In this talk, I will introduce a new statistic on Weyl groups called the atomic length, and clarify this terminology by drawing parallels with the usual Coxeter length. It turns out that the atomic length has a natural Lie-theoretic interpretation, based on crystal combinatorics, that I will present. Last but not least, I will explain how this can be used as a tool for tackling a broad range of enumeration problems arising from modular representation theory (and related to the study of core partitions).
Part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

Let V be a representation of a connected complex reductive group G. The group acts on the ring of regular functions on V: the asymptotic support of this representation is a closed convex polyhedral cone, called moment cone. We will present an algorithm that determines the minimal list of linear inequalities for this cone. Some aspect are relevant from algorithm and convex geometry and others from algebraic geometry.
Part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

NB:This colloquium is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We consider linear stable operators L of order 2s whose spectral measure is positive only in a relative open subset of the unit sphere, the aim being to present Liouville type results, in a half space, for the inequality -Lu ≥ u^p. In particular we will show that u≡0 is the only nonnegative solution for 1 ≤ p ≤ (N+s)/(N-s). The optimality of the exponent (N+s)/(N-s) will also be discussed. Based on a joint work with I. Birindelli and L. Du (Sapienza Università di Roma)
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result can be seen as a Lagrangian counterpart of the problem of Lyapunov stability of fixed points of area-preserving diffeomorphisms. An application is given to the model of a bouncing ball. Joint work with Rafael Ortega.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

The inverse spectral problem asks to what extent one can recover the geometry of a manifold from knowledge of either its Laplace spectrum or dynamical counterparts, e.g., the (marked) length spectrum. While counterexamples do exist in general, there are certain symmetry and nondegeneracy conditions under which spectral uniqueness holds. Perhaps the most tantalizing unsolved case is that of strictly convex planar domains, known as Birkhoff billiard tables. It turns out that there is a deep relationship between the Laplace and length spectra, which is encoded in the Poisson relation. In this talk, I will describe my work on both Laplace and length spectral invariants as well as limitations in using the Poisson relation for inverse problems.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

We present a construction which associates to differential equations discrete groups. In order to establish this relation we use automata and random walks on ultra discrete limits. We discuss related results concerning von Neumann dimension and L2 Betti numbers of closed manifolds.

The Reeb graph R(f) of a C^1-function f from M to the real numbers with isolated critical points is a quotient object by the identification of connected components of function levels which has a natural structure of graph. The quotient map p from M to R(f) induces a homomorphism p* from the fundamental group of M to the fundamental group of R(f) which is equal to F_r, the free group of r generators. This leads to the natural question whether every epimorphism from a finitely presented group G to F_r can be represented as the Reeb epimorphism p* for a suitable Reeb (or even Morse) function f. We present a positive answer to this question. This is done by use of a construction of correspondence between epimorphisms from the fundamental group of M to F_r and systems of r framed non-separating hypersurfaces in M, which induces a bijection onto their framed cobordism classes. As applications we provide new purely geometrical-topological proofs of some algebraic facts.

Logarithmic and orbifold structures provide two different paths to the enumeration of algebraic curves with fixed tangencies along a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will explain how the two systems of invariants can be identified by passing to an appropriate blowup. This identifies “birational invariance” as the key property distinguishing the two theories. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors. Time permitting, I will discuss extensions of this result to the setting of negative tangencies, where the pathological geometry of the moduli space is controlled using tropical geometry.

This is joint work with Luca Battistella and Dhruv Ranganathan.

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

I will give an introduction to the arithmetic of rational points on surfaces that can be fibred into conics. In the end I will talk about upcoming work with Christopher Frei that uses arguments from analysis to answer some basic questions.

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