Seminari periodici DIPARTIMENTO DI MATEMATICA
Topics in Mathematics 2019/2020

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18
Mar

2020
Davide Bolognini
Around Simon's Conjecture

seminario di algebra e geometria

Simon's conjecture is an interesting open problem in Combinatorics. In the first part of the seminar, we provide all the necessary notions to formulate it. Recently, several authors gave new improvements on this topic. In the second part, we present some of these results, pointing out the main ideas and giving some new questions arising from them.

12
Feb

2020
Matteo Franca
Una breve introduzione alla teoria di Melnikov e al chaos in sistemi dinamici non-autonomi a tempo continuo

seminario di analisi matematica

In questo seminario inizierò parlando brevemente di alcuni meccanismi che stanno alla base di particolari fenomeni caotici, per poi concentrarmi sulla teoria di Melnikov. In particolare analizzerò il caso di un sistema dinamico a tempo continuo autonomo che presenta una traiettoria omoclina (che quindi converge ad un punto critico sia nel passato che nel futuro), soggetto ad una perturbazione non-autonoma. La teoria di Melnikov fornisce condizioni che garantiscono la persistenza dell’omoclina e la nascita di fenomeni caotici. Il modello più noto per questa tipologia di fenomeni è il pendolo (non-lineare) perturbato. Si vedranno brevemente estensioni al caso multidimensionale e a quello discontinuo (piecewise smooth) che trova applicazione nella modellizzazione dei rimbalzi o dell’attrito strisciante.

18
Dic

2019
Berardo Ruffini
Un invito alle disuguaglianze di riarrangiamento

seminario di analisi matematica

Nella prima parte del seminario parlerò di varie disuguaglianze geometriche di tipo isoperimetrico. Partendo dalla disuguaglianza isoperimetrica classica (quella che ci dice perché le bolle di sapone all'equilibrio devono essere sferiche), menzionerò alcune sue controparti funzionali e come si possono sfruttare per dimostrare disuguaglianze (di nuovo) geometriche ma definite tramite EDP. Nella seconda parte del seminario ci concentreremo su una delle svariate evoluzioni delle disuguaglianze di riarrangiamento: lo studio della loro rigidità quantitativa. Oltre a spiegare questo concetto, e citare alcuni dei casi più celebri risolti recentemente, vedremo alcune applicazioni (ancor più recenti) di tali risultati.

26
Nov

2019
Alessio Savini
Euler number for measurable cocycles of surface groups

seminario di algebra e geometria

A quite useful philosophy in mathematics is to use the sharpness of an inequality regarding the "shape" of a topological space in order to detect a precise geometry on it: more precisely, the maximal value of the inequality usually allows to identify a specific geometric structure. Think for instance either to the applications of arithmetic/geometric mean inequality or to the isoperimetric inequality on the plane. Something similar happens in the world of Zimmer's cocycle theory. In this seminar we are going to focus our attention on Zimmer's cocycles associated to the fundamental group a surface S with genus bigger than or equal to 2. If such a measurable cocycle admits a (generalized) boundary map, one can define the notion of Euler number. The latter well behaves along cohomology classes and its absolute value is bounded by the modulus of the Euler characteristic of S. Remarkably the maximal value is attained if and only if the cocycle is cohomologous to a hyperbolization. The first part of the talk will be a gentle introduction to measurable cocycles and boundary theory. Then, we are going to introduce the orientation cocycle on the circle. Finally we will define the Euler number of a measurable cocycle and we will discuss its rigidity property. This is a joint work with Marco Moraschini.

11
Nov

2019
Peter Donelan
Singularities and Robotics
The idea of singularity is found in many parts of mathematics, capturing the idea of a position where some regular behavior breaks down. A standard situation is in linear algebra where for a linear transformation or matrix rank deficiency corresponds to singularity. This provides a basic model for other settings, especially for differentiable functions between Euclidean spaces. Results of Marston Morse (the Morse Lemma) and Hassler Whitney (stable singularities of mappings from the plane to the plane) led to pioneering work in differential topology by René Thom, whose interest in biological morphogenesis gave rise to Elementary Catastrophe Theory and a wide interest in mathematical models of singularity. The foundations of singularity theory were developed, in which the concepts of transversality and stratification played an important role. In the first part of the seminar I will outline some of this history and ideas from singularity theory. In robotics, kinematic mappings relate inputs, outputs and constraints. The impact of singularities on robotic control systems was recognised in the 1960s. Subsequent interest in the variety of ways that singular phenomena occur in robot kinematics has led to a large literature on the subject. In the second part of the seminar, I will discuss some of my research on kinematic singularities.