# Topics in Mathematics

## Ph.D. Program in Mathematics (academic year 2015/2016)

### Program

The 2-hours seminars are split in two parts of 45 minutes each: dissemination + in-depth.

Partial Program: seminars will be scheduled till May 2016.

### Archive

Wednesday 14th October 2015

Giuseppe Mingione Università degli studi di Parma

Alcuni elementi di teoria del potenziale non lineare

La teoria del potenziale non lineare si occupa di estendere al caso di equazioni non lineari, e possibilmente degeneri, i classici risultati della teoria lineare sulle proprietà fini delle funzioni armoniche e più in generale delle soluzioni di equazioni ellittiche e paraboliche lineari. Malgrado l'assenza di alcuni strumenti essenziali come le soluzioni fondamentali si possono tracciare dei paralleli sorprendentemente precisi. A questo proposito, farò una panoramica di risultati classici e recenti.

Tuesday 27th October 2015

Philippe Briet Université de Toulon

An introduction to the spectral theory of waveguides

In this talk I will discuss the wave propagation (acoustic, electromagnetic or quatum waves) in waveguides. I will present results about existence of trapped mode solutions which may occur in waguides due to some geometric deformations. Tow geometrical effects are discussed: bending and twisting.

Wednesday 28th October 2015

Dimitri Yafaev Université de Rennes 1

Rational approximation of singular functions

We consider functions $\omega$ on the unit circle $\bf T$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions. We find an asymptotic formula for the distance in the BMO norm between $\omega$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.

Tuesday 3rd November 2015

Piotr G. Grinevich Moscow State University, LITP

Quantum problems with special singular potentials and Hilbert spaces with indefinite scalar products

Plan of the talk. 1. Classical Kepler problem. The difference between 1/r and 1/r^2 potentials: in the second case the particle may fall to the centre. 2. Spin zero particle in quantum mechanics. Schrodinger equation and Hilbert space of states. Connection between the conservation of mass and self-adjointness of the quantum Hamiltonian. 3. Eigenfunction decomposition for Schrodinger operator with sufficiently good potentials. The spectral measure. 4. Quantum Kepler problem. For strong attractors the Schrodinger operator becomes non-self-adjoint. 5. Special class of singular 1-dimensional Schrodinger operators - spectrally meromorphic operators. The natural indefinite scalar product for this problem.

Wednesday 18th November 2015

Aristide Mingozzi Università di Bologna

Ottimizzazione combinatoria: applicazioni e metodi

I problemi di ottimizzazione possono essere suddivisi in due categorie: quelli con variabili /continue/ e quelli con variabili /discrete/, che vengono piu comunemente denominati come problemi di ottimizzazione combinatoria. Nei problemi continui la soluzione corrisponde a un insieme di numeri reali, mentre nei problemi combinatori la soluzione e un insieme di numeri interi. Molti problemi reali come, ad esempio, nei campi della logistica e della supply chain sono formulabili come problemi di ottimizzazione combinatoria la cui soluzione richiede lo sviluppo di algoritmi specifici. Nella prima parte del seminario verranno illustrati alcuni dei problemi applicativi tipici dell'ottimizzazione combinatoria quali il Traveling Salesman Problem, alcune varianti del Vehicle Routing Problem ed il Fixed Charge Transportation Problem. Nella seconda parte del seminario verranno illustrati alcuni dei metodi generali di soluzione dell'ottimizzazione combinatoria che vengono impiegati per la risoluzione dei problemi descritti nella prima parte, quali: gli algoritmi branch and bound, i metodi cutting plane, i metodi column generation e gli algoritmi branch-and-cut-and price.

Friday 18th December 2015

Alberto Romagnoni Collège de France

Mathematical and computational neuroscience: the example of the visual cortex

In the last decades it became clear that the living systems represent the new frontiers of the “exact sciences”. Mathematics and physics on one side, and biology on the other, are more and more interconnecting and taking mutual advantage from this interplay. In this talk I will discuss mathematical and computational neuroscience, by focusing on the case of the visual system and pointing out different approaches to its study. In particular, in the second part of the talk, I will discuss more specifically some recent results on functional maps in the visual cortex of higher mammals, from both the experimental and the theoretical point of view.

