Topics in Mathematics 2015/2016

In 1976 Paul Malliavin proposed in the paper "Stochastic calculus of variation and hypoelliptic operators" a probabilistic proof of Hormander's "sum of square" theorem. His proof was based on a new infinite dimensional differential calculus on the Wiener space. The theory was further developed by Stroock, Bismut and Watanabe, among others, to become what is nowadays known as the Malliavin calculus. This calculus has become a fundamental tool in the theory of stochastic (partial) differential equations and has found important applications in mathematical finance. This short course aims to provide a coincise introduction to the subject together with a sketch of Malliavin's proof of Hormander's theorem. Few remarks on the applications in mathematical finance will also be provided

In 1976 Paul Malliavin proposed in the paper "Stochastic calculus of variation and hypoelliptic operators" a probabilistic proof of Hormander's "sum of square" theorem. His proof was based on a new infinite dimensional differential calculus on the Wiener space. The theory was further developed by Stroock, Bismut and Watanabe, among others, to become what is nowadays known as the Malliavin calculus. This calculus has become a fundamental tool in the theory of stochastic (partial) differential equations and has found important applications in mathematical finance. This short course aims to provide a coincise introduction to the subject together with a sketch of Malliavin's proof of Hormander's theorem. Few remarks on the applications in mathematical finance will also be provided.

The minimizers of integral functionals of the Calculus of Variations and the solutions of PDE's 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 regularity theory are due to De Giorgi. In 1957 he proved the local Holder continuity of solutions to linear elliptic equations in divergence form with measurable coefficients. An example by 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 and in PDE's. In this "Topics" lecture, I will describe some general facts and some results.

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.

Abstract: 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

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.

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.

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.

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.

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 .

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.

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.

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.

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.

n 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.

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.

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.

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.