# Pure Mathematics Seminar 2018 Semester 2

The Pure Mathematics Seminar meets Fridays 3:15pm-4:15pm in Peter Hall Room 213.

Organisers: Jesse Gell-Redman, Ting Xue

### Schedule

Dec 7 Masoud Kamgarpour (Queensland)

Title: Topology of the moduli space via arithmetic geometry and representation theory

Abstract: Let G be a complex reductive group and X a Riemann surface. It is known that the following moduli spaces are isomorphic:

1. Solutions to the Hitchin (aka 2d Yang-Mills) equations

2. Flat G-bundles

3. Higgs bundle

4. Representations of the fundamental group (aka character variety)

Thus, there has been a huge interest in understanding the topology of this space by people from diverse parts of mathematics. Despite some recent breakthroughs (e.g. Schiffmann’s formula for the Betti numbers of moduli space for $G=GL_n$ in the non-singular case), much remains to be done. For instance, we know very little about the singular case or when $G=Sp_4$.

Inspired by the pioneering work of Hausel and Rodriguez-Villegas  we count points over finite fields and use Weil’s conjecture to determine the Euler characteristic of the moduli space for $G=GL_n$ in the singular case. The point of departure for us is that we interpret the word “space” to mean “stack” as oppose to “variety”. The approach will work for general group $G$, provided one can extract the relevant information from representations theory of the finite group $G(\mathbb{F}_q)$.

Based on a joint project with David Baraglia.

Nov 23 Greg Martin (The University of British Columbia)

Title: Prime number races

Abstract: This talk is a survey of “prime number races”. Around 1850, Chebyshev noticed that for any given value of $x$, there always seem to be more primes of the form $4n+3$ less than $x$ than there are of the form $4n+1$. Similar observations have been made with primes of the form $3n+2$ and $3n+1$, primes of the form $10n+3,10n+7$ and $10n+1,10n+9$, and many others besides. More generally, one can consider primes of the form $qn+a,qn+b,qn+c,\dots$ for our favorite constants $q,a,b,c,\dots$ and try to figure out which forms are “preferred” over the others—not to mention figuring out what, precisely, being “preferred” means. All of these “races” are related to the function $\pi(x)$ that counts the number of primes up to $x$, which has both an asymptotic formula with a wonderful proof and an associated “race” of its own; and the attempts to analyze these races are closely related to the Riemann hypothesis—the most famous open problem in mathematics.

Nov 9 Gregoire Leoper (Monash)

Title: Reconstruction of missing data by optimal transport: applications in cosmology and finance

Abstract: Optimal Transport is an old optimisation problem that goes back to Gaspard Monge in 1781. I will give some historical perspective of the problem and its solutions, and then present some recent results where techniques from Optimal Transport can be used, going from the problem of reconstruction of the early universe, to a problem of model calibration in finance.

Nov 2 James Borger (ANU)

Title: Lambda-rings and Hilbert’s 12th Problem

Abstract: Hilbert’s 12th Problem asks whether it’s possible to generate in an explicit way all the extensions L of a number field K which are Galois with abelian Galois group. It had been known since Kronecker that one could do this using special values of the exponential function (roots of unity) when K is the field of rational numbers, or special values of elliptic and modular functions when K is an imaginary quadratic field. Since the mid-20th century, this has typically been expressed in the language of commutative algebraic groups with large endomorphism rings, and partial results were established in some new cases by Shimura and others. In this talk, I’ll review some of this history and then explain a new framework for Hilbert’s 12th Problem based on a generalization of the notion of lambda-ring in algebraic K-theory. This talk reports on joint work with Bart de Smit.

Oct 26 Anna Beliakova (UZH)

Title: Quantum annular Khovanov homology

Abstract: In 1999 Khovanov assigned to a knot (or link) a complex whose homotopy type is a link invariant and whose Euler characteristic is the Jones polynomial.This construction is known as a categorification of the Jones polynomial. In the talk I will explain Khovanov’s approach and show that it can be extended to annular links in a way sensible to annular link cobordisms. This is a joint work with K. Putyra and S. Wehrli.

Oct 19 Amnon Neeman (ANU)

Title: Approximable triangulated categories

Abstract: We will begin the talk with a theorem in algebraic geometry, about the relation between the derived categories $D^b_{coh}(X)$ and $D^{perf}(X)$ where $X$ is a noetherian scheme. The theorem represents a major improvement over what was known.

It turns out that the result is a relatively straightforward corollary of the (somewhat technical) theorem that the category $D_{qc}(X)$ is approximable. Approximability is a new notion, a recently introduced useful tool we will explain.

