measures (strongly closed measures) we define the Mather set for sub-Riemannian control systems and we show that such a set is included in the Aubry set.
Joint work with P. Cannarsa
[1] P. Cannarsa, C. Mendico, "Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space", J. Differential Equations 332 (2022), 83-122.
[2] P. Cannarsa, C. Mendico: “Aubry set for sub-Riemannian control systems”, Arxiv:2204.12544.

Andrea Santi
Title: Symmetry superalgebras in parabolic supergeometries
Abstract: I will present an overview of recent results on Lie superalgebras of symmetries for geometric structures on supermanifolds related to nonholonomic distributions and their structure reductions. I will discuss a novel extension of Tanaka theory to the context of supermanifolds, giving an upper bound on the supersymmetry dimension, and some examples of distributions with flag supervarieties as models with maximum symmetry. In particular, I will show realizations of the exceptional simple Lie superalgebras G(3) and F(4) that generalize the first realizations by E. Cartan and F. Engel in 1893 of the simple Lie algebra G(2).

Sara Scaramuccia
Title: A glimpse of Topological Data Analysis
Abstract: In the last decades, Topological Data Analysis - TDA is gaining much attention due to, in principle, the scale-free and robust-to-noise nature of topological descriptors. The basic goal of TDA is to apply topological tools to compare and classify data based on their shape. We quickly review the main pipeline of TDA: data, filtrations of simplicial complexes, topological signatures provided by persistent homology, and pseudometrics on topological signatures. The main purpose will be that of providing a general introduction to TDA so that to show weak and strength points of those approaches and future developings both in terms of applied and more theoretical results.

Stefano Vigogna
Title: Understanding Neural Networks with Reproducing Kernel Banach Spaces
Abstract: Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this talk I will discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, I will show a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, I will show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure.

Elia Bruè (Institute for Advanced Studies, USA)
Title: Instability and non-uniqueness in fluid dynamics.
Abstract: The incompressible Navier-Stokes and Euler equations are fundamental for understanding basic phenomena in fluid dynamics, but many fundamental aspects of these equations still need to be fully understood. In particular, the question of well-posedness and uniqueness within physically relevant classes of solutions is not entirely resolved. Of particular importance are the class of Leray solutions to the Navier-Stokes equations and the class of solutions to the 2d Euler equations with L^p vorticity. This talk outlines recent progress toward their understanding and discusses newly discovered nonunique solutions.

Luis Ferroni (KTH Royal Institute of Technology)
Title: The Ehrhart theory of matroid polytopes
Abstract: In this presentation I will give an overview of the topics and the results of my PhD Thesis at the University of Bologna, where I worked under the supervision of Prof. Luca Moci. I will discuss two fundamental objects in combinatorics, namely the notion of Ehrhart polynomial and the notion of matroid. I will report on the negative solution to conjectures of De Loera, Haws, and Köppe (2007) and Castillo and Liu (2015) asserting the positivity of the Ehrhart polynomial of all matroids and generalized permutohedra.

Massimo Fornasier (T.U. München, Germany)
Title: Breaking Nonconvexity: Consensus Based Optimization
Abstract: Consensus-based optimization (CBO) is a multi-agent metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. In fact, optimizing agents (particles) move on the optimization domain driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace’s principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. Based on an experimentally supported intuition that CBO always performs a gradient descent of the squared Euclidean distance to the global minimizer, we show a novel technique for proving the global convergence to the global minimizer in mean-field law for a rich class of objective functions. The result unveils internal mechanisms of CBO that are responsible for the success of the method. In particular, we present the proof that CBO performs a convexification of a very large class of optimization problems as the number of optimizing agents goes to infinity. We further present formulations of CBO over compact hypersurfaces and the proof of convergence to global minimizers for nonconvex nonsmooth optimizations on the hypersphere. We conclude the talk with several numerical experiments, which show that CBO scales well with the dimension and is extremely versatile. To quantify the performances of such a novel approach, we show that CBO is able to perform essentially as good as ad hoc state of the art methods using higher order information in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.

Maurizia Rossi (Università di Milano Bicocca, Italy)
Title: The nodal length of random spherical harmonics.
Abstract: In this talk we investigate the behavior of the "typical" Laplacian eigenfunction of a compact smooth Riemannian manifold. In particular, motivated by both Yau's conjecture on nodal sets and Berry's ansatz on planar random waves, we consider Gaussian eigenfunctions on the sphere and study the distribution of the length of their nodal lines in the high energy limit. The results we obtain raise several questions regarding both the distribution of other geometric functionals and the behavior of nodal statistics of random eigenfunctions on a "generic" manifold. In this talk, mainly based on a joint work with D. Marinucci and I. Wigman, we answer some of these questions, relying on recent developments by I. Nourdin and G. Peccati in the theory of Normal approximations for Wiener chaos.

Catharina Stroppel (Universität Bonn, Germany)
Title: From Platonic solids to Springer theory and Fukaya categories
Abstract: In this talk I want to give a small tour starting from Platonic solids and explain how one might naturally construct spaces/manifolds/varieties which arise in representation theory, more precisely Springer theory and sketch why representation theorists care. From that spaces we will construct a Fukaya type category