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

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).

Charles, Goren and Lauter in 2006 proposed a cryptographic hash function based on walks in the ℓ-isogeny graph of supersingular elliptic curves in large characteristic p. In 2020 Eisenträger et al. showed that such hash functions allow for an efficient computation of second pre-images (and hence are dramatically broken) as soon as the endomorphism ring of the starting vertex is known. In this attack, the main auxiliary tool is the so-called "KLPT algorithm" for finding a connecting ideal between two maximal orders in a positive definite quaternion algebra. Since all known methods for constructing supersingular elliptic curves implicitly leak the endomorphism ring, secure instantiations of the CGL hash function should be set up by a trusted party, or through multi-party computation. In this talk I will present a similar result for hash functions from (ℓ, ℓ)-isogenies between superspecial principally polarized abelian surfaces in characteristic p: if the principal polarization on the starting surface is sufficiently well-understood, then collisions can be produced in polynomial time, and therefore the hash function should be considered broken (but in a weaker sense than in the case of elliptic curves). The main auxiliary tool is a generalization to dimension 2 of the KLPT algorithm. It is likely that all known methods for generating a starting surface implicitly reveal the information needed for producing collisions, so it seems that, here again, a trusted set-up is needed. This is joint work with Thomas Decru, Péter Kutas, Abel Laval, Christophe Petit and Yan Bo Ti.

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

For the PS-12 split introduced by Powell and Sabin in 1977 we present an optimal symmetric 4 point quadrature rule and a collection of weighted rules. These are useful for an efficient formation of the linear system arising in Galerkin discretization on this split. We use the S-spline version of simplex splines introduced by Cohen, L., Riesenfeld in 2013, and a global basis based on the theory of minimal determining sets adapted to S-splines on the PS-12 split. This is joint work with Salah Eddargani, Carla Manni, and Hendrik Speleers. The talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

Physicists in condensed matter physics realize certain tensor networks appear in their studies of two-dimensional topological order. We have identified their 4-tensors with bi-unitary connections appearing in subfactor theory before. We now identify their 3-tensors satisfying the zipper condition with intertwiners between bimodules arising from bi-unitary connections. This completes identification of their fusion categories and ours in subfactor theory.

Over the complex numbers, generic vanishing theory is useful for studying the geometry of irregular varieties. A standard application of this theory is that if X is a smooth, complex projective variety of maximal Albanese dimension (i.e. dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is non-negative. This relies on vanishing theorems of analytic nature.

In this talk, we will show that the same statement holds in positive characteristic, assuming further that the Frobenius morphism acts bijectively on the cohomology (such a hypothesis tends to be true for "most" varieties). If we also assume that our variety is not of general type, then we also show that its Euler characteristic is in fact zero, and that the image of the Albanese morphism is fibered by abelian varieties (which are well-known statements over the complex numbers).

The proof relies on a positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, together with a Witt vector version of the Grauert-Riemenschneider vanishing theorem.

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

Fourier analysis is a powerful tool in analysis. In the setting of abelian schemes, Fourier-Mukai transformation and the weight decomposition play a similar role. For degenerate abelian fibrations, the relative group structure disappears and understanding the intersection theory leads to many interesting questions, such as the P=W conjecture, χ-independence phenomenon, and multiplicative splitting of the perverse filtration for the Beauville-Mukai system. In this talk, I will connect Fourier transform between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, I will compute the pushforward of monomials of divisor classes on compactified Jacobians via the twisted double ramification formula. Along the way, we will encounter instances of χ-independence and the multiplicativity of perverse filtration for compactified Jacobians. This is a joint work with Samouil Molcho and Aaron Pixton.

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

This talk presents recent results on the generic properties of conjugate points in viscosity solutions to first-order Hamilton–Jacobi equations. In this context, we introduce a quantitative version of the transversality theorem and apply it to estimate the total number of shock curves in weak entropy solutions to scalar conservation laws. Additionally, we establish sharp quantitative bounds on the critical sets of smooth functions and provide an explicit upper bound for the (d−1)-dimensional Hausdorff measure of the zero set of nontrivial multivariable polynomials.
N.B.This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006