In March 2023, I will be a speaker and panelist at the Frontiers of Science Forum in Sydney.

15 Sept 2022: Properties of families of projective varieties via their moduli, Behrouz Taji (UNSW Sydney)

Abstract: A celebrated conjecture of Mordell predicted that the genus of a smooth projective curve controls the density of rational solutions for the underlying polynomial equation. In the 1980s, following the footsteps of Shafarevich, Arakelov and later on Faltings observed that, thanks to Deligne-Mumford's striking construction of the moduli of high-genus curves, one can reduce Mordell's conjecture to the problem of establishing a single natural inequality over curves. This inequality, which was fundamental for Faltings' solution to Mordell's conjecture, is nowadays referred to as the Arakelov inequality. My aim in this talk is to give an overview of these interconnected themes and discuss some new advances and open questions in this area arising from recent breakthroughs in moduli theory of higher dimensional varieties. This talk is partially based on a joint work with S. Kovacs.

20 Sept 2022: Varieties with trivial canonical class, Henri Guenacia (CNRS)

Abstract: Compact Kahler manifolds with trivial canonical class play a major role in complex geometry. I will give an historic overview of these manifolds and discuss seminal results like the Calabi-Yau theorem and the Beauville-Bogomolov decomposition theorem. If time permits, I will explain why it is natural and desirable to consider singular versions of these objects and results and I will briefly talk about recent progress in that direction.

27 Sept 2022: Asymptotics of operator semigroups, Abraham Ng (Wollongong)

Abstract: Operator semigroups on Banach or Hilbert spaces can be used to model both continuous and discrete systems that evolve through time. In this talk, I will introduce both strongly continuous and discrete operator semigroups and discuss their asymptotic, i.e. long-term, behaviour. A particular focus will be given to quantifying rates of decay.

4 Oct 2022: Zeta functions: from classical generating functions to abstract harmonic analysis, Sean Lynch (UNSW Sydney)

Abstract: The modern practitioner of zeta functions tends to work with Haar integrals over locally compact groups. This is a far cry from our familiar representation of the Riemann zeta function as a Dirichlet series or as an Euler product. Nonetheless, Tate's Thesis (1950) makes a convincing case for adopting this more abstract perspective. It turns out that the study of many other classical generating functions can be enriched through abstract harmonic analysis e.g. Cauchy's q-binomial series, Eisenstein-Hermite's integral lattice generating function, Euler's partition generating function, and Erdos-Szekeres' abelian group generating function. We see this by relating our classical generating functions to zeta functions of arithmetic modules, which are originally defined as Dirichlet series.

11 Oct 2022: Quantitative local to global principles, Bryce Kerr (UNSW Canberra)

Abstract: This talk surveys techniques and applications related to the following problem: Given an algebraic set in nonzero characteristic, when can we lift to zero characteristic and preserve the algebraic structure? A notion of size/height is important for determining when this problem has a positive solution. Techniques for investigating the extent to which we can perform this kind of lifting have previously been applied to problems in additive combinatorics, distribution of primes, Rudin's conjecture and computer science.

18 Oct 2022: Term break (no seminar)

25 Oct 2022: Seminar cancelled

1 Nov 2022: Seminar cancelled

8 Nov 2022: Representation theory: from finite groups to supergroups, Alex Sherman (University of Sydney)

Abstract: We offer a certain perspective on representation theory with an eye toward motivating supergroups. We will begin with a discussion of the more familiar case of finite groups, highlighting the importance of Sylow p-subgroups in their representation theory. Then we will introduce supergroups, with an emphasis on examples, and explain how one can analogise the definition of Sylow subgroups to the super setting. We finish with a recent theorem which describes explicit analogues of Sylow subgroups for the general linear and orthosymplectic supergroups.

15-17 Nov 2022: Honours presentations (UNSW Sydney)

22 Nov 2022: Noncommutative Fourier Analysis through von Neumann Algebras, Thomas Scheckter (UNSW Sydney)

Abstract: We are interested in the natural extension from Fourier analysis for abelian groups and the corresponding function spaces to non-commutative groups. Unfortunately, this quickly becomes difficult, as the study of the duality of such groups leads to Tannaka-Krein duality and quantum groups. Here we consider only profinite groups, and restrict our attention to those for which the left regular representation generates a semifinite von Neumann algebra. In this setting, we have powerful functional analytic tools built from the spectral theory of these operators which allow us to extend the ideas of classical Fourier analysis. We will discuss convergence of Fourier series in this setting, possible extensions of the Carleson-Hunt theorem, and the difficulties that arise from non-unimodular groups and non-doubling metrics.

24 Nov 2022: Mahler Lecture: The Arithmetic of Power Series, Frank Calegari (University of Chicago)

Abstract: A holomorphic function P(z) of a complex variable around z = 0 has a power series expansion P(z) = \sum a_n z^n. What constraints are imposed on P(z) by assuming that all the coefficients a_n are integers? We discuss some variations on this problem starting with some very elementary observations and leading up to a resolution of a 50 year old conjecture, as well as the surprising links to differential equations and group theory.

