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This workshop is a collaborative initiative between the Centre de recherches mathématiques (CRM) in Montreal and the Centre de Recerca Matemàtica in Barcelona.

CRM Exploratory Workshop

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Workshop
July 15, 2024
  • Room: A1 (CRM)
  • Dates: 15th of July, 2024

Registration for the activity is free but mandatory.

Registered participants from CRM-MONTREAL will receive the zoom link to follow the sessions by July 12, 2024.

Registration deadline 10 / 07 / 2024
Introduction

Since the seminal works of the MIT professors Gian-Carlo Rota in the 1960’s, and later Richard Stanley, combinatorics has become an important branch of modern mathematics where interplay of different areas arise. In particular, the Rota-Hero-Welsh’s Conjecture, proposed by Gian-Carlo Rota in the 1960s for graphs and later by Hero and Welsh for matroids, has long stood as one of the most tantalizing problems in combinatorics and algebraic geometry. It posited a deep connection between the combinatorial structure of matroids and the algebraic structure of projective varieties. Specifically, it conjectured that certain relations, now known as the Hodge Riemann relations, hold true for the intersection numbers of divisors on a projective variety defined by the matroid. Despite decades of effort from mathematicians around the world, the conjecture remained stubbornly unresolved, serving as a formidable challenge at the intersection of these two fields.

June Huh’s initial breakthrough came with his groundbreaking proof of Rota’s Conjecture [1], achieved through a novel synthesis of techniques from algebraic geometry, combinatorics, and homological algebra. His work unveiled the intricate connections between combinatorial objects such as matroids and algebraic varieties, providing a deep understanding of the underlying structures governing their intersection. Through meticulous analysis and innovative reasoning, Huh demonstrated that the Hodge Riemann relations indeed hold true, thereby resolving a decades-old mystery and opening up new avenues for exploration in combinatorial algebraic geometry.

Subsequently, Huh’s collaboration with Karim Adiprasito and Eric Katz [2] further illuminated the rich mathematical landscape that emerged from their initial breakthrough. Together, they delved into the study of positivity for matroids and polytopes, uncovering deep connections between combinatorial structures and algebraic geometry. Their research yielded not only elegant proofs of fundamental results but also novel mathematical objects that bridge the gap between discrete and continuous mathematics. Through their collaborative efforts, they uncovered new perspectives on toric varieties, matroid theory, and tropical geometry, among other areas.

On another direction it has become nowadays more and more popular formal logic systems used to formalize mathematical proofs. In this context, LEAN is getting more and more popular due to its is versatile, allowing for a wide range of applications and uses. This tool is growing day by day and we believe it would be good to show to a wide mathematical audience the state of the art of this tool.

Organising Committee
Marc Masdeu | UAB-CRM
Juanjo Rué | UPC-CRM
SPEAKERS

Session 1 | Morning

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

TBA

Abstract: TBA

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

Abstract: The notion of Kähler Package  will be presented and illustrated with several examples, particularly those introduced by June Huh and collaborators with which they could prove various long-standing combinatorial conjectures, but also including its first appearance in Grothendieck’s standard conjectures –a scheme to approach the Weil conjectures about algebraic varieties defined over a finite field. Then the theory of Lorentzian polynomials will be considered, with emphasis on its role as a bridge to Kähler packages and on how it is used to settle or suggest conjectures. In the final part, a sample of recent contributions in closely related themes will be discussed.

Sebastià Xambó

Universitat Politècnica de Catalunya

Session 2 | Afternoon

 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

TBA

Abstract: TBA

Yaël Dillies

University of Cambridge

Geometry in mathlib

Abstract: We will give an overview of the status of geometry (both algebraic and analytic) in mathlib, the official mathematical library of Lean. We will explain what is already in mathlib and what is missing, focusing on what can be added rather easily. We will also explain what are the difficulties in formalizing geometric notions following mathlib’s philosophy of being as general as possible and we talk about the solutions found by the mathlib community.

Riccardo Brasca

Université Paris Cité
SCHEDULE

MONDAY

July 15th

9:30

10:15

Matroids

Anna de Mier

Universitat Politècnica de Catalunya

10:20

11:05

Polytopes

Julian Pfeifle

Universitat Politècnica de Catalunya

11:10

11:40

COFFEE BREAK

11:40

12:25

Cohomology

Souvik Goswami

Universitat de Barcelona

12:30

13:15

Kähler packages and their combinatorial significance

Sebastià Xambó

Universitat Politècnica de Catalunya

13:15

15:30

LUNCH

15:30

16:30

Algebra in Lean

María Inés de Frutos Fernández

Universidad Autónoma de Madrid

16:35

17:35

Combinatorics in Lean

Yaël Dillies

University of Cambridge

17:35

18:15

COFFEE BREAK

18:15

19:15

Geometry in Lean

Riccardo Brasca

Université Paris Cité

LIST OF PARTICIPANTS
Name Institution
Vanessa Ramírez Bis Universitat de Barcelona
Huaxin Ou Universitat Autònoma de Barcelona
Eloi Torrents Universitat Autònoma de Barcelona
Jordi Saludes Universitat Politècnica de Catalunya
Marc Noy Serrano Universitat Politècnica de Catalunya
Guillem Perarnau Llobet Universitat Politècnica de Catalunya
Juan Jose Rue Perna Universitat Politècnica de Catalunya
Xavier Povill Universitat Politècnica de Catalunya
Fatemeh keivani Aalto University
INVOICE/PAYMENT INFORMATION
IF YOUR INSTITUTION COVERS YOUR REGISTRATION FEE: Please note that, in case your institution is paying for the registration via bank transfer, you will have to indicate your institution details and choose “Transfer” as the payment method at the end of the process.
UPF | UB | UPC | UAB
*If the paying institution is the UPF / UB/ UPC / UAB, after registering, please send an email to comptabilitat@crm.cat with your name and the institution internal reference number that we will need to issue the electronic invoice. Please, send us the Project code covering the registration if needed.
Paying by credit card
IF YOU PAY VIA CREDIT CARD but you need to provide the invoice to your institution to be reimbursed, please note that we will also need you to send an email to comptabilitat@crm.cat providing the internal reference number given by your institution and the code of the Project covering the registration (if necessary).
LODGING INFORMATION

ON-CAMPUS AND BELLATERRA

BARCELONA AND OFF-CAMPUS 

 

For inquiries about this event please contact the Scientific Events Coordinator Ms. Núria Hernández at nhernandez@crm.cat​​