CRM Exploratory Workshop

An Introduction to Matroid Theory

Abstract: Matroids originated in the 1930’s as an abstraction of the notion of linear independence in vector spaces. Whereas matrices and graphs provide the first examples and were the motivation for the developing of the field, matroid theory has many connections to other areas, such as matchings, optimization, geometry or criptography.  In this talk we will give an overview of matroids, focusing on those aspects that justify their alternative name, albeit not so common, of combinatorial (pre)geometries.

One special feature of matroids is that they have several equivalent definitions. We will go over the main ones (bases, indepedents sets, flats, rank, circuits…), and then we will survey the basic operations (minors and duality) and present some important classes of matroids. We will then move to matroid polynomials, with an emphasis on the characteristic polynomial and its properties. 

No previous knowledge of matroid theory will be assumed.

Anna de Mier

Universitat Politècnica de Catalunya

Matroid polytopes, tropical geometry, and log-concavity

Abstract: We continue the previous talk and examine some geometric incarnations of a matroid M, namely matroid polytopes, Bergman fans, and tropical linear spaces. More algebraically, the rational cohomology of the complement of a certain hyperplane arrangement associated to M is called the Orlik-Solomon algebra, whose Poincaré polynomial is related to the characteristic polynomial of M.

We hope that these examples will build some intuition for understanding the essence of Huh–Adiprasito–Katz’s proof of the log-concavity of the coefficients of this characteristic polynomial, namely that the tropical variety or Bergman fan associated to a matroid has the structure of the cohomology ring of a smooth projective variety.

Julian Pfeifle

Universitat Politècnica de Catalunya

Survey of Chow groups

Abstract:

The study of algebraic cycles is at the cornerstone of Algebraic Geometry, and shares common ground with Arithmetic Geometry, Algebraic K-theory, and Analytic geometry. The definition of the free group of algebraic cycles is easy to understand, and can be generalized to different categories. The non-trivial part is to define a suitable equivalence relation, so that the quotient group becomes equipped with a ring structure. Such an equivalence relation is provided by rational functions on subvarieties, and the resultant quotient is defined as the Chow group attached to an algebraic variety.

The purpose of this talk is to give an introduction and survey of this topic. More aligned towards the topic of this conference, I will introduce Chow group of a matroid.

Souvik Goswami

Universitat de Barcelona

Kähler packages and their combinatorial significance

Sebastià Xambó

Universitat Politècnica de Catalunya

 Algebra in the Lean mathematical library

Abstract: We will start by introducing the interactive theorem prover Lean and its mathematical library Mathlib. We will next give an overview of the algebra hierarchy in the Mathlib library (how groups, rings, modules, etc. are formalized, and the dependencies between these objects). Finally, we will discuss the formalizations of several topics that appear in Huh’s work, including polynomial rings and graded algebras.

María Inés de Frutos Fernández

Universidad Autónoma de Madrid

Combinatorics in Mathlib and beyond

Abstract: As the main Lean library of mathematics, Mathlib acquires new formalisations at an increasing pace. Curiously, one area of mathematics is lagging behind: combinatorics.

Yaël Dillies

University of Cambridge

Geometry in mathlib

Riccardo Brasca

Université Paris Cité