Pure Mathematics Seminar

2024 Semester 1

Upcoming talks

10 May, 3:15pm, Peter Hall 162

June Park (University of Melbourne)

Totality of rational points on modular curves over function fields

People want to count elliptic curves over global fields such as the field Q of rational numbers or the field F_q(t) of rational functions over the finite field F_q. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1, 1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1, 1}. In this talk, I will explain the exact counting formula as well as basic generalities, relevant tools and ideas.

17 May, 3:15pm, Peter Hall 162

Alex Sherman (University of Sydney)

Past talks

26 April, 3:15pm, Peter Hall 162

Will Donovan (Tsinghua University)

McKay correspondence and toric geometry

Finite subgroups of the matrix group SU(2) may be studied algebraically via their representations. They may also be studied geometrically via two-dimensional complex manifolds naturally associated to them. The McKay correspondence is a general phenomenon which, in particular, explains how these two approaches relate. I'll introduce this using diagrammatics from toric geometry, indicate how the correspondence generalizes to higher dimensions, and discuss open questions and current projects.

Tuesday 23 April, 3:15pm, Peter Hall 162

Behrouz Taji (University of New South Wales)

Boundedness problems: algebraic geometry meets arithmetic

In the 1960’s Shafarevich asked a simple question: Do families of curves of genus at least 2 have a finite number of deformation classes? Soon after Parshin gave an affirmative answer to this question and furthermore showed that this finiteness property is precisely the geometric underpinning of Mordell’s famous conjecture regarding finiteness of rational points for projective curves of genus at least 2  (over number fields). It was this very observation that led Faltings to his complete proof of Mordell’s conjecture in 1980s. For higher dimensional analogues of curves of high genus, Shafarevich’s problem was settled in 2010 by Kovács and Lieblich, partially thanks to multiple advances in moduli theory of so-called stable varieties. In this talk I will present this circle of ideas and then focus on our recent solution to Shafarevich’s question for other higher dimensional (non-stable) varieties, e.g. the Calabi-Yau case. This is based on joint work with Kenneth Ascher (UC Irvine).

12 April, 3:15pm, Peter Hall 162

Jieru Zhu (University of Queensland)

Tensor representations for the Drinfeld double of the Taft algebra

The Drinfeld double of the Talf algebra often serves as a common example of non-semisimple Hopf algebras, and is related to Lie theory being a quotient of the small quantum group. It is also a ribbon Hopf algebra where its module category is a ribbon category. We show that the braid group action on the tensor representation, introduced by Ram-Leduc, factors through the Temperley-Lieb algebra and induces an isomorphism with the centralizer algebra. This is under the assumption that the number of tensors is small. Further work includes studying a modular Schur-Weyl duality, as well as the action of the Karoubi envelope of the Temperley-Lieb category. This is joint work with Benkart-Biswal-Kirkman-Nguyen.

Wednesday 27 March, 2:00pm, Peter Hall 107

Aravind Asok (University of Southern California)

Motivic homotopy theory: what is it good for?

I will try to explain some aspects of motivic homotopy theory, culminating with a discussion of recent progress and applications to problems about when holomorphic vector bundles on complex affine varieties admit algebraic structures. This talk is based on joint work with Tom Bachmann, Jean Fasel and Mike Hopkins.

21 February, 11:00am, Evan Williams Theatre

Scott Mullane (University of Melbourne)

Teichmüller dynamics and the moduli space of curves

Integrating a differential on a Riemann surface allows the pair to be expressed as a collection of polygons in the plane with parallel side identifications. The action of GL(2,R) on the plane extends naturally to these polygons, and the orbits of the action, originally considered for their dynamical importance, have unexpected algebraic properties. In this talk, we'll introduce these ideas and discuss ways that this new perspective can be applied to questions on the birational geometry of moduli spaces of curves.

20 February, 3:30pm, Evan Williams Theatre

Arunima Ray (Max-Planck Institute Bonn)

Knots, links, and 4-dimensional spaces

Manifolds are fundamental objects in topology since they locally model Euclidean space. A central problem of interest is the classification of low-dimensional manifolds, especially those of dimension four. I will explain how powerful techniques from high-dimensional manifold topology, such as surgery theory, can be useful in this context, e.g. in Freedman's proof of the 4-dimensional Poincare conjecture. The key step involves replacing a given map from a surface to a 4-dimensional manifold by an embedding. Remarkably, important open problems in 4-dimensional manifold topology have equivalent formulations in terms of properties of knots and links in 3-dimensional space.

20 February, 9:30am, Evan Williams Theatre

Dougal Davis (University of Melbourne)

Hodge theory and unitary representations of real groups

A classical open problem in representation theory is to determine the set of irreducible unitary representations of a non-compact Lie group. This has proven difficult partly due to a lack of tools to control the key property of unitarity. In this talk, I will discuss a new such tool based on Hodge theory. Hodge theory has its roots in the study of the cohomology of complex algebraic varieties and plays a major role in modern algebraic geometry. The main result of this talk (joint work with Kari Vilonen) is that Hodge theory also controls unitary representations for the main class of interesting Lie groups, the real groups. If time permits, I will sketch how this new perspective leads to a simple proof of unitarity for a wide class of new representations, called rigid unipotent, which are conjectured to play a pivotal role in the full classification (joint work in preparation with Lucas Mason-Brown).