Wednesday 13th January 2016

Nicola Arcozzi Università di Bologna

Lo spazio di Hardy secondo la teoria dei segnali

download abstract

Friday 15th January 2016

Michele Benzi Emory University

Some Mathematical and Computational Challenges in Network Science

Network science is a rapidly growing interdisciplinary area at the intersection of mathematics, physics, computer science, and a multitude of disciplines ranging from the life sciences to the social sciences and even the humanities. Network analysis methods are now widely used in proteomics, the study of social networks (both human and animal), finance, ecology, bibliometric studies, archeology, the evolution of cities, and a host of other fields. After giving a broad overview of network science, I will introduce the audience to some of the more fundamental mathematical and computational problems arising in the analysis of networks, with an emphasis on the basic notions of centrality, communicability, and robustness. I will show how these lead to large-scale sparse numerical linear algebra computations including the solution of linear systems and eigenvalue problems, and the evaluation of functions of matrices. The talk is intended to be accessible to a broad audience.

Wednesday 20th January 2016

Isabeau Birindelli Università La Sapienza, Roma

Principal eigenvalue, maximum principle and regularity: theory and applications

In the first part, we will show how to extend the notion of principal eigenvalue to Dirichlet problems for fullynonlinear uniformly elliptic equations using the maximum principle and some a priori regularity results, in the contest of viscosity solutions. We shall describe fundamental ideas scattering from the acclaimed work of Berestycki, Nirenberg and Varadhan and more recent results in the theory of viscosity solutions./ /In the second part we will dwell on the regularity results. Proving Holder regularity using typical viscosity technics and proving Holder regularity of the gradient a' la Caffarelli via the improvement of flatness lemma, for a class of degenerate elliptic equations.

Wednesday 27th January 2016

Giovanni Gaiffi Università di Pisa

Models of reflection arrangements and related combinatorial structures

Let us consider, in a real or complex vector space V, an hyperplane arrangement A whose hyperplanes generate a (real or complex) finite reflection group W. We will focus on the combinatorial properties of the De Concini-Procesi models associated with A (for instance, if A is the braid arrangement, and therefore W = Sn, the minimal complex De Concini-Procesi model associated to it is the moduli space of stable genus 0 curves with n + 1 points). We will point out a combinatorial action of a "big" symmetric group on the boundary strata of these models and we will show how this action leads to find non recursive formulas for the computation of Betti numbers of the models and of the faces of some polytopes (nestohedra) associated to this construction.

Tuesday 2nd February 2016

Giovanni Jona-Lasinio Università La Sapienza, Roma

Mathematics and Physics: two facets of the same path?

E. Wigner spoke of the 'unreasonable effectiveness of mathematics in the natural sciences', an expression that M. Atiyah echoed forty years later arguing about 'the unreasonable effectiveness of physics in mathematics'. In the colloquium I will discuss these statements.

Tuesday 16th February 2016

Luca Migliorini Università di Bologna

Topology of algebraic varieties: a long still-ongoing story.

In the first part I will survey the study of topological properties of algebraic varieties through three crucial steps: -the seminal work of S. Lefschetz; -the introduction of Hodge theory -the extension of the theory to singular varieties and to maps. Through the theory of toric varieties, I will illustrate how the progress made at each step yields a deeper insight on the combinatorics of convex polytopes. In the second part I will discuss some of the main techniques used in the field: 1. cohomological dimension of affine varietes, 2. semisimplicity of monodromy representations, 3. the yoga of weights .

Tuesday 23rd February 2016

Simonetta Abenda Università di Bologna

Total positivity and integrable systems

Totally positive matrices were introduced in 1930 by Schoenberg in connection with the problem of estimating the number of real zeros of a polynomial. Since then, they have arised in connection with problems from different areas of pure and applied mathematics, including smnll vibrations of mechanical problems, statistics, approximation theory, combinatorics, graph theory, quantum field theories and integrability. In the seminar I shall review some classical theorems of total positivity and explain some applications to the theory of integrable systems.

