Plenary Speakers and Morning Schedule

Mornings will be devoted to plenary lectures. The talks are in person and will be in the Lory Student Center (LSC) grand ballroom. We expect to record them and make them available live on Zoom.

July 14-18

Monday: 8:50-9 Introductory remarks and announcements, 9-9:50 Mattias Jonsson, 10:30-11:20 Karl Schwede

Tuesday: 9-9:50 Sébastien Boucksom, 10:30-11:20 Kevin Tucker

Wednesday: 9-9:50 Chenyang Xu, 10:30-11:20 Jakub Witaszek

Thursday: 9-9:50 Chenyang Xu, 10:30-11:20 Isabel Vogt

Friday: 9-9:50 Chenyang Xu, 10:30-11:20 Zsolt Patakfalvi

July 21-25

Monday: 9-9:50 June Huh, 10:30-11:20 Melody Chan

Tuesday: 9-9:50 John Pardon, 10:30-11:20 June Huh

Wednesday: 9-9:50 Junliang Shen, 10:30-11:20 June Huh

Thursday: 9-9:50 Roman Bezrukavnikov, 10:30-11:20 Junliang Shen

Friday: 9-9:50 Hannah Markwig, 10:30-11:20 Emanuele Macrì

July 28-August 1

Monday: 9-9:50 Zhiwei Yun, 10:30-11:20 Michael Hopkins

Tuesday: 9-9:50 Jacob Tsimerman, 10:30-11:20 Toby Gee

Wednesday: 9-9:50 Bhargav Bhatt, 10:30-11:20 Jacob Tsimerman

Thursday: 9-9:50 Bhargav Bhatt, 10:30-11:20 Jacob Tsimerman

Friday: 9-9:50 Olivier Wittenberg, 10:30-11:20 Bhargav Bhatt

Titles and Abstracts

Bhargav Bhatt. $p$-adic Hodge theory and algebraic geometry: a symbiotic relationship (Lecture 1, Lecture 2, Lecture 3)

These lectures will survey some developments in $p$-adic Hodge theory and their applications (due to many authors, mostly over the last decade). The first lecture will revolve around the $p$-adic Riemann--Hilbert correspondence and its uses. The second lecture will discuss integral topics, especially (related) improvements in our understanding of both Galois representations and the de Rham cohomology of algebraic varieties in characteristic $p$. The final lecture will focus on the $p$-adic (Corlette--)Simpson correspondence and its behaviour under reduction modulo $p$. The central theme of all lectures is geometrization: realizing a category of interest in $p$-adic Hodge theory as quasi-coherent sheaves on certain (close to algebraic) stacks, which are then more amenable to algebro-geometric techniques.

Roman Bezrukavnikov. Representation theoretic geometry of algebraic symplectic varieties.

The goal of the talk is to survey (my perspective on) the algebro-geometric concepts vital to the present day representation theory. Perverse (Weil) sheaves and Hodge modules have been a major theme in this area since 1980's. More recent tools include enumerative geometry, various flavors of mirror symmetry, Bridgeland stability conditions and new connections to Hodge theory.

Sébastien Boucksom. On the Yau–Tian–Donaldson conjecture (joint with M. Jonsson)

Let X be a smooth complex projective variety, and L an ample line bundle on X. The Yau-Tian-Donaldson conjecture states that the existence of a constant scalar curvature Kähler metric in the first Chern class of L is equivalent to a stability condition of algebro-geometric nature. In these two lectures, we will outline a proof of the conjecture, using non-Archimedean geometry over the field of complex numbers equipped with the trivial absolute value.

Melody Chan. Tropicalizations of locally symmetric varieties

I will give a friendly introduction to a topic at the nexus of tropical geometry, moduli spaces, and arithmetic groups: a theory of tropicalizations of locally symmetric varieties. Joint with Eran Assaf, Madeline Brandt, Juliette Bruce, and Raluca Vlad.

