As a consequence of COVID-19 safety
precautions, the seminar continues to be held in hybrid
mode.

Time & location in semester 2, 2022: Fridays 12:10pm in**Napier
G04 and Napier G03** and on Zoom. Contact: Thomas
Leistner.

#### ~ Past talks in 2022 ~

Time & location in semester 2, 2022: Fridays 12:10pm in

- See schedule
for 2023

*Timothy Trudgian (UNSW Canberra)*

**Title: The Riemann Hypothesis: severely undervalued at a meagre one million dollars**Friday, 16 September at 12:10pm on Zoom, link by e-mail, and in Napier G04

Abstract: Even though a solution to the Riemann Hypothesis lands the luck solver with a million dollars (and USD at that!) this still seems very cheap, given the difficulty of the problem. I shall outline some history of the problem and the very limited partial progress towards it.

*David Ridout (University of Melbourne)*

**Title: Wess-Zumino-Witten models: integer versus fractional levels.**

Friday, 2 September 2022 at 12:10pm on Zoom, link by e-mail, and in Napier G04

Abstract: The nonnegative-integer-level Wess-Zumino-Witten models are well known toy models for strings propagating on curved spacetimes. They have beautiful mathematical properties, for example the modules of the associated vertex operator algebra form a modular tensor category!

Less well known are their fractional-level cousins. Interest in these has surged recently because of applications to the 4D/2D correspondence of Beem et al, new approaches to affine W-algebras related to the geometric Langlands program, and ongoing interest in logarithmic conformal field theory.

I will review some of what is known about the integer and fractional cases in the simplest example. If time permits, I will discuss a new technique for fractional cases to classify modules and understand their modularity.

*Serena Dipierro (UWA)*

**Title: Boundary behaviour of nonlocal minimal surfaces**Friday, 12 August 2022 at 12:10pm on Zoom, link by e-mail, and in Napier G04

Abstract: Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors from the ones minimizing the classical perimeter. We will investigate some structural properties of nonlocal minimal surfaces, and in particular we will discuss the "stickiness phenomenon", namely the strong tendency of adhering at the boundary of the reference domain.

*Aleksander Simonic (UNSW Canberra)*

**Title: Conditional estimates for the log-der of L-functions**Friday, 29 July 2022 at 12:10pm on Zoom, link by e-mail, and in Napier G04

Abstract: In this talk I will present recent progress (joint work with A. Chirre and M. V. Hagen) in obtaining conditional (GRH) and effective estimates in the q-aspect for the logarithmic derivative of Dirichlet L-functions. Possibilities to generalize similar results to functions in the Selberg class will also be discussed (joint work in progress with N. Palojarvi).

*Kevin Coulembier (University of Sydney)*

**Title: Perfecting group schemes**Friday, 3 June 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: Recent results in algebraic geometry, as well as in the theory of tensor categories, motivate studying the (well-known) process of taking the inverse limit of an affine group scheme (over a field of positive characteristic) along the Frobenius homomorphism. This is the ‘perfection’ of the group scheme.

I will focus mainly on the perfection of reductive groups. In particular, I will discuss their classification in combinatorial terms, the relation with topological localisation of classifying spaces and with generic cohomology.

This is joint work with Geordie Williamson.*Pedram Hekmati (University of Auckland)*

**Title: Twisted Milnor metric for finite group actions**Friday, 27 May 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: The Milnor metric is an invariant of unitary flat vector bundles on compact manifolds. Its analytic counterpart was introduced by Ray and Singer and the equality of these norms is the celebrated Cheeger-Muller theorem. In this talk we explain how to extend these invariants to certain flat superconnections in the presence of finite group actions.

*Anna Romanov (UNSW)*

**Title: Geometric representation theory of affine Lie algebras**Friday, 13 May 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: D-modules on flag varieties are a useful tool for studying representations of (finite-dimensional) semisimple Lie algebras. In fact, thanks to the Beilinson-Bernstein localisation theorem, we can obtain a complete understanding of representations of complex semisimple Lie algebras using D-modules techniques. However, when we shift our attention to (infinite-dimensional) affine Lie algebras, the situation is not so cut and dried. In this talk, I’ll discuss some approaches for studying representations of affine Lie algebras using D-modules on ind-schemes and their limitations. In particular, I’ll describe a class of representations of an affine Lie algebra that I am especially interested in – Whittaker modules – and I will explain recent work with Emily Cliff (Sherbrooke) and Gurbir Dhillon (Yale) which lays the foundations for a D-module approach to classifying such representations.

*Ole Warnaar (University of Queensland)*

**Title: Virtual Koornwinder Integrals**Friday, 6 May 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre,

Abstract: Virtual Koornwinder integrals are deformations of integrals over classical group characters that can be used to express characters of affine Lie algebras combinatorially. In this talk I will first explain the classical theory and its connection to Gelfand pairs, and then talk about some open problems and conjectures concerning Virtual Koornwinder integrals.

*Fei Han (National University of Singapore)*

**Title: Characteristic numbers and index theoretic invariants for 24 dimensional string manifolds**

Friday, 22 April 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: A manifold M is called string manifold is its free loop space LM is spin. There are many studies on the string geometry. Dimension 24 is in particular interesting for string geometry. In the talk, I will report our work on the study of characteristic numbers and index theoretic invariants for 24 dimensional string manifolds and string cobordism following Mahowald-Hopkins. This represents our joint work with Ruizhi Huang.

*Michael Albanese**(**CIRGET Montreal**)*

**Title: Spin^h and further generalisations of spin**Friday, 1 April 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: The question of which manifolds are spin or spin^c has a simple and complete answer. In this talk we address the same question for the lesser known spin^h manifolds which have appeared in geometry and physics in recent decades. We determine the first obstruction to being spin^h and use this to provide an example of an orientable manifold which is not spin^h. The existence of such an example leads us to consider an infinite sequence of generalised spin structures. In doing so, we determine an answer to the following question: is there an integer k such that every manifold embeds in a spin manifold with codimension at most k? This is joint work with Aleksandar Milivojevic.

Link to the recording (Passcode: D!C8Uhb+)

*David Baraglia (University of Adelaide)*

**Title: Non-trivial smooth families of K3 surfaces**Friday, 18 March 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: We will show that the fundamental group of the diffeomorphism group of a K3 surface contains a free abelian group of countably infinite rank as a direct summand. Our construction relies on some deep results concerning Einstein metrics on K3, such as the global Torelli theorem. Non-triviality is detected using a families version of the Seiberg-Witten invariants.

*Michael Albanese**(**CIRGET Montreal**)*

**Title: The Yamabe Invariant of Non-Kähler Surfaces**Friday, 4 March 2022 at 12:10pm on Zoom, link by e-mail, and in Marjoribanks 126 SANTOS Lecture Theatre

Abstract: The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We consider the extent to which this remains true when the Kähler hypothesis is removed.