Topics in Mathematics 2017/2018

I will use time dependent Schroedinger equation on the two dimensional torus as a ``laboratory" for describing control and observability for time dependent PDE. The plan of the lectures is 1. General description of the problem and Lions's argument for the equivalence of observability and control. 2. Introduction to semiclassical methods and defect measures. 3. Control by non-empty open sets on tori (a proof of now classical results of Haraux, Jaffard and Komornik using defect measures). 4. A review of modern advances: control by sets of positive Lebesgue measure (Bourgain--Burq--Z, Burq--Z), control on discs (Anantharaman--Leautaud--Macia), control by non-empty open sets on hyperbolic surfaces (Bourgain--Dyatlov, Dyatlov--Jin, Jin).

I will use time dependent Schroedinger equation on the two dimensional torus as a ``laboratory" for describing control and observability for time dependent PDE. The plan of the lectures is 1. General description of the problem and Lions's argument for the equivalence of observability and control. 2. Introduction to semiclassical methods and defect measures. 3. Control by non-empty open sets on tori (a proof of now classical results of Haraux, Jaffard and Komornik using defect measures). 4. A review of modern advances: control by sets of positive Lebesgue measure (Bourgain--Burq--Z, Burq--Z), control on discs (Anantharaman--Leautaud--Macia), control by non-empty open sets on hyperbolic surfaces (Bourgain--Dyatlov, Dyatlov--Jin, Jin).

In this talk we will illustrate a new line of research concerning the persistent homology of regular functions from a closed manifold to R^2. In particular, we will describe the phenomenon of monodromy for 2D persistence diagrams, and a recent mathematical model based on the concepts of extended Pareto grid and coherent transport of matchings. Some new theoretical results and open problems will be presented.

In these lectures the basic notions (such as irreducibility, complete reducibility, indecomposability, induction…) of representation theory are introduced. We will develop the theory following basic examples of finite groups, algebraic groups, associative algebras, Lie algebras and quivers, focusing on similarities and discrepancies.

In this talk we will present a metric approach to topological data analysis that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. In particular, we will illustrate the concept of G-equivariant non-expansive operator and how it can be used to build G-invariant persistent homology. The exposition will remain at an elementary and non-technical level, limiting itself to a description of the main ideas in our mathematical setting.

In these lectures the basic notions (such as irreducibility, complete reducibility, indecomposability, induction…) of representation theory are introduced. We will develop the theory following basic examples of finite groups, algebraic groups, associative algebras, Lie algebras and quivers, focusing on similarities and discrepancies.

The maximal regularity in L^p (1<p<∞) for the solution of a linear abstract Cauchy problem (1) u'(t) + Lu(t) = f(t), (2) u(0) = 0 where the unknown function u and the given function f are defined on [0,T] with values in a Banach space X, is the requirement that for any f∈L^p(0,T ; X) the Cauchy problem (1) - (2) has a unique solution and that u' and Au belong to L^p(0,T; X) and depend continuously on f in L^p(0,T; X). This problem can be stated in a more abstract form as the problem of solving the equation Au + Bu = f in the space Y = L^p(0,T; X) for appropriate operators A and B acting in Y. In these two seminars I will speak of a result that gives conditions on A, B and X to ensure the bounded invertibility of the operator A+B, and hence the maximal regularity for the solutions of the Cauchy problem.

The maximal regularity in L^p (1<p<∞) for the solution of a linear abstract Cauchy problem (1) u'(t) + Lu(t) = f(t), (2) u(0) = 0 where the unknown function u and the given function f are defined on [0,T] with values in a Banach space X, is the requirement that for any f∈L^p(0,T ; X) the Cauchy problem (1) - (2) has a unique solution and that u' and Au belong to L^p(0,T; X) and depend continuously on f in L^p(0,T; X). This problem can be stated in a more abstract form as the problem of solving the equation Au + Bu = f in the space Y = L^p(0,T; X) for appropriate operators A and B acting in Y. In these two seminars I will speak of a result that gives conditions on A, B and X to ensure the bounded invertibility of the operator A+B, and hence the maximal regularity for the solutions of the Cauchy problem.

Le leggi naturali e i dispositivi artificiali "lineari" il cui funzionamento non varia col tempo sono esprimibili in termini di convoluzione, quindi le trasformate di Fourier, che traducono le convoluzioni in prodotti, hanno una particolare utilità pratica e concettuale. Una particolare incarnazione della trasformata di Fourier è quella che associa a una successione a=(a_n: n=0,...) (un segnale discreto) la funzione olomorfa f=Za avente quella come successione dei coefficienti (in combinatoria Za è la "funzione generatrice dalla successione"). La legge naturale o il dispositivo artificiale possono essere quindi compresi in termini di operazioni algebriche tra funzioni olomorfe, se del caso introducendo operatori di proiezione. Molte proprietà delle funzioni olomorfe diventano proprietà del sistema sotto esame. A partire dalla fine degli anni '70, queste idee sono state applicate al "controllo" di un sistema lineare. Lo scopo di questi seminari è di percorrere questo circolo di idee, tenendo a mente anche le connessioni con diverse aree della matematica pura (analisi in spazi di Hilbert di funzioni olomorfe, combinatoria, teoria dei numeri), privilegiando gli esempi (da ingegneria e matematica) e le interazioni alla teoria generale.

Le leggi naturali e i dispositivi artificiali "lineari" il cui funzionamento non varia col tempo sono esprimibili in termini di convoluzione, quindi le trasformate di Fourier, che traducono le convoluzioni in prodotti, hanno una particolare utilità pratica e concettuale. Una particolare incarnazione della trasformata di Fourier è quella che associa a una successione a=(a_n: n=0,...) (un segnale discreto) la funzione olomorfa f=Za avente quella come successione dei coefficienti (in combinatoria Za è la "funzione generatrice dalla successione"). La legge naturale o il dispositivo artificiale possono essere quindi compresi in termini di operazioni algebriche tra funzioni olomorfe, se del caso introducendo operatori di proiezione. Molte proprietà delle funzioni olomorfe diventano proprietà del sistema sotto esame. A partire dalla fine degli anni '70, queste idee sono state applicate al "controllo" di un sistema lineare. Lo scopo di questi seminari è di percorrere questo circolo di idee, tenendo a mente anche le connessioni con diverse aree della matematica pura (analisi in spazi di Hilbert di funzioni olomorfe, combinatoria, teoria dei numeri), privilegiando gli esempi (da ingegneria e matematica) e le interazioni alla teoria generale.