Pure Mathematics Seminar

2023 Semester 2

Past talks

21 November, 11am, Peter Hall 107:

J.S. Lemay (Macquarie University)

This talk will be an introduction to differential/tangent categories and how they link to differential geometry, algebraic geometry, operads, etc. This talk will be introductory and should be accessible to students who want to know about category theory.

14 November, 3:15 - 4:15pm:

Miles Reid (Univeristy of Warwick)

Fano 3-folds and the Graded Ring Database

Fano varieties occupy the "positive curvature" niche in the classification of varieties. They include many of the familiar varieties such as projective space, low degree hypersurfaces and homogeneous spaces used in representation theory of algebraic groups. 

Fano varieties are a key area of progress in current algebraic geometry. I will give a colloquial presentation of the general state of the subject, and a few nice special cases. 

It goes without saying that the modern theory goes beyond the simple first cases. The Graded Ring Database (work of Gavin Brown and Al Kasprzyk) contains lists of candidate constructions of Fano 3-folds. A few hundred cases can be settled either by explicit constructions or by impossibility proofs, and while it is hard to reach convincing conclusions in the majority of cases GRDB provides a huge source of challenging research problems.

10 November:

Yalong Cao (RIKEN)

Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

BPS invariants were introduced by Gopakumar-Vafa on Calabi-Yau 3-folds, Klemm-Pandharipande on CY 4-folds and Pandharipande-Zinger on CY 5-folds. They are conjectured to be integers (proven in many cases) and have correspondence with Gromov-Witten invariants. On holomorphic symplectic 4-folds, (ordinary) GW and hence BPS invariants vanish, one can consider reduced GW invariants which are usually nontrivial rational numbers. In this talk, we will introduce BPS invariants for such a reduced theory. Joint works with Georg Oberdieck and Yukinobu Toda. 

3 November:

Justine Fasquel (University of Melbourne)

Building blocks for W-algebras

W-algebras are a large family of vertex algebras associated to nilpotent orbits of simple Lie algebras. For classical Lie algebras, they are parametrized by certain partitions. Among the W-algebras of type sl(n) those with nilpotent orbits corresponding to hook partitions (m,1,1,…) of n are the most understood ones. In this talk, we will show that in fact any W-algebras of type sl(n) should be expressed by using several hook-type W-algebras. We will illustrate with examples in small ranks. It’s a work in progress with T. Creutzig, A. Linshaw and N. Nakatsuka.

20 October:

Valeriia Starichkova (UNSW Canberra)

Primes in short intervals

This talk is inspired by my main thesis project on primes in short intervals. I would like to talk about the main ingredients used in the last works in the area which involve sieves, zero-density estimates and some other combinatorial ideas. In particular, we will introduce and talk about sieve methods (such as the linear sieve and Harman's sieve), which play an important role in multiplicative number theory.

13 October:

Christian Haesemeyer (University of Melbourne)

K-theory of singularities, revisited 

Algebraic K-theory is an invariant that reflects algebraic and geometric information in a complicated mix. It has been known since the inception of the field that while K-theory behaves like a (co)homology theory in the regular case, homotopy invariance can fail in the presence of singularities. T Vorst conjectured that this failure can be used to detect singularities over fields; this conjecture has been proved in both characteristic zero and positive characteristic using trace methods, and the algebraic geometry of varieties. I will discuss this - by now classical - work, and then talk about work in progress with Weibel trying to understand failure of homotopy invariance as an expression of (homological) algebraic instead of (algebraic) geometric properties.

2 October:

Bryce Kerr (UNSW Canberra)

Quantitative local to global principles and the sum product problem

I’ll describe some joint work with Jorge Mello and Igor Shparlinski which makes progress on some additive combinatorics problems in prime fields by applying quantitative local-to-global principles.

22 September:

Nora Ganter (University of Melbourne)

An introduction to Grothendieck duality

I will speak about joint work with Simon Willerton and explain how Grothendieck duality, in its simplest setting, naturally arises from Fourier-Mukai theory.

15 September:

Brian Krummel (University of Melbourne)

Analysis of singularities of area minimizing currents

In his monumental work in the early 1980s, Almgren showed that the singular set of an n-dimensional locally area minimizing submanifold T has Hausdorff dimension at most n - 2.  We will discuss a new approach to this problem (joint work with Neshan Wickramasekera) in which we first prove certain regularity properties such as uniqueness of tangent cones at H^(n - 2)-a.e. singular point of T.  We then prove results about the fine structure of the singular set, namely that the singular set is an n-2-dimensional countably rectifiable set and T is asymptotic to a unique homogeneous multi-valued harmonic function at H^(n - 2)-a.e. branch point of T.

8 September:

Joint Pure Maths/Mathematical Physics seminar, Evan Williams Theatre

Ole Warnaar (University of Queensland)

Virtual Koornwinder integrals

Virtual Koornwinder integrals are deformations of integrals over classical group characters that can be used to obtain combinatorial expressions for characters of affine Lie algebras. In this talk I will first describe the main ingredients of the classical theory and its connection to Gelfand pairs, and then discuss generalisations and applications to characters of affine Lie algebras.

1 September:

Gufang Zhao (University of Melbourne)

Towards a cohomological field theory via Lagrangian correspondences

This talk aims to propose a construction of a cohomological field theory using the category of Lagrangian correspondences, and the bi-category of matrix factorizations. A motivating example comes from the GIT quotient of a vector space by an abelian group, in the presence of an equivariant regular function (potential).  In the example, virtual counts of quasimaps from prestable curves to the critical locus of the potential are defined, drawing ideas from the theory of gauged linear sigma models as well as recent developments in shifted symplectic geometry and the Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials, based on work in collaboration with Yalong Cao, are discussed.

25 August:

Yau Wing Li (University of Melbourne)

Endoscopy for affine Hecke categories

Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H.

22 August:

Lisa Carbone (Rutgers University)

Lie group analogs for infinite dimensional Lie algebras

We discuss the question of associating analogs of Lie groups to certain classes of infinite dimensional Lie algebras. We are particularly interested in Kac-Moody algebras, which are the infinite dimensional analogs of simple Lie algebras, and further generalizations of Kac-Moody algebras known as Borcherds algebras. The Monster Lie algebra is an example of a Borcherds algebra and it admits an action of the Monster finite simple group. We discuss recent developments in the construction of a Lie group analog for the Monster Lie algebra.

11 August:

Angus McAndrew (Australian National University)

A descent theorem for K3 surfaces

Descent problems have fascinated mathematicians since ancient times. A modern descent question asks for the field of definition of a given algebraic variety, i.e. whether there is a criterion for when it can be descended from a field to a smaller one. A theorem of Grothendieck gives an answer to this question in the case of abelian varieties and transcendental field extensions. We will discuss a general conjecture inspired by this, and prove it in the case of K3 surfaces, under some hypotheses. The proof uses Madapusi-Pera's work on the Kuga-Satake construction.

4 August:

Daniele Celoria (University of Melbourne)

The 3D index and Dehn surgery

After giving a general introduction to Dimofte-Gaiotto-Gukov's 3D index for cusped hyperbolic 3-manifolds, we'll dive into some of its relations with basic hypergeometric series. Then we'll describe an ongoing effort to prove how the 3D index changes under Dehn surgery. This is work in progress with Profs C. Hodgson and H. Rubinstein.