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

  A theorem of Rajan says that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines individual constituents uniquely. This is analogous to the uniqueness of prime factorization of natural numbers. We discuss a more general question of determining all the pairs (V₁ , V₂) consisting of two finite dimensional irreducible representations of a semisimple Lie algebra g such that Res(g₀)|V₁ ≅ Res(g₀)|V₂ , where g₀ is the fixed point subalgebra of g with respect to a finite order automorphism.
  We will also discuss the above tensor product problem in the category of typical representations of basic classical Lie superalgebras.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

In this talk we study chaotic dynamics generated by analytic convex billiards. We consider the set S of analytic billiards with negative curvature satisfying the following property: for any rational rotation number, there exists a hyperbolic periodic orbit whose stable and unstable manifolds intersect tansversally along a homolinic orbit. And we prove that the set S is residual among analytic billiards with negative curvature with the ususal analytic topology. This result is a consequence of the Baire property and the main result of this work, which reads: Fixing a rational rotation number p/q, we can prove that the set of analytic billiards with negative curvature having a hyperbolic periodic orbit of rotation umber p/q whose stable and unstable manifolds intersect tansversally along a homolinic orbit, is open and dense.

As a consequence of our results, we have that chaotic billiards are dense among analytic biliards.

Our proof combines Aubry-Mather theory to study periodic orbits of any rotation number as well as their heteroclinic trajectories, with the work by Zehnder on planar twist maps with elliptic points in the 1970's, which provides a methodology for constructing analytic perturbations of maps in order to obtain transversality between the invariant manifolds of hyperbolic periodic orbits.

This is a joint work with Imma Baldomá, Anna Florio and Martin Leguil.

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

In this talk I will describe the normal stable surfaces with K²=2p_g-3 whose only non canonical singularity is a cyclic quotient singularity of type 1 4k  (1,2k-1) and the corresponding locus 𝔇 inside the KSBA moduli space of stable surfaces. The main result is the following: for p_g≥ 15,   (1) a general point of any irreducible component of 𝔇 corresponds to a surface with a singularity of type 1 4k  (1,1),   (2) the closure of 𝔇 is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K²=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of 𝔇. Moreover 𝔇 has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4. This is joint work with Rita Pardini.

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 study the evolution in time of smooth sets in the n–dimensional flat torus, such that their boundaries, which are smooth hypersurfaces, move by surface diffusion flow (i.e. the H^{-1}gradient flow of the Area functional). More precisely, in this talk we present two different proofs of the stability of strictly stable critical sets for the volume–constrained Area functional, under the surface diffusion flow. The first approach is based on suitable energy estimates and compactness arguments, whereas the second one relies on the gradient flow structure of the evolution, in particular, the main tool is the Alexandrov-type inequality combined with the quantitative isoperimetric inequality. Hence, assuming different hypotheses, we prove that if the initial set is sufficiently “close” to a strictly stable critical set, then the flow actually exists globally in time and exponentially converges to a “translation” of the critical set. This is based on joint works with N. Fusco (Univ. Federico II di Napoli & SSM), C. Mantegazza (Univ. Federico II di Napoli & SSM), D. De Gennaro (Univ. Bocconi), A. Kubin (TU Wien) and A. Kubin (Jyväskylä University)
NB : This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

The aim of this seminar is to present the regularity results I obtained during my PhD journey for the gradient of solutions to some classes of strongly singular or degenerate elliptic and parabolic problems.

The elliptic problem under consideration arises as the Euler-Lagrange equation of an integral functional in the Calculus of Variations. The energy density of this functional satisfies standard $p$-growth and $p$-ellipticity conditions, for $p > 1$, with respect to the gradient variable -- but only outside a ball with radius $lambda > 0$ centered at the origin.

As for the parabolic problems in question, a motivation for studying them stems from gas filtration models taking into account the initial pressure gradient.

The presentation will focus mainly on the higher differentiability of solutions, both of integer and fractional order.

This talk is based on joint work with Fabian Bäuerlein, Antonio Giuseppe Grimaldi and Antonia Passarelli di Napoli.

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