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

  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

In Kähler geometry, the Calabi-Yau theorem establishes the existence and uniqueness of Kähler metrics with prescribed volume form. In this talk, I will present a non-Archimedean version of this theorem for general Kähler manifolds, extending a theorem of Boucksom-Jonsson from the projective setting. This is joint work with David Witt Nyström.
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 Weyl energy on four-manifolds is a geometric functional related to the Chern-Gauss-Bonnet formula. Similarly to Willmore’s functional for surfaces embedded in the three-dimensional Euclidean space, it enjoys conformal invariance properties. We are interested in the interaction of the Weyl’s functionals of two manifolds under the operation of connected sum, showing conditions that decrease the total energy. Such estimates might be useful in understanding compactness properties of minimizing or critical families of metrics. This is joint work with F.Malizia
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

In this seminar, I will present the problem of the propagation of singularities for differential equations and the corresponding problem of propagation of polarization for systems of differential equations. The problem has been solved for operators of principal type, but the theory is still incomplete for more general operators. Finally, I would like to present an example of an extremely degenerate operator that does not fall within the classical theory, yet for which propagation phenomena still occur. This example arises from differential geometry and concerns certain systems of PDEs related to the curvature tensors of a Riemannian or Lorentzian metric. N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

In his seminal classification paper from 1976, Connes remarked that every separable tracial von Neumann algebra ought to be embeddable into an ultrapower of the hyperfinite type II_1 factor, or, in other words, be approximable by matrices. Over the following decades, the Connes Embedding Problem (CEP) remained unsolved, but many interesting and deep reformulations were discovered. Prominently, Kirchberg proved in his famous 1991 Inventiones paper that CEP is equivalent to several questions concerning C*-algebras and their tensor product, including his QWEP conjecture. He also showed that CEP holds if and only if there is a unique C*-norm of the tensor product of two copies of the full group C*-algebra of the free group. The latter was shown (by several authors) to be equivalent to Tsirelson’s conjecture about quantum correlations. CEP also relates to the open problems in group theory if all infinite discrete groups are sofic. Recently, Ji-Natarajan-Vidick-Wright-Yuen announced a negative solution to Tsirelson’s conjecture, and hence also a negative answer to CEP by proving that two complexity classes are the same

The YTD conjecture predicts that the existence of "canonical" Kähler metrics on a polarized projective manifold is governed by a purely algebro-geometric notion known as K-stability. Recently, solutions to (variants of) this conjecture have been proposed, and the purpose of this talk is to review this recent progress.
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