Fridays 3:15 - 4:15pm
Peter Hall Building, Room 162
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25 October (Week 13):
Pure Mathematics Seminar: Jack Hall (University of Melbourne)
Full faithfulness for Deligne-Mumford stacks
In the mid-90s, Bondal and Orlov gave a simple criterion to check for when a functor between bounded derived categories of smooth projective complex varieties is fully-faithful. This type of result is frequently used to relate derived categories of smooth projective varieties with derived categories of moduli spaces of sheaves on them. I will describe an extension of Bondal-Orlov full faithfulness to smooth Deligne-Mumford stacks over fields of characteristic 0, which is a mild generalization of some recent work of Lim-Polischuk.
11 October (Week 11):
Student Pure Mathematics Seminar: Adam Monteleone (University of Melbourne)
Fourier-Mukai Transforms and Derived Equivalences
The theory of derived categories, introduced by Grothendieck and Verdier, provides a powerful tool for studying the geometry of algebraic varieties via their categories of coherent sheaves. In 1981, Mukai introduced the Fourier-Mukai transform while studying abelian varieties, and it has since become a fundamental tool used in understanding when the derived categories of two varieties are equivalent.In this talk, I will setup the basic theory of derived categories, introducing the Fourier-Mukai transform and the Bondal-Orlov full faithfulness criterion. If time permits, I will discuss the role of the Fourier-Mukai transform in Kontsevich's homological mirror symmetry conjecture.
4 October (Week 10):
Pure Mathematics Seminar: Chenyan Wu (University of Melbourne)
Explicit relation between invariants from Eisenstein series and theta lifts, with an application to Arthur packets
To a cuspidal automorphic representation of a classical group or a metaplectic group and a conjugate self-dual character, we associate an Eisenstein series and a family of representations that are called theta lifts. We establish a precise relation between the poles of the Eisenstein series and the lowest occurrence index among the theta lifts.
As an application, we show that certain Arthur packets cannot contain cuspidal automorphic representations.
20 September (Week 9):
Pure Mathematics Seminar: Thorsten Hertl (University of Melbourne)
Moduli spaces of Riemannian metrics with positive curvature
A manifold, which a priori a topological object, can be turned into a geometric object by equipping it with a Riemannian metric. This choice, however, is completely arbitrary, and one may very well wonder how geometric meaningful quantities, like the diameter, the volume, the total curvature or the geometric mass, depend on the Riemannian metric. Usually, these (global) quantities are invariant under the group of diffeomorphisms, so they should be studied as function on the quotient, which we refer to as the moduli space.
After a quick reminder of the notions of Riemannian metrics and the different notions of curvature, I will quickly outline the history of research concerning the moduli space of positive curvature metrics, and I will explain why it is better from a topologists perspective to study the observer moduli space. Finally, I will provide examples coming from current research.
13 September (Week 8):
Student Pure Mathematics Seminar: Chengjing Zhang (University of Melbourne)
An introduction to the theta correspondence
In 1964, to simplify Siegel’s analytical study of quadratic forms over integers, Weil generalised the oscillator representation in quantum mechanics to other settings and introduced an automorphic reincarnation of Jacobi’s theta function. In 1979, inspired by the classical invariant theory, Howe used the oscillator representation and the adelic theta function to construct the so-called theta correspondence.
There are two closely related versions of the theta correspondence: the local version is a ‘duality' between irreducible smooth representations of certain pairs of algebraic groups (for example, symplectic groups and orthogonal groups of even-dimensional quadratic spaces), while the global version is a method to transfer cuspidal automorphic representations between certain pairs of algebraic groups.
In this talk, I will discuss Weil’s generalisation of the oscillator representation and the theta function, and how they are used to construct the theta correspondence.
6 September (Week 7):
Student Pure Mathematics Seminar: Louie Bernhardt (University of Melbourne)
The Role of Curvature in Geometry
For thousands of years, the notion of curvature has been a touchstone of mathematics. From Archimedes to Einstein, curvature has not just been of intrinsic interest to mathematicians, but has fundamentally shaped the way we understand the (curvy) world around us. Nowadays, the idea of curvature lies in the disciplines of differential geometry and topology. These are large, active fields of research which interact with many areas of mathematics, physics, and other physical sciences. In this talk I will discuss the notion of curvature, and the important role it plays in the study of geometry. I will begin with a brief history of the field, touching in particular on Gauss' genius insights into the geometry of surfaces. Then, after a recap of some important concepts from differential geometry, I will discuss the relevance of curvature in modern mathematical research. This will include the study of general relativity, and of Riemannian manifolds with positive curvature.
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.
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.
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
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.