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