There is a beautiful "positroid" stratification of the Grassmannian, studied by Lusztig, Postnkov, Rietsch, Williams, K-Lam-Speyer, and many others (with strata most naturally indexed by juggling patterns!). In [KLS] we gave a formula representing the class of a positroid variety, which [Fan-Guo-Su-Xiong '25] recently expressed as a sum over "periodic pipe dreams". In this talk I'll explain all that, and introduce "matrix positroid varieties", which are what the FGSX formula is really for. Time permitting, I'll explain the necessary upgrades for Chern-Schwartz-MacPherson classes. This work is joint with Paul Zinn-Justin.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  In this talk, I will report on work in progress, joint with B. Elias, H. Ko, and N. Libe-dinsky, which aims at giving a diagrammatic presentation of the category of singular Soergel bimodules.
  Soergel bimodules categorify the Hecke algebra of a Coxeter group and play a central role in modern representation theory. Their singular counterparts categorify the Hecke algebroid, which is governed by the rich combinatorics of double cosets in Coxeter groups. I will describe this category and explain how its diagrammatic presentation will allow us to make explicit computations and to work in great generality, for example over Z.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

  Around 1985 three groups of jugglers came up with the same system for recording, studying, and creating juggling patterns. I'll explain and demonstrate this system, and explain 21st-century connections to probability and algebraic geometry.
  Note:   this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV 2023/2027 (CUP E83C23000330006)

  A permutation   a₁ , a₂ , ... , a_n   of   1 , 2 , ... , n   is alternating if   a₁ > a₂ < a₃ > a₄ < ...   The number E_n of alternating permutations of   1 , 2 , ... , n   is called an Euler number. We will survey the theory of alternating permutations and Euler numbers, beginning with the famous formula of Désiré André:   ∑_n E_n xⁿ/n! = sec(x) + tan(x) .   Connections will be given to such topics as convex polytopes, tridiagonal matrices, probability theory, and the representation theory of the symmetric group. We will explain how the enumeration of alternating permutations that are also fixed-point-free involutions is related to an asymptotic result in one of Ramanujan's notebooks.
  Note: this talk is part of the activity of the MUR Excellence Department Project CUP MatMod@TOV 2023-27 (CUP E83C23000330006)

Does the heterogeneous Fisher-KPP equation admit a unique stationary solution? We will present a positive result which is valid up to dimension 6. Then we will see that some surprises may occur in higher dimension. These results shed a light on the question of long-time convergence of solutions of the reaction-diffusion equation. This is a joint work with O. Tough.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Starting from a prescribed spline space, basis construction naturally leads to two questions: how to determine its dimension and how to construct a complete, linearly independent family of functions. In this talk, I will present recent results on both questions for polynomial splines of maximal smoothness over T-meshes. I will first introduce the smoothing cofactor method, which translates smoothness requirements into algebraic compatibility conditions. By decomposing the T-connected component into diagonalizable and non-diagonalizable parts, we reduce dimension computation and stability analysis to the rank of a smaller matrix associated with the multi-vertices of the remaining component. I will then describe a basis construction for arbitrary regular T-meshes. The method first builds local tensor-product B-splines on a suitable extended diagonalizable mesh, guided by its dimension formula, and then uses Extended Edge Elimination to obtain a basis on the original mesh. Examples will illustrate how changes in knot geometry affect both the dimension and the resulting basis. The talk is based on joint work with Falai Chen and Shicong Zhong. It is part of the activity of the MUR Excellence Department Project MatMod@TOV.

Core Entropy is a function aimed at measuring the complexity of the iterations of post critically finite polynomials. In this talk we first recall the basic notions of polynomial dynamics: Julia sets, Hubbard Trees, and Core Entropy, and then we explore the core entropy as a function on the parameter space of polynomials. We then introduce new techniques to understand the maxima of core entropy.

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

I will show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of $mathbbm{Q}$-ample line bundles on higher models. As already known by previous works, I will then explain how this implies a solution to the Yau-Tian-Donaldson conjecture, connecting the $K$-stability of a smooth polarized projective variety $(X,L)$ to the existence of constant scalar curvature Kähler metrics in $c_1(L)$.
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

Perturbation theory is the construction of local quasi-integrals through the composition of a series of canonical or near-identity transformations, called perturbation steps. It has been, historically, the fundamental analytical method for the study of nearly integrable Hamiltonian systems, as it allows the study of the local behavior in an otherwise complex dynamical setting, as well as the construction of local analytical solutions, and the study of the effective stability of the trajectories in the associated phase space. Naturally, the construction of such quasi-integrals is a non-convergent process , with the rate of non-convergence heavily depending from the domain that is considered. In celestial mechanics, the domains of fast divergence are called resonances. In their vicinity, stable and unstable manifolds are formed, splitting the phase space into rotation regions, foliated with KAM rotational tori, libration regions, which contain secondary tori, and chaotic domains, where the tori have broken down. In the non resonant domain, quasi-integrals allow the study of the secular dynamics of celestial bodies, their classification into groups of common origin, and the study of their stability . In contrast, resonant regions (Dogkas & Guido 2026) restrict the applicability domain of such methods. Nevertheless, effective stability estimates can be found in their vicinity (Celletti, Dogkas, et al. 2026), when an optimized selection of the associated parameters is considered, while quasi-integrals can still be defined in the libration regions, deep inside the resonant domain (Dogkas & Vartolomei 2026). N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).