Wednesday 2nd March 2016

Maurizio Falcone Università di Roma, La Sapienza

An Introduction to Optimal Control Problems and Games

The analysis of non linear optimal control problems and differential games via the solution of the Hamilton--Jacobi--Bellman or Isaacs equations was initiated by R. Bellman in the 60s. The method is based on Dynamic Programming and leads to the analysis and approximation of some non linear PDEs. The advantage of this approach is that it stands on solid mathematical grounds, the drawback is the difficulty to use it for large scale problems due to the curse of dimensionality. In the first part, I will present the basic ideas and show how the characterization of the value function can be derived via Dynamic Programming. I will also present very briefly the concept of weak solution in the viscosity sense, a notion which is very useful in this framework and has a great impact also in many other applications. The second part will be devoted to the numerical approximation of these problems, to the construction of the algorithms and to their analysis. I will present some numerical tests to show the main features (and limits) of this approach. No previous knowledge of control theory is required for this lectures.

Wednesday 16th March 2016

Andrea Bonfiglioli Università di Bologna

Mean Value Formulas for degenerate-elliptic PDOs: applications to Potential Theory

During this Topics in Mathematics Seminar, a gallery of applications is given for the Potential Theory associated with second-order PDOs L, possibly elliptic-degenerate, possessing a positive and global fundamental solution. In the absence of such a fundamental solution, it is nonetheless outlined the proof of the Harnack inequality for L.

Tuesday 22nd March 2016

Setsuro Fujiié Ritsumeikan University of Kyoto

Quantization of resonances in the semiclassical limit

This talk will be concerned with the quantum resonances, the poles of the resolvent of the Schroedinger operator. The asymptotic distribution of the resonances close to the real axis in the semiclassical limit is closely related with the trapped trajectories of the underlying classical mechanics. I will begin with the background of the problem and some elementary examples, and end with a very recent work about the resonances created by many homoclinic and heteroclitic trajectories.

Tuesday 5th April 2016

Ivan Dimitrov Queen's University, Canada

Weight representations of $gl(\infty)$

The Lie algebra $gl(\infty)$ consisting of infinite matrices with finitely many nonzero entries is an infinite dimensional Lie algebra which is a natural generalization of (simple) finite dimensional Lie algebras. It inherits a lot of the properties of the Lie algebras $gl_n$ but also exhibit many new features. One significant difference is that there are several possible generalizations of finite dimensional representations. The goal of this talk is to motivate the study of integrable weight modules, to discuss their properties, and to provide a classifications of the irreducible integrable weight modules with finite dimensional weight spaces.

Tuesday 12th April 2016

Fabrizio Caselli Università di Bologna

Il sollevamento generalizzato negli intervalli di Bruhat

Motivati dallo studio di un sistema Hamiltoniano integrabile detto gerarchia di Kostant-Toda, viene introdotto un politopo convesso detto politopo (dell’intervallo) di Bruhat. Lo studio di questo politopo porta a considerare una nuova proprietà di tali intervalli rimasta fino ad ora inosservata che estende la classica proprietà di sollevamento e viene quindi detta di sollevamento generalizzato. Verrà mostrato come tale proprietà caratterizza i gruppi di Coxeter finiti semplicemente allacciati.

Tuesday 19th April 2016

Giovanni Cupini Università di Bologna

General Facts and particular Results in the Theory of Regularity in Calculus of Variations and in PDEs

The minimizers of integral functionals of the Calculus of Variations and the solutions of PDEs in divergence form are related by the Euler's equation. The main areas of research concern the proof of the existence of minimizers/solutions and the study of their regularity. Two milestones in the regularitytheory are due to De Giorgi. In 1957 he proves the local Holder continuity of solutions to linear elliptic equations in divergence form with measurable coefficients. An example of De Giorgi himself, in 1968, shows that linear elliptic systems can have solutions not only discontinuous, but even locally unbounded. Since then, the theory of regularity has been hugely developed, in many directions, both in Calculus of Variations an in PDEs. In this "Topics" lecture, I will describe some general facts and some results.