Topics in Mathematics 2021/2022

We introduce, hopefully in a friendly way, some classical free boundary problems. Then, possibly, we shall discuss the main difficulties arising in studying the regularity of the solutions of such problems as well as some tools used for solving them.

The theory of elliptic equations and systems of m equations in divergence form, is strictly related to the theory of minimization of integral functionals. After a review on the existence issue, we will focus on the regularity problem: under which conditions the solutions are regular? The ideal process to prove that a (weak) solution, apriori only in a Sobolev space W^{1,p}, is C^{\infty} will be sketched. Unfortunately, the gain in regularity is not for free, and it is guaranteed only if particular conditions are met. In the past years, counterexamples have shown that: 1) under certain growth conditions the regularity can be lost, even in the scalar case; 2) in the vectorial case the situation is far worse, since even solutions to linear and uniformly elliptic systems may be locally unbounded (!). The main effort is to find conditions that force the regularity of the solutions. We will focus in particular to the vectorial case; i.e. the local regularity of weak solutions to elliptic systems. The main and most common structure condition, that forces, in general, regularity in the vectorial setting, is the so called Uhlenbeck’s structure (dependence on the modulus of the gradient). Meier, in 1982, introduced another assumption, related to a so called Indicator function: a more general condition than Uhlenbeck’s one, that allows to include more general systems. For them, Meier proved the local boundedness of the solutions. We will exhibit examples of systems that do not satisfy the Meier’s condition, but for which, in a recent result in collaboration with F. Leonetti (L’Aquila) and E. Mascolo (Firenze), we proved the boundedness of the solutions. The crucial structure assumption is the componentwise coercivity introduced by Bjorn in 2001.

Many geometric structures on algebraic varieties are best studied by considering the collection of structures, modulo some natural equivalence, and giving it a geometric structure itself. We will a gentle introduction, focused on examples. Stesso link per il Seminario del Prof. M. Thaddeus, stessa ora

The numerical solution of possibly large dimensional algebraic linear systems permeates scientific modelling. Often systems with multiple right-hand sides arise, whose efficient numerical solution usually requires ad-hoc procedures. In the past decades a new class of linear equations has shown to be the natural algebraic framework in the discretization of mathematical models in a variety of scientific applications. These problems are given by multiterm linear matrix equations of the form A_1 X B_1 + A_2 X B_2 + ... + A_k X B_k = C where all appearing terms are matrices of conforming dimensions, and X is an unknown matrix. The case k=2 is called the Sylvester equation, and computational methods for its solution are well established, especially for small dimensions. The general multiterm case turns out to be a key ingredient in problems such as time-space, stochastic and parametric partial differential equations. Its numerical solution is the current challenge, though little is known also about its algebraic properties. In this lecture we give a gentle introduction to the problem, and discuss various attempts to numerically solve it.

I survey some results and open questions regarding the applications of optimal stopping theory to real option analysis. The main focus is on the issue of obtaining explicit solutions for the related free-boundary problems. First, some elementary examples are presented which are of interest for economic applications. Then an explicit expression for the value function in the two- dimensional (and n-dimensional ) case is obtained. The value function is written in terms of a modified Bessel function of second kind. Some useful formulas for the one-dimensional case are presented as well.

Totally non-negative Grassmannians are a special case of G. Lusztig extension of the classical notion of total positivity, and have been combinatorially characterized in a seminal paper by A. Postnikov. They also appear in many relevant problems of mathematical and theoretical physics. The Kadomtsev-Petviashvili (KP) equation is the first non-trivial flow of the most relevant classical integrable hierarchy, and was originally introduced to study the stability of soliton solutions of another integrable system, the Kortweg-de Vries equation. Kasteleyn theorem represents the number of dimer configurations in planar graphs as determinants of sign matrices. In this talk I shall explain the role of totally non-negative Grassmannians in the characterization of the asymptotic behavior in space-time of a class of solutions of the Kadomtsev-Petviashvili equation, in the solution of a spectral problem for the same equation and in counting dimer configurations in planar bipartite graphs in the disc. The presentation will be elementary and self-contained.

In the first part of the seminar we introduce Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion, and we show a classical well-posedness result for this class of equations. In the second part, we give an overview on jump measures and related stochastic calculus. Then, we consider BSDEs driven by a general random measure, and we show how additional conditions have to be imposed in order to recover existence and uniqueness for the corresponding solutions.

In the first​ part of this seminar we will introduce Coxeter groups, fully commutative elements and the Temperley-Lieb algebra, by illustrating some classical examples. In the second part, we will recall a recent construction of a diagrammatic representation of the Temperly-Lieb algebra of affine type C due to Dana Ernst, and we will show that this representation is faithfull in a new combinatorial way.