# Topics in Mathematics 2021/2022

## Seminari passati

2022
19 Maggio
Stacks, Stability conditions.
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022
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

2022
10 Maggio
Valeria Simoncini
Matrix equations: from theory to (computational) practice
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

seminario di analisi numerica

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.

2022
07 Marzo
Rossella Agliardi
Optimal stopping problems arising in real option theory
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

seminario di finanza matematica

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.

2022
01 Febbraio
Simonetta Abenda
KP soliton theory, dimer models in the disc and totally non-negative Grassmannians
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

seminario di analisi matematica

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.

2021
20 Dicembre
Elena Bandini
BSDEs driven by general random measures
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

seminario di probabilità

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.

2021
13 Dicembre
Riccardo Biagioli
Fully commutative elements in Coxeter groups and the Temperley-Lieb algebra
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

seminario di algebra e geometria

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.