Seminars by PhD students: Informal Events @ Dini (S.P.I.E.Dini)
Seminars by PhD students: Informal Events @ Dini (S.P.I.E.Dini)
Benvenuti sulla pagina di S.P.I.E.Dini.
Questo è il sito del ciclo di seminari organizzati dai dottorandi di Matematica dell'Università di Firenze che ha il proposito di condividere e confrontare il lavoro sviluppato nei diversi ambiti della matematica, sia all'interno del nostro dipartimento, sia presso altre Università. Lo scopo di questi seminari, di natura informale, è presentare una visione divulgativa e accessibile degli argomenti studiati dai dottorandi.
Per rimanere aggiornati sui prossimi eventi, iscrivetevi al nostro gruppo o mandate una e-mail a spiedini-seminars-group+subscribe@unifi.it
Welcome to the S.P.I.E.Dini webpage.
This is the website for the seminar series organized by the Mathematics PhD students at the University of Florence, aimed at sharing and discussing research carried out in various areas of mathematics, both within our department and at other universities. The goal of these informal seminars is to provide an accessible and engaging overview of the topics studied by the PhD students.
To receive updates on next events, join our group or send an email to spiedini-seminars-group+subscribe@unifi.it
Titolo: Energetic behaviour of dislocations of semicoherent interfaces
Abstract: A dislocation is a one-dimensional singularity in an elastic deformation that encodes a mismatch in the underlying displacement field.
Mathematically, it arises as a defect set where strain incompatibility concentrates, and it represents a fundamental example of line defects in continuum models of solids.
In the applied analysis literature, several approaches have been developed; here we focus on the one introduced by Lauteri and Luckhaus in their study of the interaction between two crystalline lattices rotated by an angle [Lauteri–Luckhaus, preprint (2016)].
In their seminal work, they established the scaling predicted by Read and Shockley in the 1950s [Read–Shockley, Phys. Rev. 1950], without assumptions on the defect distribution and introducing several deep ideas.
We propose a Lauteri–Luckhaus–type energy tailored to semi-coherent interfaces.
In this setting, the analogue of the Read–Shockley law is given by the classical van der Merwe model [van der Merwe, Proc. Phys. Soc. 1963], whose energetic behavior we rigorously confirm.
We also introduce a linearized model that enables us to prove the analogous logarithmic scaling in higher dimensions through a slicing procedure.
Based on joint work with Emanuele Spadaro (Sapienza).
Questions? Remarks? Contact: federico.thiella@unifi.it matteo.calosci@unifi.it
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.
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.
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.
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.
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.
Titolo: Hadamard ranks and Tropical Geometry
Abstract: Given a complex projective variety X, the X-rank of a point p is a measure of the complexity of decomposing p as a sum of points in X with as few summands as possible. An example of this problem is writing a symmetric matrix as a sum of (as few as possible) rank 1 symmetric matrices. The variety X in this example is a Veronese variety. In general, the points of X-rank less or equal to a fixed positive integer r form the r-th secant variety of X. It is the smallest variety containing the union of all (r-1)-dimensional secant planes to X. Motivated by these important objects in classical algebraic geometry, we shift from additive to multiplicative decomposition by introducing Hadamard X-ranks. The task is to decompose a point as a coordinate-wise product of points in X (Hadamard product) with as few factors as possible. In this context, the r-th Hadamard power of X plays the role of its r-th secant variety. In the first part of the seminar, I will give an overview of these topics. Then, I will focus on generic Hadamard X-ranks. We study their finiteness using the combinatorial tools provided by Tropical Geometry. This is joint work in progress with Alessandro Oneto and Guido Montúfar.
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.
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.
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.
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.
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.
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.
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.
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.
Titolo: Interpolazione di flussi di dati 3D tramite spline quintiche PH ed applicazione alla pianificazione di traiettorie.
Titolo: Induzione matematica e catene di inferenze logiche: un'analisi cognitivo-didattica.
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.
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.