Titles and Abstracts
Reto Buzano: Singularities of geometric flows
Abstract:
In this talk, we will give a gentle introduction to the fascinating research area of geometric flows. We will focus mainly on Ricci flow and on mean curvature flow. We will try to explain why these flows have been so successful in recent years and why it is important to understand their singularity behaviour in order to make further progress. This talk is aimed at a general audience and will be rather non-technical.
Francesca Colasuonno: Radial solutions to second-order and fourth-order elliptic problems
Abstract:
In this talk, we will present two different radially symmetric elliptic problems set in a ball of R^N: an eigenvalue optimization fourth-order problem, and a second-order supercritical Neumann problem. From a physical point of view, in two dimensions, the first problem corresponds to building a plate, of prescribed shape and mass, out of materials with different densities -varying in a certain range of values- in such a way to minimize the lowest frequency of the body. On the other hand, the second problem corresponds to finding spatially inhomogeneous stationary solutions of the Keller-Segel system that models the biological phenomenon of chemotaxis, i.e., the oriented motion of cells towards higher concentrations of a certain chemical substance. In our case, the solutions to the problem represent the concentration of the chemical substance in the considered region.
In both cases, the radial symmetry of the problem is crucial to overcome the difficulties arising either from the high order of the equation or from the lack of compactness in the supercritical regime. Using variational methods and the shooting method for ODEs, we obtain existence, multiplicity or uniqueness, and also some qualitative properties of solutions.
This talk is based on some joint papers with Alberto Boscaggin (Torino), Eleonora Cinti (Bologna), Benedetta Noris (Amiens), and Eugenio Vecchi (Trento).
Luciano Mari: On some problems and methods in Geometric Analysis
Abstract:
This is meant to be a non-tecnical introduction to some problems in Differential Geometry and Geometric Analysis that I am currently working on. The unifying principle, foundational for Geometric Analysts, is the idea of associating, to given geometrical objects, some natural analytical ones (usually giving rise to PDEs invariantly defined on the space under consideration) in such a way that their control allows to have a better understanding of the original problem.
Nadir Murro: Some advancements in number theory and cryptography
Abstract:
In 1839, Hermite posed to Jacobi the problem of representing algebraic irrationalities by means of periodic sequences of integers in order to highlight algebraic properties and provide rational approximations. Continued fractions completely solve this problem for quadratic irrationalities, but the problem for algebraic numbers of degree greater than 2 is still open. In this context Jacobi, and subsequently Perron, introduced multidimensional continued fractions (MCFs). We will see some results regarding convergence and periodicity of the Jacobi-Perron algorithm. Moreover, we have recently introduced MCFs in the field of p-adic numbers, starting a study about their finiteness, periodicity and approximation's properties.
In the second part, we will talk about cryptography and specifically about RSA-like cryptosystems. RSA is one of the most famous and used public-key cryptosystems, however it leaks some vulnerabilities and it can not be used in broadcast scenarios. RSA-like schemes have been proposed in order to avoid these problems and they also turn to have a faster decryption procedure. We will see an RSA-like scheme based on the properties of the Pell conic and RĂ©dei rational functions.
Gianluca Paolini: On the Admissibility of a Polish Group Topology
Abstract:
In [Sh771] Shelah rediscovered an old result of Dudley on the non-admissibility of a Polish group topology on an uncountable free group. Crucial to his proof is a so-called Completeness Lemma for Polish Groups, concerning satisfaction of algebraic equations for certain sequences of group elements converging to the identity. In joint work with Shelah we prove that the techniques of [Sh771] apply to many other groups and group constructions, most notably the combinatorial construction known as graph product of groups, a construction that generalises simultaneously the notion of direct sum and free product of groups. In particular, we characterise the graph product of groups G(\Gamma, G_a) admitting a Polish group topology under the assumption that all the factor groups G_a are countable, showing that these groups are essentially direct sums of canonical building blocks in abelian group theory. In my talk I will briefly introduce Polish groups, stressing the crucial role that they play in logic and set theory, and then survey the above-mentioned results on the algebraic restrictions imposed by the admissibility of a Polish topology.
Bruno Toaldo: Initial value problems: from analysis to modeling through random processes
Abstract:
In this talk we discuss the probabilistic representation of the solution to several (integro-)differential equations. Firstly we review several aspects of the classical theory concerning the interplay between classical Cauchy problems and Markov processes. Then we present some recent results on the interplay between integro-differential equations in the time-variable and semi-Markov processes. Since these equations are very popular in applications, several modellistic aspects will be discussed (e.g. anomalous diffusive phenomena, generalized kinetic models, applications to neurophysiology and many others)