19 February, 3:30pm, Evan Williams Theatre

Lior Yanovski (Hebrew University Jerusalem)

Descent in algebraic K-theory and the telescope conjecture

Algebraic K-theory is a fundamental invariant of rings and categories with applications to various fields of mathematics including number theory, differential topology, and algebraic geometry. It is however very hard to compute, largely because algebraic K-theory fails to have good descent properties. I.e., it does not satisfy a good "local-to-global principle" on the source of the construction. Stable homotopy theory suggests a different ''local-to-global principle'' on the target of the construction, such that, remarkably, the local pieces of algebraic K-theory satisfy much better descent properties on the source. In this talk, I will discuss a recent work of Ben-Moshe, Carmeli, Shclank, and myself showing that these local pieces of algebraic K-theory satisfy in fact much stronger ''higher descent'' properties. I will also outline an application of these results to the resolution of the celebrated long-standing telescope conjecture by Burklund, Hahn, Levy, and Schlank, and the relationship to the Ausoni-Rognes redshift conjecture. I will review the relevant background material, so no prior knowledge of stable homotopy theory is required.

19 February, 2:00pm, Evan Williams Theatre

Toni Annala (IAS, Princeton)

Homotopy theory of varieties

In topology, homotopy theory is an important tool for understanding cohomology groups of topological spaces. Morel and Voevodsky set up an analogous theory in algebraic geometry, essentially by declaring homotopies to be parameterized by the affine line A^1. This theory is called A^1-homotopy theory, and it was used in a crucial way in Voevodsky's proof of the Milnor and Bloch-Kato conjectures. Unfortunately, A^1-homotopy theory has a flaw built into its heart: by design, the affine line A^1 is contractible in A^1-homotopy theory, meaning that A^1-homotopy theory can only treat cohomology theories for which the cohomology groups of A^1 are isomorphic to those of a point. Many important cohomology theories (prismatic cohomology, algebraic K-theory) violate this assumption, and this fact has sparked efforts to discover another, more fundamental form of homotopy theory of varieties. I will describe my program, together with Marc Hoyois and Ryomei Iwasa, whose purpose is to tackle exactly this issue by constructing "the correct" stable homotopy theory of varieties.

16 February

Sabino Di Trani (University of Rome)

On irreducible representations in exterior algebra and their multiplicities

Let g be a simple Lie algebra over C. The adjoint action of g on itself induces a structure of a g- representation on Λg, the exterior algebra  over g.   During the talk I will provide an overview of known results and open problems concerning irreducible representations appearing in Λg and their graded multiplicities.  

7 February, 2:00pm, Evan Williams Theatre

Giovanni Inchiostro (University of Washington)

Wall crossings and moduli spaces

Moduli spaces provide an important tool to study how certain objects, say Riemann surfaces, relate to each other. I will start by discussing a few moduli spaces of compact and open Riemann surfaces, compare them, and explain generalizations to studying arbitrary dimensional spaces.

7 February, 11:00am, Zoom

Anne Dranowski (University of Southern California)

Refinements in representation theory

In representation theory, groups are studied via their actions. An action is a way of associating to the elements of a group symmetries of a vector space. Symmetries are classified with the help of eigenbases. Good eigenbases are those which are compatible with certain natural filtrations. Perfect eigenbases are those which are approximately permutated by generators of the group. We present three examples of perfect bases constructed using very different methods (geometric, algebraic) as well as original results and works in progress comparing them.

6 February, 2:00pm, Evan Williams Theatre

Dr Jian Wang (University of North Carolina)

Mathematical theory of internal waves

Internal waves are a central topic in oceanography and the theory of rotating fluids. They are gravity waves in density-stratified fluids. In a two-dimensional aquarium, the velocity of linear internal waves can concentrate on certain attractors. Locations of internal wave attractors are related to periodic orbits of homeomorphisms of the circle, given by a nonlinear "chess billiard" dynamical system. This relation provides a surprising "quantum--classical correspondence" in fluid dynamics. In this talk, I will explain connections between homeomorphisms of circles, spectral theory, and internal wave dynamics. This talk is based on joint work with Semyon Dyatlov and Maciej Zworski.

1 February, 3:15 - 4:15pm

Slava Futorny (SUSTech Shenzhen)

Representations of Lie algebras of vector fields on algebraic varieties

Derivations of a ring of polynomial functions on affine algebraic varieties is a rich source of simple infinite-dimensional Lie algebras. We will discuss the state of the art of the representation theory of these Lie algebras.

25 January, 3:15 - 4:15pm, Evan Williams Theatre

Erik Carlsson (UC Davis)

A descent basis for the Garcia-Procesi module

The Tanisaki ideal encodes the generators and relations in the cohomology ring of the regular nilpotent Springer fiber, whose graded character with respect to a certain action of the symmetric group was famously shown by Springer to be the Hall-Littlewood polynomial. The corresponding quotient ring was studied as a module by Garsia and Procesi, who gave an explicit basis of monomials, which is a subset of the Artin basis of the coinvariant algebra. I'll present a new basis consisting of certain Garsia-Stanton descent monomials, as well as some connections to the Garsia-Haiman ring, which encodes the modified Macdonald polynomial. This is joint work with Ray Chou.

Previous semesters

Semester 2 2023 

Semester 1 2023 

Semester 1 2019