Martedì 25 Novembre 2025, 14:30 - 15:30, Aula Tricerri 

Sara Cebellan Debon (Università di Murcia)

Titolo: From rational groups to groups with quadratic values on the character table

Abstract: Rational groups have been an object of interest in representation theory for many years. Recall that a group is said to be rational if its character table is rational valued. There are some families of groups that arise as generalizations of rational groups, for instance, semi-rational or character quadratic groups. In these examples, the values of each row in the character table are contained in some quadratic field, not necessarily the same. In the case where the field is exactly the same we say that the group is quadratic.

In this seminar I will talk about these groups (when they are also solvable) and present some related problems.

Martedì 8 Luglio 2025, Aula Tricerri 

Marco Miceli (SISSA)

Titolo: The McKay Correspondence as an equivalence of Derived Categories

Abstract: The classical McKay correspondence relates the representation theory of a finite group G in SL(2,C) to the geometry of a resolution of the correspondent Kleinian singularity.

Given a quasiprojective nonsingular complex variety M and a finite group of automorphisms G, we can extend, under suitable hypotheses, this construction to an equivalence between the G-equivariant derived category of M and the the derived category of a crepant resolution of M/G, obtained via G-Hilbert schemes.

Since these conditions are always satisfied in dimension ≤ 3, we prove along the way a conjecture by Nakamura.

In this talk, we describe the projective case following the paper "The McKay correspondence as an equivalence of Derived Categories" by Bridgeland, King, and Reid, and defining many of the objects above.

Martedì 10 giugno 2025, Aula Tricerri

Giacomo Gavelli (Università di Tubinga)

Titolo: Laplace and Length Spectra  for closed hyperbolic surfaces of large volume

Abstract: Two of the most studied invariants of a closed hyperbolic surface are the Laplace and the Length spectrum. Even though a complete description of these spectra is difficult to obtain in general, one can rely on good asymptotics for counting the number of eigenvalues of the Laplacian (respectively, length of closed geodesics) in a certain window, as this window expands to infinity. Changing the perspective, one can fix an interval and ask for asymptotics on the number of eigenvalues (respectively, length of closed geodesics) in that interval as the volume grows to infinity. This bears the question on how to obtain sequences of surfaces with increasing volume which give good asymptotics. In this talk I will introduce two such notions, namely Benjamini-Schramm and Plancherel convergence, and discuss their relation.

This talk is based on a joint work with C. Kamp.

Giovedì 10 Aprile 2025,
Aula 201 - Plesso Didattico Morgagni

Letizia Branca (Università degli Studi di Milano)

Titolo: Canonical Riemannian metrics in dimension n ≥ 4

Abstract: Given a Riemannian manifold (M, g), the Riemann curvature tensor encodes all the information about the curvature. It is well known that in dimension n ≥ 3, it splits as a sum of three different parts. Many of the so-called canonical metrics arise as critical points of suitable Lp-norms of the components of the curvature tensor. Throughout the talk, we will provide some background on Riemannian manifolds and discuss some known results and open problems concerning Riemannian functionals and critical metrics in dimension 4 and beyond.
The last part of the talk is based on a joint work with G. Catino, D. Dameno and P. Mastrolia.

Giovedì 13 Marzo 2025, Aula Tricerri

Marco Vergani (Università degli Studi di Firenze)

Titolo: Rationality of finite groups

Abstract: The main topic of this seminar is families of groups that have a characterization of their integral central units inside the rational group algebra. Using representation theory it is possible to consider groups as acting over vector spaces in a natural way, relating the irreducibile actions to the field generated by the trace of the representation. Those fields give us a lot of information about the group itself. In this seminar we will focus on groups with field of values that are quadratic extensions of the rationals and we will define tools that allow us to detect how far the group is from a “rational” action.

Lunedì 14 Ottobre 2024, Aula Tricerri,

Emanuele Di Bella (Università degli studi di Trento)

Titolo: Classificazione delle orbite della rappresentazione naturale di SL(8,C) su ⋀^4 C

Abstract: Se si considera l'azione di un gruppo algebrico complesso su un generico spazio vettoriale, spesso è molto complicato riuscire a descriverne le orbite. Tuttavia, una notevole eccezione sono le $\theta$-rappresentazioni, definite da Vinberg negli anni Settanta. Tali rappresentazioni sono costruite da un'algebra di Lie graduata semisemplice, così che, al fine di classificare le orbite della rappresentazione, si possano utilizzare le varie tecniche derivanti dalla teoria di Lie. Un esempio noto è la rappresentazione naturale di SL$(8,\mathbb{C})$ su $\bigwedge^4 \mathbb{C}^8$. In particolare, la somma diretta SL$(8,\mathbb{C}) \oplus \bigwedge^4 \mathbb{C}^8$ ha una struttura di algebra di Lie 2-graduata. Le orbite complesse di tale rappresentazione sono già state classificate negli anni Ottanta da Antonyan, nel suo paper 'Classification of Four-Vectors of an Eight-Dimensional Space'. In questo seminario, l'obiettivo è quello di descrivere dettagliatamente le strutture algebriche coinvolte e l'approccio di Antonyan, per poi analizzare il problema di classificare le orbite reali.

