It is well-known that submanifolds of least area for a fixed boundary (Plateau problem) or in a fixed homology class (homological Plateau problem) shall not be smoothly embedded in general, but rather exhibit a singular set (as first noted by Simons and then justified by Bombieri-De Giorgi-Giusti half a century ago). The first singular example(s) of minimizers were in fact extremely rigid: cones with an isolated singularity at the origin. As it is now clear, the occurrence of singularities is an intriguing and partly elusive pathology that may be imputable to diverse causes, ranging from topological obstructions (related e.g. to pioneering work by Thom) to basic complex-analytic phenomena. But how wild may the singular set possibly be, and how frequently will it be observable as one varies the boundary in question or, respectively, the background metric? Over the past five years we have witnessed striking advances on both fronts. In this lecture I will present the general state of the art and my contributions to the latter question(s), known as the generic regularity problem, as well as some surprising geometric applications. Based on joint works with Yangyang Li and Zhihan Wang.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We will start with a brief introduction to topology of real algebraic curves, and then will discuss in more details the case of curves of degree 6 in the real projective plane. The main purpose of the talk is to present an equisingular deformation classification of simple real plane sextic curves with smooth real part. In particular, we will show that the equisingular deformation type of such a curve is determined by its real homological type, that is, the polarization, exceptional divisors, and real structure recorded in the homology of the covering K3-surface. (This is a joint work with Alex Degtyarev.)
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

We will discuss the period index questions for 2-torsion Brauer elements of function field of a hyperelliptic curve over global field of char 0. As a consequence, we deduce finiteness of the u-invariant of totally imaginary number fields.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

The question of whether a Hamiltonian system is typically integrable or chaotic is a central topic in dynamical systems, which traces back to the pioneering works of Poincaré in Celestial Mechanics. A satisfactory picture of the typical dynamics of such systems did not emerge until the 1970s, when Markus and Meyer established that a generic (in the Baire category sense) Hamiltonian system on a compact symplectic manifold is neither integrable nor ergodic. On the contrary, the case of natural Hamiltonian systems is much less studied, in spite of its central relevance in mathematical physics. Specifically, a natural Hamiltonian corresponds to the situation in which the symplectic manifold is the cotangent bundle of a manifold M , and the Hamiltonian is given by the sum of a fixed kinetic energy term and a potential field V in C^{infty}(M ; R). It is known that a generic potential field on a compact manifold is non-ergodic. Moreover, near the potential maximum, the system may exhibit positive topological entropy under (non-generic) suitable conditions. Nevertheless, the fundamental question of whether motion at low energy levels is typically integrable or chaotic remains open to date. This difficulty arises because standard transversality methods are no longer applicable, raising the conjecture of whether classical results on generic non-integrability extend to the setting of potential fields. In this talk we shall show that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential V on T^2 exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. To put this result in perspective, the existence of hyperbolic periodic orbits is the natural starting point to establish the presence of chaos in dynamical systems.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

I'll report on progress on a program to understand global hypoellipticity of certain geometric (degenerate) Laplacians. Global hypoellipticity means that globally smooth data correspond to globally smooth solutions. This is a very weak property, which is compatible with propagation of singularities. Results from the 1990s about the d-bar Neumann problem (due to Boas, Straube, Christ etc.) hint to the existence of topological obstructions to global hypoellipticity. This talk is partially based on joint work with A. Martini.

I will give a gentle introduction with several examples to the following topic: Yau-Tian-Donaldson conjecture states that a polarised manifold $(X,L)$ admits a cscK metric in $c_1(L)$ if and only if $(X,L)$ is K-polystable. Matsushima proved in 1957 that existence of such cscK metric implies reductively of the automorphism group of $X$. In a positive direction on the YTD conjecture, we show that K-polystability implies reductively of the automorphism group of $X$, for smooth Fano threefolds. This is joint work with Paolo Cascini and Ivan Cheltsov.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

  C^*-algebras are fundamental objects in the mathematical descriptions of quantum mechanics and field theory. Cuntz algebras are a family of C^*-algebras first defined in [Comm. Math. Phys. 57 (1977), 173-185]. In this talk I will survey the main combinatorial concepts and results arising from, and related to, these algebras. In particular, I will define certain permutations and review the state of the art on their characterization, enumeration, and construction. I will also explain how these results can be applied to the construction of subgroups of the automorphism, and outer automorphism, groups of these algebras. I will conclude with some open problems and directions for further research..
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

&nbsp; An involution system (<em>W</em>,<em>S</em>), that is a group <em>W</em> generated by a set of involutions <em>S</em>, is naturally endowed with a weak order arising from orienting its Cayley graph. If (<em>W</em>,<em>S</em>) is a Coxeter system, Bj&ouml;rner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. <br> &nbsp; In this talk, we discuss the following question: For which involution systems is the weak order a complete meet-semilattice? <br> &nbsp; The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. If time allows, we will also discuss open problems (e.g. in relation to automatic structures, geometric representations,…). <br> &nbsp; This is joint work with Fabricio Dos Santos and Aleksandr Trufanov. <br> &nbsp; &nbsp; <em><small><small> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </small></small></em>

Topologically ordered phases of matter have interesting features, such as the existence of quasi-particles with braid statistics. These quasi-particles can be studied using an AQFT-inspired approach along the lines of the celebrated Doplicher-Haag-Roberts programme on superselection sectors. In this talk I will introduce an axiomatisation, called local topological order, of such quantum models. These axioms are defined in terms of nets of (ground state) projections satisfying certain conditions. They allow us to define a physical boundary algebra, and I will outline how in concrete models (such as Kitaev's toric code or Levin-Wen models) the bulk superselection sector (''DHR'') category can be recovered from the boundary algebra, giving a mathematical framework for topological holography. If time permits, I will explain how these axioms can be extended to included models with topological boundaries, and outline how this can be used to study, for example, Walker-Wang bulk-boundary systems. Based on joint work with Corey Jones, Dave Penneys and Daniel Wallick (arXiv:2307.12552 and arXiv:2506.19969)

We investigate the distribution and clustering of extreme events of stochastic processes constructed by sampling the solution of a Stochastic Differential Equation on R^n. We do so by studying the action of an annealed transfer operators on suitable spaces of densities. The spectral properties of such operators are obtained by employing a mixture of techniques coming from SDE theory and a functional analytic approach to dynamical systems (joint work with F. Flandoli, S. Galatolo and S. Vaienti).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006