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.

We present some recent results on optimal control/differential games in cases where the state equation is a stochastic delay differential equation (SDDE) with delay in the state and/or in the control. We look at two possible approaches: one based on partial smoothing properties of the transition semigroup associated with the uncontrolled SDDE (papers on optimal control , mean field control, mean field games without common noise with G. Bolli, F. Masiero, M. Ricciardi, M. Rosestolato), one based on regularization of viscosity solutions (paper with F. De Feo, A. Swiech, L. Wessels, on mean field control with only common noise).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Motivated by the Green-Griffiths and Lang-Vojta conjectures, it is expected that the algebraic exceptional set of a log-surface $(X,B)$ of log-general type - which parametrizes rational curves on $X$ meeting $B$ in at most two points - is finite. In this talk, I will discuss recent results on this set for the cases where $X$ is the projective plane or a Hirzebruch surface, and is a curve with three irreducible components. This talk is based on [arXiv.2507.13280] and work in progress.
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 introduce Lp spectral triples and give an overview of their properties. I will also discuss spectral triple isometries. On a different note, I will introduce twisted crossed products of Lp and Banach algebras by locally compact groups and outline some of their properties.
Time permitting, I will present a generalization of the so-called Packer-Raeburn trick to the Lp setting.

We start by reviewing the classical spectral theory of the Dirichlet-Laplacian, on a general open set. It is well-known that the spectrum may fail to be purely discrete, in this generality. We then turn our attention to a nonlinear variant of this problem, by considering the case of the $p-$Laplacian with Dirichlet homogeneous conditions. More precisely, we analyze the minmax levels of the constrained $p-$Dirichlet integral: we show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ''at infinity'', it actually defines an eigenvalue. We also prove a quantitative exponential fall-off at infinity for the relevant eigenfunctions: this can be seen as a generalization of v{S}nol-Simon--type estimates to the nonlinear case. Some of the results presented have been obtained in collaboration with Luca Briani (TUM Monaco) and Francesca Prinari (Pisa). <br> <b>NB</b>: <i> This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006 </i>