Topics in Mathematics

Ph.D. Program in Mathematics (academic year 2012/2013)

Organizers

Giampaolo Cristadoro and Giovanni Cupini

Program

Archive

giovedì 14 marzo

André Martinez

Larghezze di risonanze molecolari (Widths of molecular resonances)

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giovedì 21 marzo

Giorgio Bolondi

An odd triangle: research, mathematics, education

giovedì 11 aprile

Patrizio Frosini

A quick trip through geometrical shape comparison

Geometrical shape comparison is formally based on metrics that measure how much two vector-valued functions defined on a given topological space are "similar" to each other, up to the action of a given group.
In the first part of this talk we will introduce and justify this approach through several examples taken from Pattern Recognition. After this introduction, we will illustrate the concept of natural pseudo-distance and some related results and open problems. Finally, we will show how lower bounds for the natural pseudo-distance can be obtained via the concept of persistent homology and how this fact can be used in applications.
The presentation will be maintained at an elementary level, trying to focus on the main ideas and skipping the technical details as much as possible.

giovedì 18 aprile

Ermanno Lanconelli

The Mean Value Theorem in second order PDE theory

The Cauchy integral formula and the function $z \mapsto \frac{1}{z}$ play a crucial role in the theory of holomorphic functions. In linear second order PDE theory a similar role can be played by the Mean Value Theorem - whose ancestor is the Gauss Mean Value Theorem for harmonic functions - and by the fundamental solutions. Aim of this "Topics" is to present the main basic ideas of this approach, starting form the classical analysis of the Laplace, Heat and Wave equations. Recent developments are also presented, together with a list of open problems and some applications to the Riemannian geometry.

La formula integrale di Cauchy e la funzione $z \mapsto \frac{1}{z}$ costituiscono i "mattoni fondamentali" della teoria delle fuzioni olomorfe. Teorie esaustive per varie classi di Equazioni alle Derivate Parziali lineari del secondo ordine, possono svilupparsi in analogia con quella delle fuzioni olomorfe, sostituendo alla formula integrale di Cauchy opportune formule di media integrali, e alla funzione $z \mapsto \frac{1}{z}$ le soluzioni fondamentali degli operatori coinvolti. Nel presente "Topics" verranno presentate le idee principale di questo approccio, a partire dall'analisi dei classici operatori di Laplace, del calore, e delle onde. Verranno illustrati sviluppi recenti, insieme con alcuni problemi aperti ed applicazioni alla geometria riemanniana.

giovedì 9 maggio

Valeria Simoncini

The wonder of matrices in scientific computing

Matrices are ubiquitous in scientific computations. Either small and dense matrices, or large and possibly very sparse matrices arise in a huge variety of apparently unrelated application problems, and in particular whenever the quantity of interest needs to be approximated.
Structural and spectral matrix properties can effectively describe and in some cases reveal characteristic phenomena of the underlying problems. Knowing these properties thus provides an important tool in many scientific research areas.
In this talk we discuss a few examples where matrices play a key, and sometimes unexpected, role.

venerdì 17 maggio

Luca Migliorini

Moduli spaces in analytic and algebraic geometry

It has been known since Riemann and Weierstrass that analytic varieties have "moduli", namely, that the different analytic structures on a fixed differentiable manifolds such as for instance an orientable compact surface, are naturally parameterized by a complex variety, the "moduli space". In the case of a surface, these moduli varieties come in different, related, flavours: Teichmuller spaces, Torelli spaces, bonafide moduli spaces. In general, and somewhat vaguely, moduli spaces are parameters spaces for geometric structures underlying a given topological structure.
In the talk I will first discuss the classical case of compact Riemann surfaces, starting with the completely explicit case of 2-dimensional tori (aka elliptic curves).
In the second part I will discuss the moduli problem I am mostly interested in right now, that is moduli space of Higgs vector bundles (I will explain what these are, no boson involved) on a fixed Riemann surface, and moduli spaces of representations of the fundamental group of a surface.
If time permits, I will discuss some open problems, in particular the "P=W" conjecture, which, for groups of type A_1, was proved by de Cataldo Hausel and myself.

giovedì 23 maggio

Daniele Morbidelli

Orbits of families of vector fields and their regularity: classical and modern results.

Given a family of vector fields in the Euclidean space, we talk about the regularity of the subset reachable from a given point by means of piecewise integral curves of vector fields of the family. In particular, the discussion will include the classical Frobenius theorem, the more recent Sussmann's theorem, and the geometric ideas contained in their proof. Several related examples and counterexamples will be described.

giovedì 30 maggio

Pierluigi Contucci

The Statistical Mechanics formalism and its applications.

The seminar will introduce, from an elementary perspective, a statistical mechanics system undergoing a phase transition. The relevance of the statistical mechanics method in social science will be discussed and a case study illustrated.

giovedì 6 giugno

Cristian Gutiérrez

Fully nonlinear equations and geometric optics - Lecture n.1

giovedì 13 giugno

Cristian Gutiérrez

Fully nonlinear equations and geometric optics - Lecture n.2

giovedì 20 giugno

Cristian Gutiérrez

Fully nonlinear equations and geometric optics - Lecture n.3

giovedì 27 giugno

Cristian Gutiérrez

Fully nonlinear equations and geometric optics - Lecture n.4

Giovedì 7 marzo, ore 14.30, seminario II

prof. Pierluigi Contucci

TBA

Giovedì 14 marzo, ore 14.30, seminario II

prof. André Martinez

Larghezze di risonanze quantistiche (Widths of molecular resonances)

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Giovedì 21 marzo, ore 14.30, aula da definire

prof. Giorgio Bolondi

An odd triangle: research, mathematics, education

Giovedì 16 maggio, ore 14.30, aula da definire

prof. Luca Migliorini

Moduli spaces in analytic and algebraic geometry

It has been known since Riemann and Weierstrass that analytic varieties have "moduli", namely, that the different analytic structures on a fixed differentiable manifolds such as for instance an orientable compact surface, are naturally parameterized by a complex variety, the "moduli space". In the case of a surface, these moduli varieties come in different, related, flavours: Teichmuller spaces, Torelli spaces, bonafide moduli spaces. In general, and somewhat vaguely, moduli spaces are parameters spaces for geometric structures underlying a given topological structure.
In the talk I will first discuss the classical case of compact Riemann surfaces, starting with the completely explicit case of 2-dimensional tori (aka elliptic curves). In the second part I will discuss the moduli problem I am mostly interested in right now, that is moduli space of Higgs vector bundles (I will explain what these are, no boson involved) on a fixed Riemann surface, and moduli spaces of representations of the fundamental group of a surface. If time permits, I will discuss some open problems, in particular the "P=W" conjecture, which, for groups of type A_1, was proved by de Cataldo Hausel and myself.

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