To illustrate the power of the new technique we will end the talk with several more applications, which we will then compare with what was known.

Oct 12 Alex Ghitza (Melbourne)

Title: Differential operators on modular forms, and Galois representations.

Abstract: Since a modular form is a holomorphic function, it is tempting to take its derivative. However, this destroys the modularity property. Several approaches exist for “fixing” this problem, and the resulting objects have many arithmetic applications.

I will discuss such differential operators on various types of modular forms (mod p), indicate a few ways of constructing them, and describe the effect of these operators on the Galois representations attached to Hecke eigenforms.

(This is an amalgamation of various projects joint with Owen Colman, Ellen Eischen, Max Flander, Elena Mantovan, Angus McAndrew, and Takuya Yamauchi.)

Sep 27, 3:15-4:15pm, Alexander Stoimenov (Korea, Gwankju Institute of Science and Technology)

Special location: Evan Williams Theatre

Title: Exchange moves and non-conjugate braid representatives of links

Abstract: We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braid representatives of links. As a consequence, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move. We discuss a project to give some fairly general conditions on the linking numbers of a link, so that it has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move.

Sep 7 Mumtaz Hussain (La Trobe University)

Title: The Generalised Baker–Schmidt Problem (1970)

Abstract: The Generalised Baker–Schmidt Problem, inspired by the pioneering work of Alan Baker and Wolfgang Schmidt (1970),  is a central problem in metric Diophantine approximation on manifolds. It concerns the estimation of $f$-dimensional Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold.  In this talk, I will explain resolution of this problem for a parabola, planar curves and hypersurfaces. These result are the first of their kind.

This is a joint work with David Simmons (York) and Johannes Schleischitz (Ottawa).

Aug 31 Chenyan Wu (Melbourne)

Title: Arthur parameters, Theta Correspondence and Period Integrals

Abstract: We give a brief overview of the theory of theta correspondence and show how it manifests in the Arthur parameter attached to an irreducible cuspidal representation of a symplectic group. We also propose a refinement in terms of period integrals.

Aug 24 Xinwen Zhu (Caltech)

Title: Hilbert’s twenty-first problem for p-adic varieties

Abstract: Hilbert’s twenty-first problem, formulated to generalize Riemann’s work on hypergeometric equations, concerns the existence of linear differential equations of Fuchsian type on the complex plane with specified singular points and monodromic group. Its modern solution, due to Deligne and known as the Riemann-Hilbert correspondence, establishes an equivalence between two different types of data on a complex algebraic manifold X: the representations of the fundamental group of X (topological data) and the linear systems of algebraic differential equations on X with regular singularieties (algebraic data).

I’ll review this classical theory, and discuss some recent progress to solve similar problems for p-adic manifolds.

Aug 17 Tony Licata (ANU)

Title: Categorical taffy

Abstract:  The Artin braid group appears prominently in several mathematical subjects, including both the geometry of surfaces and the representation theory of Lie algebras and quantum groups.  The goal of this talk will be to motivate the study of higher, categorical representations of braid groups by illustrating how some of the structure of interest on the geometric side of braid theory (Teichmuller space, dynamics…) also emerges from higher representation theory.

Aug 10 Iva Halacheva (Melbourne)

Title: Schur-Weyl duality and Lie superalgebras

Abstract: In the classical setting, Schur-Weyl duality describes an interaction between the symmetric group on d elements and the general linear Lie algebra gl(n), in terms of their action on d tensor copies of the vector representation of gl(n). This approach has been extended by Arakawa and Suzuki, and later Brundan and Kleshchev, to more general gl(n)-representations by upgrading the symmetric group to the degenerate affine Hecke algebra. A further generalization includes replacing gl(n) by sp(2n) or so(n), and the symmetric group by the Brauer algebra respectively. I will review some of these constructions and then discuss another instance of Schur-Weyl duality for the periplectic Lie superalgebra. One aspect which makes this case more unusual is the trivial action of the center of the universal enveloping algebra, and so a more elaborate construction than the standard Casimir element is required.

August 2, 3-4pm,  Jack Hall (Arizona)

Special location Building 165 (Chem/Bio), G20

August 3, 3:15-4:15pm, Deepam Patel (Purdue)

4:15-5:15pm,  Graeme Wilkin (National U. Singapore)

July 6 Francois Petit (University of Luxembourg)

Title: Quantization of spectral curves and DQ-modules.

Abstract: In this talk, I will explain how the quantization of spectral curves associated to Higgs bundles can be studied from the standpoint of DQ-modules. In particular,  I will prove that given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, there exists an holonomic DQ-module supported by the  spectral curve associated to this bundle.