Toby Gee. Modularity theorems for abelian surfaces

I will give a gentle introduction to my joint work with George Boxer, Frank Calegari, and Vincent Pilloni in which we prove the modularity of a positive proportion of abelian surfaces over the rational numbers.

Michael Hopkins. Algebraic constructions of projection operators

This talk concerns joint work with Aravind Asok and Tom Bachmann. Suppose that P is a finite rank projection operator over a ring R, and consider the problem of constructing new projection operators whose entries are polynomials in the entries of P. Beyond the Shcur-Weyl method of tensoring P with itself and exploiting the action of the symmetric group, it is difficult to think of examples. In this talk I will describe recent progress in motivic homotopy theory permitting one to give a more or less complete answer to this question.

June Huh. Lorentzian polynomials, volume polynomials, and matroids over triangular hyperfields (Lecture 1, Lecture 2, Lecture 3)

In the first talk, I will focus on two concrete questions about projections of geometric objects in 4-dimensional spaces. The first question concerns convex bodies in R^4: Let (p_12, p_13, p_14, p_23, p_24, p_34) be the areas of the six coordinate projections of a convex body in R^4 to R^2. Which six numbers arise in this way? The second question concerns algebraic surfaces in (P^1)^4: Let (p_12, p_13, p_14, p_23, p_24, p_34) be the degrees of the six coordinate projections from an irreducible surface in (P^1)^4 to (P^1)^2. Which six numbers arise in this way? The answers to these questions are governed by the Plücker relations for the Grassmannian Gr(2,4) over the triangular hyperfield T_2. These results suggest a general conjecture regarding homology classes of irreducible surfaces in smooth projective varieties (based on joint work with Daoji Huang, Mateusz Michalek, Botong Wang, Shouda Wang).

In the second talk, I will give an overview of the intricate relationships among Grassmannians over hyperfields and the theory of Lorentzian polynomials. The main result is the identification of the space of Lorentzian polynomials with a given support and the corresponding matroid strata in the Grassmannain over the triangular hyperfield, up to homeomorphism (based on joint work with Matt Baker, Mario Kummer, Oliver Lorscheid).

In the third talk, I will focus on volume polynomials, a distinguished class of Lorentzian polynomials with remarkable analytic and combinatorial properties that arise from projective varieties. I will discuss their applications to algebraic matroids, introduce the new class of analytic matroids, and pose a number of questions (based on joint work with Lukas Grund, Mateusz Michalek, Henrik Süss, Botong Wang).

Mattias Jonsson. On the Yau–Tian–Donaldson conjecture (joint with S. Boucksom)

Hannah Markwig. Tropical curve counting

In enumerative geometry, we fix geometric objects and conditions and count how many objects satisfy the conditions. For example, there are 2 plane conics passing through 4 points and tangent to a given line. Tropical geometry can be viewed as a degenerate version of algebraic geometry and has proved to be a successful tool for enumerative problems. We review tropical curve counting problems. In particular, we show how tropical methods can be applied for quadratically enriched counts, which can be viewed as generalizations that allow results over any ground field.

Davesh Maulik and Junliang Shen. Moduli of Higgs bundles and the P=W Conjecture (Lecture 1, Lecture 2)

In these lectures, I will survey some recent questions and progress on the cohomology of the moduli space of Higgs bundles on a curve, focusing on the recently proven P=W conjecture of de Cataldo-Hausel-Migliorini. In the first lecture, I will introduce the basic objects of study. In the second lecture, I will discuss ideas related to the conjecture. In the third lecture, I will raise some related questions regarding lifts to algebraic cycles.

Emanuele Macri. A walk through hyper-Kähler geometry

A hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics. The theory in dimension two, namely K3 surfaces, is well understood. The aim of the seminar is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from the interrelated viewpoints of Lagrangian fibrations and moduli spaces of sheaves.

John Pardon. Universally counting curves in Calabi-Yau Threefolds

Zsolt Patakfalvi. Stable varieties

I will give an overview on the construction of the moduli space of stable varieties, on the study of its properties, and on its applications.

