The homotopy groups of CW complexes and of the mapping spaces between them are notoriously difficult to compute. However, if one disregards torsion, rational homotopy theory becomes very effective and can easily solve such problems. Moreover, it produces efficient invariants of homotopy classes of maps, called Maurer-Cartan elements, which encode the rational type of path components. I will give a couple of examples and then explain how this extends to embedding spaces.
  Based on joint work with Benoit Fresse and Thomas Willwacher.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Assuming g_1, g_2, ..., g_k are measure preserving transformations on a probability space X, we require that a function f from X to a measurable space Y satisfies that f(x) is in F(x, f(g_1 x), .... f(g_k x) ) almost everywhere for an upper semi continuous correspondence F defined on X x Y^k. If there exists such functions however NONE of them are measurable with respect to any finitely additive extension of the probability measure for which the g_i are still measure preserving, we say that the correspondence F is paradoxical. We demonstrate some paradoxical correspondences that are also convex valued and nowhere empty. We are curious if there are applications beyond optimization and economics.

This talk addresses Kac’s famous question, “Can one hear the shape of a drum?”—that is, whether the spectrum of the Laplacian on a domain uniquely determines its shape— in the context of convex planar billiard tables. While non-convex counterexamples are known (Gordon–Webb–Wolpert), the problem remains open for strictly convex domains with smooth boundaries. As shown by Anderson, Melrose, and Guillemin, the spectral question is deeply connected to its dynamical analogue: whether the length spectrum—the set of lengths of all periodic billiard trajectories—determines the domain up to isometry. In joint work with Vadim Kaloshin and Alfonso Sorrentino, we show that this is indeed the case for domains that are sufficiently close to a general ellipse and possess dihedral symmetry.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

Hénon-like maps are invertible holomorphic maps, defined on some convex bounded domain of $mathbb C^k$, that have (non-uniform) expanding behaviour in $p$ directions and contracting behaviour in the remaining $k-p$ directions. They form a large class of dynamical systems in any dimension. In dimension 2, they contain the Hénon maps, which are among the most studied dynamical systems. In this talk, I will give an overview of the main dynamical properties of these maps. In particular, I will focus on how tools from pluripotential theory can allow one to go beyond the algebraic setting of the Hénon maps. The talk is based on joint works with Tien-Cuong Dinh and Karim Rakhimov.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

We consider a notion of fractional s-area for codimension 2 surfaces in a closed Riemannian manifold or the Euclidean space, which can be seen as an extension of the fractional perimeter to higher codimension. The definition involves a minimum problem over a class of circle-valued maps having prescribed singularities on the given surface. We discuss various properties of the s-area when s is fixed, and we show that when s tends to 1 it Gamma-converges, with coercivity, to the classical area in the framework of currents. The talk is based on a joint project with Michele Caselli and Mattia Freguglia.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous count over the reals and extensions thereof. This is joint work with Yoav Len, Hannah Markwig and Yue Ren (arXiv:2512.24277).
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 study of Fourier coefficients of modular forms has a long and rich history, from Ramanujan’s conjectures to the modularity theorem relating modular forms to elliptic curves. In this talk, we first present the arithmetic–geometric viewpoint on modular forms provided by modular curves, and use it to study certain congruences. These may first appear as numerical coincidences, but in fact hide structures arising from the geometry and cohomology of modular curves in p-adic setting. This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

  Gentle algebras, introduced by Assem and Skowro?ski, are a well-loved class of algebras. They are string algebras, so their module categories are combinatorially described in terms of strings and bands, they are tiling algebras associated with dissections of surfaces, and they have many other remarkable properties. Furthermore, Jacobian algebras arising from triangulations of unpunctured marked surfaces are gentle.
  In the talk, I will present a multiplication formula for cluster characters induced by generating extensions in a gentle algebra A. This formula generalizes a previous result of Cerulli Irelli, Esposito, Franzen, Reineke. Moreover, in the case where A comes from a triangulation T, it provides a representation-theoretic interpretation of the exchange relations in the cluster algebra with principal coefficients in T.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  Many objects of interest in algebraic topology can be computed by graph complexes. This includes homotopy groups of embedding spaces and diffeomorphism groups, and in particular (parts of) the cohomology of the moduli space of curves. Unfortunately, the graph homology itself is still a mysterious object and far from being fully understood. In my talk, I will introduce a smaller quasi-isomorphic variant of the most basic graph complex (the commutative graph complex of Kontsevich), and discuss the present state of knowledge of the graph homology.
  The talk is based on arXiv:2503.17131 and arXiv:2508.13724.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

In this talk I will present a recent construction of equilibrium states at positive temperature, with and without Bose-Einstein condensation, for a non-relativistic Bosonic QFT (gas of Bose particles) in the infinite volume limit, interacting through a localised two body interaction. In order to obtain this result, we use methods of quantum field theory in the algebraic formulation and of quantum statistical mechanics in the operator algebraic setting. The convergence of the interacting correlation functions is obtained constructing an equivalent perturbative series expansion introducing an auxiliary stochastic Gaussian field which mediates the interaction. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination with other regimes. This talk is based on a collaboration with Nicola Pinamonti.