Mercoledì 2 Ottobre 2024 Aula Tricerri

Lucrezia Prosperi (Università di Galway)

Titolo: The double coset zeta function of some p-groups of maximal class

Abstract: The study of zeta functions has long been a central theme in number theory and group theory, providing deep insights into the structure and properties of various algebraic objects. Starting with the subgroup zeta function, a possible question is what happens if we replace the number of cosets with the number of double cosets. This question leads to the definition of the double coset zeta function. The aim is to study the double coset zeta function of a given group or of some (families of) finite groups. At first we will recall some known results on the subgroup zeta function, in particular we will analyse two examples: the direct product of d infinite cyclic groups, where d is a natural number, and the discrete Heisenberg group. Then, after an introduction to double cosets, we will give a way to define the double coset zeta function and we will study some examples in detail. In particular, we will analyse the dihedral groups of order powers of two, the semidihedral groups and the quaternion groups of the same order. Moreover, we will move to the pro-2-dihedral group and then we will generalise the result for the pro-p-dihedral groups with a general prime p.

Lunedì 1 Luglio 2024, 

Aula Anfiteatro, Ex Farmacologia (viale Morgagni 65)

Giulio Binosi (Università degli studi di Firenze)

Titolo: Harmonicity in Slice Analysis: Almansi decomposition and Fueter theorem for several hypercomplex variables

Abstract: We broaden some definitions and give new results about the theory of slice functions of several variables in a general *-algebra. We investigate partial slice properties of slice functions, characterizing the sets of slice, slice regular and circular functions w.r.t. specific variables. We introduce the notions of partial spherical value and derivative for functions of several variables, that extend those of one variable, recovering some of their properties and discovering new ones. Focusing on quaternions and Clifford algebras, we derive explicit formulas for the iteration of the Laplacian applied to slice regular functions and to their spherical derivative. The formulas enlighten harmonic and polyharmonic properties, which depend on the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables, where the components are given explicitly through partial spherical derivatives. We find some applications in the quaternionic setting, such as mean value and Poisson formulas. Furthermore, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of Fueter and Fueter-Sce theorems in the several variables context. We then establish that regular polynomials of suciently low degree are the unique slice regular functions within the kernel of the Laplacian iteration, provided its power is less than the Sce exponent, which turns to be a critic index. Finally, time permitting, we analyze some relations between slice and Dunkl analysis.

Giovedì 27 Giugno 2024, Aula Tricerri

Sem K. Miller (University of California, Santa Cruz)

Titolo: Endotrivial Complexes

Abstract: Let G be a finite group and k be a field of characteristic p > 0. Endotrivial complexes are the invertible objects of the tensor triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation $kG$-modules. These chain complexes are connected to a variety of other well-known structures in group and modular representation theory, including splendid Rickard equivalences, endopermutation and endotrivial modules, and the trivial source ring.

In this talk, we will motivate and build up to the definition of these chain complexes, as well as discuss some of the aforementioned connections to the other objects of interest. We will also introduce a relative notion of endotriviality, analogous to Lassueur's construction of relatively endotrivial modules. If time permits, we will finally describe how we use both endotriviality and relative endotriviality to completely classify all endotrivial complexes!

Giovedì 14 Marzo 2024, Aula Tricerri

Matteo Tarocchi (Università degli Studi di Milano-Bicocca)

Titolo: Groups acting on fractals and graph approximations

Abstract: This talk is about rearrangement groups: a family of groups of certain homeomorphisms of self-similar fractals that permute finitely many self-similar pieces.
Homeomorphism groups of fractals are often gargantuan (uncountable) topological groups that are hard to study algebraically, and many of them are still mysterious. While only being countable, rearrangement groups are often rich enough to be dense in the homeomorphism groups of the fractals on which they act, yet they are manageable enough that algorithmic problems on them are treatable, and they have plenty of interesting algebraic properties of their own. Being aimed at a general audience in mathematics and dealing with an intersection of topology, algebra and computability theory, this broad talk will focus on a general description of rearrangements and, as an application, will show how to tackle their conjugacy problem with a graphical approach.

Martedì 12 Dicembre 2023, Aula Tricerri

Michela Ascolese (Università degli Studi di Firenze)

Titolo: Polyominoes and Tilings