Giovedì 27 Giugno 2024, Aula Tricerri, 16-17
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!
Lunedì 1 Luglio 2024,
Aula Anfiteatro Ex Farmacologia (viale Morgagni 65), 14:30-15:30
Giulio Binosi (Università 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.
Seminars by PhD students: Informal Events @ Dini (S.P.I.E.Dini)
Benvenuti sulla pagina di S.P.I.E.Dini. Questa è il sito del ciclo di seminari tenuti dai dottorandi di Matematica dell'UniFi che ha il proposito di condividere e confrontare il lavoro sviluppato nei diversi ambiti della matematica nel dipartimento. Lo scopo di questi seminari, di natura informale, è presentare una visione divulgativa e accessibile degli argomenti studiati dai dottorandi.
Welcome to the webpage of S.P.I.E.Dini. This is the website dedicated to the seminars held from Math PhD students of UniFi with the aim to share our work in the different areas of mathematics to the general public. The purpuse of those informal seminars is to present an easy-to-access version of the arguments studied during the PhD.
Questions? Remarks? Contact: marco.vergani@unifi.it.
Martedì 26 Aprile 2022, 16.30-17.30 aula Tricerri
Angel Garcia Blazquez (Murcia)
Titolo: The Isomorphism Problem for Rational Group Algebras of Metacyclic Groups
Abstract: The Isomorphism Problem for group rings with coefficients in a ring R asks whether the isomorphism type of a group G is determined by its group ring RG. We will introduce this problem in general and we will discuss the particular case of rational group rings of metacyclic groups.
Martedì 31 Maggio 2022, 16.30-17.30 aula Tricerri
Giada Volpato (Firenze)
Titolo: Representation theory of the symmetric groups
Abstract: Representations of the symmetric groups are particularly interesting because there's a nice combinatorics theory that gives us several tools to work with. We will talk about these tools and we will introduce some research problems on this topic.
Martedì 17 Maggio 2022, 16.30-17.30 aula Tricerri
Lapo Cioni (Firenze)
Titolo: Preimages of sorting algorithms
Abstract: Bubblesort, Queuesort and Stacksort are well known sorting algorithms, which have interesting properties from a combinatorial point of view. We will talk about some of those properties, focusing in particular on the problem of studying the preimages of the functions associated to the sorting algorithms.
Giovedì 9 Giugno 2022, 14.30-15.30 aula Tricerri
Camilla Brizzi (Firenze)
Titolo: The optimal transport problem: the classical and the supremal setting
Abstract: In my talk, I will present the problem of Optimal Transport, the first formulation by Monge and then the relaxed version due to Kantorovich, trying to explain the main properties and results. I will then mention some fields of research, with particular attention at the formulation of OT in a "supremal" setting.
Martedì 28 Giugno 2022, 16.30-17.30 aula Tricerri
Alice Dell'Arciprete (University of East Anglia, Norwich UK)
Titolo: Decomposition numbers of the symmetric group and related algebras
Abstract: Representations of the symmetric group are quite well understood, mainly thanks to James who developed the use of combinatorial tools, such as diagrams, tableaux and abacuses. This constructive approach can be generalised to give techniques for studying representations of related algebras including the Ariki-Koike algebras.
In particular, we will talk about the decomposition numbers of the symmetric group and sketch how we can generalise some results for the Ariki-Koike algebras.
Martedì 12 Luglio 2022, 16.30-17.30 aula Tricerri
Luca Briani (Pisa)
Titolo: Is there an optimal shape?
Abstract: How to construct a rod of maximum rigidity? Which body moves in a fluid with the least resistance? Among sets of given area, which has the smallest perimeter? In a shape optimization problem, the objective is to deform and modify the shape of a given object to minimize (or maximize) a cost function.
From a mathematical point of view, the most intriguing feature is that the competing objects are shapes (i.e. subsets of R^N) rather than functions. We will discuss some classical problems (some of which are still open) and introduce the mathematical framework that can be used to obtain existence results.
Martedì 27 Settembre 2022, 14.30-16.00 aula Tricerri
Ettore Teixeira Turatti (Firenze)
Titolo: Multilinear spectral theory
Abstract: The spectral theory of matrices is a classical concept that has many applications in image processing, signal processing, biodiversity estimation, etc. The extended notion of eigenvectors to higher order tensors has been introduced recently in 2005, we will study this concept and understand its similarities and differences to the matrix case.
Martedì 11 Ottobre 2022, 14.30-16.00 aula Tricerri
Lorenzo Putignano (Firenze)
Titolo: Something about Representation Theory
Abstract: In this talk I will introduce basic concepts and ideas about representation and character theory of finite groups. My focus will be in particular on the case of symmetric groups where combinatorics plays a fundamental role. In the final part I briefly present the concept of centralizer algebra arised in order to attack important conjectures in representation theory. Again I will put the attention on symmetric groups touching the heart of my research project and showing first improvements on it.
Martedì 25 Ottobre 2022, 14.30-16.00 aula Tricerri
Lorenzo Sacco (Firenze)
Titolo: Interpolazione di flussi di dati 3D tramite spline quintiche PH ed applicazione alla pianificazione di traiettorie.
Martedì 8 Novembre 2022, 16.30-18.00 aula Tricerri
Bernardo Nannini (Firenze)
Titolo: Induzione matematica e catene di inferenze logiche: un'analisi cognitivo-didattica.
Martedì 22 Novembre 2022, 14.30-16.00 aula Tricerri
Niccolò Di Marco (Firenze)
Titolo: The null label problem and its relation to the 2-intersection graph
Abstract: A 3-uniform hypergraph H consists of a set V of vertices, and a subset of triples of V, called set of edges E. Let a null labeling be an assignment of +1 or -1 to the triples such that each vertex has a signed degree equal to zero. If a null labeling exists, we say that the hypergraph is null. Assumed as necessary condition the degree of every vertex of H to be even, the Null Labeling Problem consists in determining whether H has a null labeling. It is remarkable that null hypergraphs arise considering two hypergraphs with the same degree sequence. In particular, the symmetric differences of these two hypergraphs give a new hypergraph that is null. From a discrete tomography point of view, null hypergraphs arise from matrices with the same projections, i.e. solutions of the same reconstruction problem. Therefore they allow modeling of switching components, a very used notion in this field of research.
Although the problem is NP-complete, the subclasses where the problem turns out to be polynomially solvable are of interest. We defined the notion of 2-intersection graph related to a 3-uniform hypergraph and we prove that if it is Hamiltonian then the related 3-hypergraph has a null labeling. Then we aimed to deepen the knowledge of the structural properties of 2-intersection graphs. Going into details, we studied when, given a graph G, it is possible to find a 3-hypergraph such that its 2-intersection graph is G. If it is possible, we say that G is reconstructable or equivalently, it has the 2-intersection property. It’s easy to see that the question is relatively straightforward for some classes. However, using some suitable gadgets, we proved that the problem in its general form is NP-Complete.
Martedì 13 Dicembre 2022, 14.30-16.00 aula Tricerri
Corentin Henriet (IRIF, Université de Paris)
Titolo: A swim with fighting fish
Abstract: In this talk, I will propose you an excursion into the world of bijective combinatorics. This is an area of mathematics where we find a variety of discrete objects arising in other mathematical domains, and try to establish bijective links between them in order to understand better their structure and their relations. The central objects of my PhD are an exotic generalization of parallelogram polyominoes called fighting fish : I will present them to you, draw their connections with planar maps, intervals in a lattice of Dyck paths (and maybe more, if time allows), and what we can learn from that.