Göran Gustafsson Symposium

The Göran Gustafsson Lectures in Mathematics 2022 is combined with a Göran Gustafsson Symposium organized at Institut Mittag-Leffler. The first lecture of June Huh takes place at KTH and the ensuing lectures and research talks at Institut Mittag-Leffler.

Research talks

Schedule

Monday, May 30 (at KTH/Institut Mittag-Leffler)
9.30 Coffee, Albanova, KTH
10.00 First lecture of June Huh at Oskar Klein (FR4), Albanova, KTH
11.30 Transportation to Institut Mittag-Leffler
12.15 Lunch
13.45 Felipe Rincón
15.00 Kris Shaw
16.15 Sam Payne (online)
 
Tuesday, May 31 (at Institut Mittag-Leffler)
9.45 Second lecture of June Huh
11.00 Matthieu Piquerez
12.00 Lunch
13.30 Alex Fink
14.45 Roberto Pagaria
16.00 Oliver Lorscheid
 
Wednesday, June 1 (at Institut Mittag-Leffler)
9.45 Third lecture of June Huh
11.00 Swee Hong Chan
12.00 Lunch
13.30 Andrew Berget
14.40 Melody Chan

The lectures will also be available on Zoom with the following Meeting IDs:

All lectures at Institut Mittag-Leffler will be in Kuskvillan (Wallenbergsalen). More information are available at Institut Mittag-Leffler.

 

Abstracts

June Huh, lecture 1: Lorentzian polynomials

Monday, May 30, 10.00‑11.00 am
Oskar Klein Auditorium FR4, Albanova, Roslagstullsbacken 21
Zoom Meeting ID: 635 0702 6779

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. The talk will be accessible to a general audience: No specific background beyond linear algebra and multivariable calculus are required for most of the presentation. In addition, I advertise the talk to people with interests in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint work with Petter Brändén.

Video recording of lecture

Coffee is served between 9.30 and 10.00.

June Huh, lecture 2: Open problems on Lorentzian polynomials

Tuesday, May 31, 9.45‑10.45 am
Wallenbergsalen, Kuskvillan, Institut Mittag-Leffler
Zoom Meeting ID: 921 756 1880

Conjecturally, skew-Schur polynomials, Schur P polynomials, Schubert polynomials, homogeneous components of Grothendieck polynomials, key polynomials, and homogenized basis generating functions of morphisms of matroids all become Lorentzian after normalizations. I will present these and some other open problems on Lorentzian polynomials. Joint work with Jacob Matherne, Karola Mészáros, and Avery St. Dizier, and with Chris Eur.

Video recording of lecture

June Huh, lecture 3: Kazhdan–Lusztig theory and Hodge theory for matroids

Wednesday, June 1, 9.45‑10.45 am
Wallenbergsalen, Kuskvillan, Institut Mittag-Leffler
Zoom Meeting ID: 921 756 1880

We explore the Hodge theory behind the fact that the basis generating polynomial of a matroid is Lorentzian. The story reveals a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or vector configuration). After giving an overview of the similarity, I will outline proofs of two combinatorial conjectures, the nonnegativity conjecture for Kazhdan–Lusztig polynomials of matroids and the top-heavy conjecture for the number of flats of matroids. The key step is to formulate and prove an analogue of the decomposition theorem in a combinatorial setup. The talk will be accessible to graduate students. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.

Video recording of lecture

Felipe Rincón: Tropical Ideals

Tropical ideals are ideals in the tropical polynomial semiring in which any bounded-degree piece is “matroidal”. They were introduced as a sensible class of objects for developing algebraic foundations in tropical geometry. In this talk I will introduce and motivate the notion of tropical ideals, and I will discuss work studying some of their main properties and their possible associated varieties.

Video recording of talk

Kris Shaw: Chern–Schwartz–MacPherson cycles of matroids and beyond

I will describe the Chern–Schwartz–MacPherson (CSM) cycles of matroid, which are a collection of polyhedral fans introduced by López de Medrano, Rincón, and myself. When the matroid arises from a hyperplane arrangement, these fans encode the CSM class of the complement in its wonderful compactification. For arbitrary matroids these classes have found further applications. For instance the CSM cycles of matroids are an ingredient in Ardila, Denham, and Huh’s proof of Brylawski's and Dawson’s conjectures and they also provide a Chow theoretic description of Speyer’s g-polynomial of a matroid arising from K-theory. Generalising the approach for matroids, I will explain how CSM classes can also be defined and used to study more general objects in tropical geometry.

Video recording of talk

Sam Payne: Odd cohomology vanishing and polynomial point counts on moduli spaces of stable curves

Video recording of talk

Matthieu Piquerez: Hodge theory for tropical fans

On the introduction to the present symposium, one can read “But the [Heron-Rota-Welsh] conjecture was for an arbitrary matroid, which might not be associated to any type of geometry at all! The proof by Adiprasito–Huh–Katz builds an object from combinatorics, which ought to play the role of the cohomology ring, and proves Poincaré duality, Hard Lefschetz and the Hodge–Riemann bilinear relations for this object directly.”