29 Nov 2022: 100 years of the Schur-Weyl Duality, Jie Du (UNSW Sydney)

Abstract: Schur algebras are certain finite dimensional algebras introduced by Issai Schur, one of the pioneers in representation theory, at the beginning of last century to relate representations of the general linear and symmetric groups. This theory is also known as the Schur-Weyl duality. Over its history of one hundred years, Schur algebras continue to make profound influence in several areas of mathematics such as Lie theory, representation theory, invariant theory, combinatorics, and so on. I will discuss its latest developments from the quantum version of the duality in late 80s to the affine/super analogues and its recent generalization to types other than A via quantum symmetric pairs.

2 June 2022 (online): A tour through design theory, Santiago Barrera Acevedo (Monash University)

Abstract: In this talk, we will visit three types of designs (sequences with perfect autocorrelation, relative difference sets, and Hadamard matrices). The periodic autocorrelation of a sequence is a measure for how much the sequence differs from its cyclic shifts. If the autocorrelation values for all nontrivial cyclic shifts are 0, then the sequence is perfect. It is very difficult to construct perfect sequences over 2nd, 4th, and in general over n-th roots of unity. It is conjectured that perfect sequences over n-th roots of unity do not exist for lengths greater than n^2. Due to the difficulty to construct perfect sequences over n-th roots of unity, there has been some focus on other classes of sequences with good autocorrelation. One of these classes is the family of perfect sequences over the quaternions. In this talk, I will introduce perfect sequences over the quaternion groups Q_8 and Q_{24}. Such sequences exhibit interesting symmetry patterns, which I aim to explain via a connection with relative difference sets and Hadamard matrices.

9 June 2022: The theorem with the longest proof (??), Peter Donovan (UNSW)

Abstract: This talk begins with the theorem, probably due to J.-P.Serre, 'Each element g of a finite simple group G is a commutator, that is can be written as g=aba^{-1}b^{-1} for some a,b in G'. The proof needs the classification of finite simple groups (CFSG) and much lengthy representation theory as well. All of this leads to a retrospect on some of my old work on modular representations.

16 June 2022: Diophantine approximation and the weighted products of partial quotients in continued fractions, Ayreena Bakhtawar (UNSW)

Abstract: The fundamental objective of the theory of Diophantine approximation is to seek answer to a simple question "how well irrational numbers can be approximated by rational numbers?" In this regard the theory of continued fractions provides quick and efficient way for finding good rational approximations to irrational numbers. In this talk, first I will discuss the relationship between Diophantine approximation and the theory of continued fractions. I relate the three fundamental theories in metric Diophantine approximation (Dirichlet's theorem, Khintchine's theorem and Jarnik's theorem) to the questions in continued fractions. Then I will describe some metrical properties of the product of consecutive partial quotients raised to different powers in continued fractions.

30 June 2022: Representation theory of monoids and monoidal categories, Daniel Tubbenhauer (University of Sydney)

Abstract: There is a MathOverflow question with title "Why aren't representations of monoids studied so much?". Well, I do not know. But it can't be because the representation theory is bad as a theory. In fact, there it is well-developed that is about 80 years old, and this theory is, imho, almost as smooth and beautiful as group representation theory itself. There are even categorifications of it.

This talk is a friendly overview of the representation theory of monoids and how to categorify bits and pieces of this story.

14 July 2022: Braid Group Representations, Nancy Scherich (ICERM)

Abstract: This talk will be a friendly introduction to braid groups and the representation theory of these groups. Braid groups are one of those objects that show up in many different areas of mathematics. This talk will focus on the topological intuition and applications of braid group theory to knot theory and some quantum computing.

21 July 2022: Holonomic generating functions and groups, Alex Bishop (University of Sydney)

Abstract: A recurring theme in group theory is to understand the asymptotics and structure of groups with the use of generating functions that have nice descriptions and closed forms. For example, it is well known that the volume growth series of every hyperbolic/Coxeter group is rational, that is, it can be written as a fraction of two polynomials; and that the cogrowth (i.e. the growth of the word problem) for a hyperbolic group is algebraic (i.e. is a solution to a polynomial equation).

In this talk, we are interested in (multivariable) generating functions coming from groups that can be described using the more general framework of holonomic functions. In particular, we cover several cases where this class of functions arrives naturally. During this talk, we cover some lesser-known techniques and computational models from which holonomic power series arise.

28 July 2022 (online): Braid varieties and cluster structures, Ian Le (Australian National University)

Abstract: I will introduce braid varieties and explain how they generalize Richardson varieties. I will motivate cluster structures on braid varieties through two perspectives. 1) Braid varieties turn out to be log Calabi-Yau, so that one can formulate mirror symmetry between braid varieties. This leads to statements like the Fock-Goncharov duality conjectures. 2) On the other hand, braid varieties are a kind of generalization of moduli spaces of local systems, so that they can be studied in the context of Hitchin systems. This leads one to study weave calculus, which is a degenerate version of Soergel calculus.

