I am going to discuss a certain characterization of Birkhoff Billiards inside discs which is related to the expansion of the formal Lazutkin conjugacy at the boundary. Based on the joint work with Jacopo De Simoi, Andrew Gad and Marco Poon.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

Noncommutative function theory and dilation theory for semigroups of completely positive maps have led to an interesting class of algebras generalizing Cuntz and Cuntz-Pimsner algebras. While formally they are defined in a familiar way, using creation operators on Fock-type spaces, what makes them difficult to study is that it is not clear how to obtain mixed commutation relations between the creation and annihilation operators. As a result until recently these algebras have been understood only in a few special cases, and there are still no general methods of obtaining relations in them, analyzing the ideal structure or computing the K-theory. Some of these algebras admit large quantum symmetries, and quantum groups provide then tools to analyze them, which are not accessible from a more classical point of view. The talk will review some results, examples and open problems in this area.

Finite Element Exterior Calculus (FEEC) is a powerful framework for developing stable discretizations of partial differential equations, providing a systematic approach to problems in computational electromagnetism and fluid mechanics. This talk explores recent developments in isogeometric versions of FEEC, with a special focus on structure-preserving discretizations for geometries with polar singularities. We will first discuss the construction of hierarchically-refined polar-spline spaces. We will then demonstrate how they form a discrete de Rham complex and provide a mathematically sound foundation for adaptive structure-preserving simulations on polar geometries. The talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

Finding smooth interpolations between probability measures is a problem of broad interest, with natural applications, e.g., in biology (trajectory inference) and computer graphics (image interpolation). In this talk, I will discuss a model in which such interpolations are obtained by minimizing an action functional of the acceleration. This minimization defines a discrepancy between measures that -- in analogy with Wasserstein distances from optimal transport theory -- admits an equivalent fluid-dynamical formulation and induces a Riemannian-like geometry on the space of measures. These results suggest possible applications to kinetic PDEs. This talk is based on arXiv:2502.15665, in collaboration with G. Brigati (ISTA) and J. Maas (ISTA), and ongoing work with G. Brigati (ISTA), G. Carlier (CEREMADE, Paris Dauphine-PSL), and J. Dolbeault (CEREMADE, Paris Dauphine University-PSL).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

I will report on a joint work in progress with I. Biswas, E. Colombo and A. Ghigi in which we describe a canonical projective structure on every etale double cover of a curve $C$ of genus $g>6$. This projective structure is the restriction to the second infinitesimal neighborhood of the diagonal in $C imes C$ of the second fundamental form of the Prym map. It gives a section of the space of projective structures on $mathbb{R}_g$ and the $(0,1)$-component of the differential of this section is proven to be the pullback via the Prym map of the Kaehler form on $A_{g-1}$. This generalises a previous result obtained in collaboration with Biswas, Colombo, and Pirola in the case of $M_g$, showing the existence of a canonical projective structure on every curve of genus $g>3$, obtained by the second fundamental form of the Torelli map.
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

La dimensione di Kodaira è un invariante bimeromorfo discreto fondamentale per le varietà kähleriane compatte. Per il teorema di uniformizzazione, due superfici di Riemann compatte hanno la stessa dimensione di Kodaira se e solo se ammettono rivestimenti universali biolomorfi. Questa equivalenza viene completamente meno in dimensione superiore, ma ciononostante, grazie alla classificazione di Enriques-Kodaira, è possibile dimostrare che due superfici kähleriane compatte aventi rivestimenti universali biolomorfi hanno necessariamente la stessa dimensione di Kodaira. Inoltre, nel 1996 H. Tsuji ha dimostrato che, in qualsiasi dimensione, ogni quoziente compatto non ramificato del rivestimento universale di una varietà proiettiva di tipo generale (ossia con dimensione di Kodaira massimale) deve essere anch'esso di tipo generale. In questo seminario presenteremo un lavoro in corso di Anna Choblet, mia studentessa di dottorato (in congiuntamente con B. Claudon), che mira a dimostrare come il risultato di Tsuji resti valido anche in dimensione tre nel caso di dimensione di Kodaira zero.

What do Pokémon, chemistry, and algebraic geometry have in common? The answer is not just that they are all fun, nor that they all have TV series about them. Rather, they are all concerned with collecting and classifying things! In this talk, I will introduce the idea of classification in algebraic geometry and explain how it can be interpreted depending on what one wants to classify, illustrating this with several examples. Finally, I will briefly describe my own contributions to the classification of Fano fourfolds. N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

In recent years, the analysis of topologically ordered ground states of 2d quantum lattice systems in the thermodynamic limit has been developed to provide a rigorous invariant of gapped phases. The methodology is based on the approach of Doplicher, Haag, and Roberts in the context of algebraic quantum field theory, and derives a braided C*-tensor category which represents the anyonic excitations above the ground state and the fusion and braiding among them.

I will report on a joint work with S. Coughlan, R. Pardini and S. Rollenske. The investigation of (minimal) surfaces of general type with low invariants and their moduli spaces started with the work of Castelnuovo and Enriques and during the last decades of the 20th century many authors continued studying these surfaces. Nowadays Gieseker's moduli space of canonical models of surfaces of general type with K^2 and χ fixed is known to admit a modular compactification, namely the KSBA moduli space, obtained considering stable surfaces. The structure of such moduli space is not completely known and studying stable surfaces with low invariants is a starting point to see concrete examples and studying its properties. The aim of this talk is to give a description of the KSBA moduli space of stable surfaces with K^2=1 and χ=3, showing different ways to construct boundary components. After an overview of the know components, I will focus on the case of 2-Gorenstein surfaces, (with particular attention to the surfaces obtained by gluing two irreducible surfaces) and to the case of normal surfaces having rational singularities.
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

There are classical results showing that if a negatively closed manifold has its horospheres of constant mean curvature then it is locally symmetric. Here we shall present a rigidity result involving the intrinsic Riemannian structure of these horospheres. More precisely if one of them is flat than the closed manifold is locally real hyperbolic. Several questions arose from the approach that we will discuss, in particular concerning replacing the hypothesis on the curvature by the assumption that there is no conjugate points. This is based on a joint work with G. Courtois and S. Hersonsky.