Pure Mathematics Seminar

Fridays 3:15 - 4:15pm

Peter Hall Building, Room 162

23 August (Week 5):

Student Pure Mathematics Seminar:  Yifan Guo (Brown University)

Matrix ensembles and zeros of the Riemann zeta function

Odlyzko showed in 1987 that the distribution of the non-trivial zeros of the Riemann zeta function can be described by the spacings of eigenvalues of certain random matrices. These random matrices belong to a class called Gaussian ensembles. In this talk, I will introduce some key examples of Gaussian ensembles and the distribution of their eigenvalues. I will also attempt to link Gaussian ensembles to other well-known problems, such as the Riemann Hypothesis. 

30 August (Week 6):

Pure Mathematics Seminar: Luca Cassia (University of Melbourne)

Virasoro constraints for β-ensembles and generalized Catalan numbers

Random matrix models involve integrals over spaces of matrices with various measures. The generating functions of correlators in these models often have a topological expansion that encodes information about enumerative geometric problems like map enumeration, Hurwitz theory, and intersection theory on moduli spaces. These models also satisfy Virasoro constraints linked to reparametrization invariance of the integrals, which can be expressed as linear differential equations for the generating function. In this talk, I will explore the connection between these aspects of random matrix models. Additionally, I will discuss a 1-parameter deformation, with the deformation parameter β related to the Virasoro algebra’s central charge, and I will show how one can derive a genus expansion of the deformed generating function, where the coefficients are polynomials in β, reducing to generalized Catalan numbers when β=1.

6 September (Week 7):

Student Pure Mathematics Seminar: Speaker TBC

Riemannian metrics, positive curvature and the shape of the universe

13 September (Week 8):

Student Pure Mathematics Seminar: Chengjing Zhang (University of Melbourne)

The theta correspondence: Blood transfusions in number theory

20 September (Week 9):

Pure Mathematics Seminar: Thorsten Hertl (University of Melbourne)

The moduli space of metrics of positive sectional curvature

27 September (Mid-semester break):

No seminar

4 October (Week 10):

Pure Mathematics Seminar: Chenyan Wu (University of Melbourne)

The theta correspondence and L-functions

11 October (Week 11):

Student Pure Mathematics Seminar: Adam Monteleone (University of Melbourne)

Compactifying stacked mirrors with Deligne-Mumford and Fourier-Mukai

18 October (Week 12):

No seminar due to MSc presentations: attend these instead!

25 October (Week 13):

Pure Mathematics Seminar: Jack Hall (University of Melbourne)

Full faithfulness for Deligne-Mumford stacks

16 August (Week 4):

Pure Mathematics Seminar: Jonathan Bowden (Regensburg) https://sites.google.com/view/jpbowden/

The tight geography problem for high dimensional contact manifolds

Contact manifolds arise naturally in the context of classical mechanics as regular level sets of Hamiltonians in phase space satisfying certain convexity properties. More precisely, a contact structure is a totally non-integrable hyperplane field, with classical examples given by left invariant fields on certain Lie groups. Relatively recently  Borman-Eliashberg-Murphy proved an h-principle for so-called overtwisted contact structures, reducing the existence and classification problem to classical obstruction theory. This leads to the problem of studying contact manifolds that exhibit rigidity, so-called tight contact structures. In this talk I will report on some progress on a programme to utilise classical surgery theory together with analytic tools such as Floer homology to construct examples of such tight contact structures. (This is part of joint work with D. Crowley and J. Hammet)

9 August (Week 3):

Student Pure Mathematics Seminar: Diarmuid Crowley

An introduction to contact topology

In this talk I'll review the basic definitions in contact topology and discuss the notions of overtwistedness and tightness for contact structures.  Then I'll take a brief look at the hierarchy of tight contact structures, including the recent work of Bowden, Gironella and Moreno.  Finally, I'll conclude by indicating how bordism theory can be used to formulate and investigate the tight contact geography problem.

This is an introduction to Jonathan Bowden's talk on August 9th.  Good background for both talks is available in the Qanta article on the work of Bowden, Gironella and Moreno

https://www.quantamagazine.org/in-the-wild-west-of-geometry-mathematicians-redefine-the-sphere-20231107/

2 August (Week 2):

Pure Mathematics Seminar: Yuxuan Li

Aldous' spectral gap conjecture and its generalizations

Aldous' spectral gap conjecture asserts that for any finite graph \Gamma with vertex set [n]=\{1, 2, \ldots, n\}, the interchange process and the random walk on \Gamma, both continuous-time Markov chains, exhibit identical spectral gaps. Notably, the random walk with the state space [n] is a subprocess of the interchange process, which operates over the larger state space S_n. After nearly two decades of being unresolved, this conjecture was conclusively validated in 2010.

This presentation aims to introduce several equivalent formulations of Aldous' conjecture from diverse perspectives. Additionally, it will explore various generalizations of the conjecture, emphasizing insights drawn from algebraic graph theory.

26 July (Week 1):

Student Pure Mathematics Seminar: David Lumsden

Card shuffling, the symmetric group and spectral gaps

In this talk I will give an overview of some results related to shuffling cards viewed as a random walk on the symmetric group. In particular the Aldous spectral gap conjecture and results surrounding generating a permutation from random transpositions will be discussed.