In this talk, I will show that, on the contrary, these results has a geometric interpretation for any matroid... in the tropical world. Indeed, one can show that the tropical cohomology of the canonical compactification of so-called tropically shellable quasi-projective fans verifies the three above properties. In particular, Bergman fans of matroids belong to those fans, hence we get a generalization of the result of Adiprasito–Huh–Katz. This is a joint work with Omid Amini.

Video recording of talk

Alex Fink: Chow classes of delta-matroids

Delta-matroids are the type BC Coxeter matroids for the short resp. long root; they can also be seen as set systems satisfying a weakening of the matroid basis exchange axiom so that bases may have different sizes. In ongoing work with Chris Eur, Matt Larson and Hunter Spink, we begin extending the intersection-theoretic perspective on matroid theory to delta-matroids, largely following the approach of Berget–Eur–Spink–Tseng. I'll present some of the combinatorial and geometric consequences.

Video recording of talk (bad focus for the first 7 minutes)

Roberto Pagaria: Hodge theory for polymatroids

Polymatroids are combinatorial objects that generalize matroids, subspace arrangements, and hypergraphs. In the case of subspace arrangements, De Concini and Procesi constructed a wonderful model and studied the Leray model associated with it. Adiprasito, Huh, and Katz defined a Chow ring for matroids and used it to prove the log-concavity conjecture.

We provide a Leray model and a Chow ring for polymatroids, which we use to generalize the Goresky-MacPhearson formula to the non-realizable setting. We also prove that the Chow ring of a polymatroid satisfies Poincaré duality and, on a certain cone, hard Lefschetz theorem and Hodge Riemann bilinear relations.

This is a joint work with Gian Marco Pezzoli.

Video recording of talk

Oliver Lorscheid: Moduli spaces in matroid theory

Families of matroids have appeared in different disguises during the last few decades: combinatorial flag varieties arise from flag matroids, Macphersonians appear as spaces of oriented matroids, Dressians consist of valuated matroids. With the advent of F1-geometry, we are able to understand these spaces through an algebro-geometric perspective as rational point sets of moduli spaces of (flag) matroids and place them into a larger landscape of geometric objects stemming from combinatorics.

In this talk, I will review my joint works with Matthew Baker and with Manoel Jarra on the topic. This includes a review of Baker–Bowler theory, a primer to F1-geometry and the construction of the moduli spaces of (flag) matroids. We explain how to recover the aforementioned geometric objects and comment on applications and future directions.

Video recording of talk

Swee Hong Chan: Combinatorial atlas for log-concave inequalities

The study of log-concave inequalities for combinatorial objects have seen much progress in recent years. One such progress is the solution to the strongest form of Mason's conjecture (independently by Anari et. al. and Bränden–Huh). In the case of graphs, this says that the sequence fk of the number of forests of the graph with k edges, form an ultra log-concave sequence. In this talk, we discuss an improved version of all these results, proved by using a new tool called the combinatorial atlas method.

Video recording of talk

Andrew Berget: Schur classes of matroids and more!

In recent joint work with Chris Eur, Hunter Spink and Dennis Tseng we defined tautological classes of matroids. These are elements of the Chow ring of the permutohedron which abstract the tautological vector bundles over torus orbits in the Grassmannian. In this talk I will define Schur classes of matroids, which are certain determinants in these tautological classes, as well as their rightful equivariant analogues. I will present a series of positivity conjectures on these classes, which are proved for complex realizable matroids and generalized to K-theory in previous joint work with Alex Fink. At the heart of these conjectures is the following question: Which equivariant classes in the Chow ring of the permutohedron have an appropriately positive equivariant degree?

Video recording of talk

Melody Chan: Graph complexes and moduli of curves

I will give an overview of some appearances of Kontsevich's graph complexes in the study of cohomology of moduli spaces. Includes joint work with Soren Galatius and Sam Payne, and with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

Video recording of talk

 

About the speaker

June Huh
Photo by Woo-Hyun Kim. June Huh received his PhD in mathematics from University of Michigan in 2014. After finishing his PhD, he was a Clay Research fellow at Princeton University and the Institute of Advanced Studies. He was an invited speaker at the International Congress of Mathematics in 2018 and received the New Horizons in Mathematics Prize in 2019 for his joint work with Karim Adiprasito and Eric Katz on combinatorial Hodge theory. In 2021, he received the Samsung Ho-Am Prize in Science. He is currently a professor at Princeton University.

 

Organizers of the symposium are Petter Brändén (KTH), Dan Petersen (Stockholm University) and David Rydh (KTH).

Listing of all Göran Gustafsson lectures in mathematics

Sponsored by the Göran Gustafsson Foundation