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Giovedì 4 Giugno 2020 ore 16:00
- Carlo Orrieri, Università di Trento
Topics in PDEs: a probability point of view
In this seminar I will discuss some classical probabilistic techniques to study elliptic and parabolic PDEs.
I will introduce basic instruments from stochastic calculus which turn out to be useful to construct/represent solutions to the Dirichlet problem, the heat equation and some semilinear equations.
The main goal of the presentation is to give different proofs and/or new insights to what is usually presented in an elementary PDE course. The seminar is intended to be accessible to students.
Giovedì 26 settembre 2019
Pablo Spiga, Universita degli Studi di Milano-Bicocca
How vertex-stabilizers grow?
Here we are interested in highly symmetric graphs. (All basic terminology will be given during the talk.) There are various natural ways to “measure” the degree of symmetry of a graph and, in this talk, we look at two possibilities. First, we consider graphs Γ having a group of automorphisms acting transitively on the paths of length s ≥ 1, starting at a given vertex. The larger the value of s is, the more symmetric the graph will be. However, we show that large values of s impose severe restrictions on the structure of Γ and on the size of the stabilizer of a vertex of Γ. This will lead us to the second perspective. We take the size of the stabilizer of a vertex of Γ as a measure of the transitivity. This measure is somehow unbiased among the
graphs having the same number of vertices. Again we present some results showing, in some very specific cases, that nature is not as diverse as one might expect: graphs have either rather small vertex stabilizers or they can be classified. Finally we give some applications of these investigations: to the enumeration problem of symmetric graphs and to the problem of creating a database of small symmetric graphs.
Martedì 29 ottobre 2019
- Giuseppe Buttazzo, Università di Pisa
Optimal reinforcing networks for elastic structures
We study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a connected one-dimensional structure. The problem consists in finding the optimal configuration for the stiffeners, the problem is then a shape optimization problem, where the admissible competing shapes are one-dimensional networks of prescribed length. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. The case where the connectedness assumption is removed is also presented. Some numerical simulations are shown to confirm the overlapping phenomenon and to illustrate the complexity of the optimal structures when their total length becomes large.
Giovedì 14 novembre 2019
- Michele Piana, Università di Genova
The many scales of oncological data: a computational perspective
This talk will describe multi-scale approaches to the mathematical modeling of oncological data provided by different experimental modalities. A speciﬁc focus will be devoted to the many computational aspects concerned with the numerical reduction of these models. Applications will involve the use of hybrid imaging methods for the assessment of leukemic patients, the investigation of glucose metabolism in cancer tissues and the simulation of a speciﬁc transition in cancer cells by means of molecular interaction maps.
Giovedì 12 dicembre 2019
- Alessandro Fonda, Università degli Studi di Trieste
On the higher dimensional Poinkaré-Birkhoff theorem for Hamiltonian flows
In a joint paper with Antonio J. Urena, I have recently obtained some higher dimensional versions of the Poincaré - Birkhoff theorem for Hamiltonian flows. This result has been then further extended in different directions, proving that multiple periodic solutions exist in a variety of situations, including systems with sublinear or superlinear growth, with singular or periodic nonlinearities, and for perturbations of completely integrable systems. Even for some infinite-dimensional Hamiltonian systems, the same theorem together with a limit procedure has been used to prove the existence of periodic solutions. The aim of the talk is to provide an overview on these results with some hints for future developments.
Giovedì 13 febbraio 2020
- Antonio Lerario, SISSA, Trieste
Probabilistic Algebraic Geometry
In this seminar I will discuss a modern point of view on Real Algebraic Geometry, which introduces ideas from Probability for approaching classical problems. The main point of this approach is the shift from the word "generic" of classical Algebraic Geometry to the word "random". This brings many interesting subjects into the picture: convex geometry, measure theory, representation theory, asymptotic analysis...
Giovedì 23 aprile 2020 ore 16:00
- José Iovino, University of Texas
Tao's Concept of Metastability as a Medium Connecting Diverse Areas of Mathematics
The concept of metastable convergence was introduced by Terry Tao as a tool for his 2008 ergodic theorem. It turns out that this concept arises naturally in many other areas of mathematics, from analysis, to topology, to theoretic computation, to logic, and connects them in unexpected ways. I will give a brief overlook. The talk is intended to be accessible to students.