Karl Schwede. Vanishing theorems and singularities in mixed characteristic

In this talk, we will learn about some analogs of Kodaira-type vanishing theorems that hold in mixed characteristic (ie, over the p-adic integers). We'll see how one can use these results in applications. We will also see how these vanishing theorems naturally lead to notions of singularities in mixed characteristic, related to but distinct from singularities from birational complex geometry, such as rational singularities.

Jacob Tsimerman. O-minimal methods in Hodge Theory (Lecture 1, Lecture 2, Lecture 3)

O-minimality is a concrete framework for dealing with `tame' structures. Peterzil and Starchenko explained how one may fruitfully apply this theory to complex geometry to obtain large classes of holomorphic functions which mimic many good properties of the algebraic category. This theory turns out to be a natural setting in which to study hodge theory, in particular giving proofs of the Griffiths conjectures, and the construction of Baily-Borel compactifications for images of period maps. We shall introduce o-minimal analytic geometry and explain these developments.

Kevin Tucker. An introduction to singularities in mixed characteristic

There has long been a rich interplay between the study of singularities in complex algebraic geometry -- defined via resolutions of singularities, as in the case of Kawamata log terminal (KLT) and rational singularities -- and their counterparts in positive characteristic, such as F-regular and F-rational singularities, defined using the Frobenius endomorphism. In recent years, this dictionary has been further expanded to encompass singularities in mixed characteristic settings, such as those arising in arithmetic geometry over the p-adic integers. In this talk, I will give an introduction to some of these developments and illustrate how they can be exploited to better understand and unify these classes of singularities. In particular, we will discuss splinters and BCM-regular singularities, which serve as mixed characteristic analogues of KLT and F-regular singularities. I will also highlight how alterations can be used to connect and compare singularities across characteristics, ultimately leading to a mixed characteristic theory of multiplier / test ideals.

Isabel Vogt. Interpolation for Brill--Noether curves

The interpolation problem is one of the oldest in mathematics. In its most broad form it asks when a given type of curve can be drawn through a given collection of points. I will discuss some of the history of this problem and my recent joint work with Eric Larson resolving the problem for general curves passing through general points in projective space.

Jakub Witaszek. The Minimal Model Program in mixed characteristic

Recent advances in mixed characteristic algebraic geometry have led to significant progress on the arithmetic Minimal Model Program. I will begin with an overview of the current state of the art, then outline the key components of the proof, and conclude with several applications of the theory.

Olivier Wittenberg. Intermediate Jacobians over non-closed fields and applications to rationality

In the 1970's, Clemens and Griffiths showed that smooth complex cubic threefolds are irrational. Their approach, based on intermediate Jacobians, soon proved instrumental in the study of rationality for threefolds over algebraically closed fields. Almost five decades later, it was discovered that the Clemens-Griffiths method can also be leveraged, to great effect, over non-closed fields. Doing so goes along with pushing further the theory of intermediate Jacobians itself. The talk will be devoted to presenting these developments.

Chenyang Xu. K-stability and Moduli of Fano Varieties (Lecture1, Lecture 2, Lecture 3)

In the last decade, the investigation of K-stability of Fano varieties has emerged as one of central topics in algebraic geometry. The notion of K-stability, which originated in complex geometers’ research on Kähler-Einstein metrics, turns out to be crucial to answer questions in higher dimensional geometry. Remarkably, it yields a construction of moduli spaces, called K-moduli, parametrizing K-stable Fano varieties. In this series of lectures, I will first introduce the notion of K-stability, and give different characterizations of it. Then I will focus on the construction of K-moduli. In the last lecture, I will discuss local stability theory for singularities with applications, as well as explicit examples.

Zhiwei Yun. Applications of Geometric Representation Theory to Algebraic Geometry

We give a survey on three applications of geometric representation theory to algebraic geometry: (1) existence of motives with exceptional motive Galois groups; (2) certain cases of the Deligne-Simpson problem (joint with K.Jakob); (3) reducedness of the ring of invariant functions on the commuting scheme of a reductive group (joint with P.Li and D.Nadler).

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