4 August 2022 (1pm): The McKay Correspondence on Kleinian Singularities, Tasman Fell (UNSW honours presentation)

Abstract: A finite subgroup G of SU(2) produces a surface singularity by its action on the complex space C^2, called a Kleinian singularity. We can study this singularity by taking its minimal resolution; that is, by constructing a smooth surface that is isomorphic to the singular surface away from the singular point, and that replaces the singular point with a collection of projective lines. Taking the intersection graph of these lines gives a graph with an ADE classification.

The action of G on C^2 also produces a different graph, called the McKay graph, which encodes information about the irreducible representations of G. In 1980, John McKay observed that these two graphs - the McKay graph and the intersection graph - are isomorphic (up to the removal of a specific vertex) for all finite subgroups G of SU(2). This gives us a connection between the geometry of the singularity (given by the intersection graph on its minimal resolution) and the algebraic structure of G (given by the McKay graph). In this talk I will present the construction of the two graphs in the case that G is cyclic, and demonstrate the isomorphism between the two graphs.

11 August 2022 (11am): The probabilistic method and positional games, Spencer Yang (UNSW honours presentation)

Abstract: My Honours project is focused on two topics: the probabilistic method and positional games. The probabilistic method is a non-constructive existence proof technique, pioneered by Paul Erdos. It is often used in combinatorics to prove the existence of an object with a given property, by defining a probability distribution on some set and proving that the property holds with positive probability. We find that the probabilistic method can used in the study of positional games and so we begin to explore the topic of positional games. Positional games give a mathematical footing for the analysis of a variety of two-player games played on a finite board including the classical game of tic-tac-toe. In my talk I will introduce both topics and give some examples.

11 August 2022: Old and New results in arithmetic statistics, Ken Ono (University of Virginia)

Abstract: Studying the statistical behavior of number theoretic quantities is presently in vogue. This lecture begins with a new look at classical results in number theory from the perspective of arithmetic statistics, and then turns to point counts for elliptic curves and K3 surfaces over finite fields. This lecture will use the celebrated Sato-Tate Conjecture (now theorem thanks to Richard Taylor and his collaborators) as motivation for refinements in several directions that arise from special properties of various types of q-series and hypergeometric functions. One of the results will feature the more exotic Batman distribution in the context of K3 surfaces.

18 August 2022: Moduli spaces in algebraic geometry , Kenny Ascher (UC Irvine)

Abstract: Moduli spaces are a tool used to study the classification problem in algebraic geometry, with the prototype being the moduli space of Riemann surfaces. Moduli spaces appearing in nature are rarely compact, and for a variety of reasons it is beneficial to find meaningful compactifications. An active area of current research with many fundamental open questions, is to study moduli spaces of higher dimensional algebraic varieties. Starting with the case of Riemann surfaces, I will survey some recent results in the higher dimensional setting.

25 August 2022: News and developments at Birkhauser and Springer , Thomas Hempfling (Birkhauser and Springer publishing)

Abstract: We present selected publications and give some insights into recent developments in publishing, particularly those using artificial intelligence / machine learning.

24 February 2022: From subfactors to actions of the R. Thompson group, Arnaud Brothier (UNSW Sydney)

Abstract: In his quest in constructing conformal field theories from subfactors efficient machine to construct actions of groups like Richard Thompson's groups. I will briefly explain the story of this discovery. I will then present a general overview of Jones' novel technology giving explicit examples and applications. Some of the results presented come from joint works with Vaughan Jones and with Dilshan Wijesena.

3 March 2022: Steinberg-Whittaker localization and affine Harish-Chandra bimodules, Gurbir Dhillon (Yale University)

Abstract: A useful technique in representation theory is localization. This relates categories of Lie algebra representations and categories of sheaves, and allows one to solve representation theoretic problems using geometry. We will begin by surveying for nonspecialists the works of Beilinson-Bernstein, Bezrukavnikov-Mirkovic-Rumynin, Kashiwara-Tanisaki, and Backelin-Kremnizer, which relate representations of semisimple and affine Lie algebras to D-modules on partial flag varieties. We will then describe some recent work, joint with Justin Campbell, which provides an alternative localization as D-modules on Whittaker flag manifolds, and time permitting discuss some applications.

10 March 2022: Weight polytopes and saturation of Demazure characters, Sam Jeralds (University of Queensland)

Abstract: Demazure modules are certain B-submodules of highest weight irreducible G-modules, for B a Borel subgroup of a semisimple Lie group G. Their characters are given by the Demazure character formula which, while straightforward to implement computationally, does not make immediately obvious which weights appear in the character. In this talk, we will first consider combinatorial incarnations of these characters - the key polynomials - and their Newton polytopes, recalling work of Monical-Tokcan-Yong and Fink-Meszaros-St. Dizier. We then situate this combinatorial perspective in a broader representation-theoretic framework by introducing weight polytopes of Demazure modules, and show how these polytopes shed light on the support of Demazure characters.

31 March 2022: The Riemann Hypothesis: worth a billion dollars? Timothy Trudgian (UNSW Canberra)

Abstract: As anyone who mentions the Riemann Hypothesis in a research grant knows, the Clay Mathematics Institute offers a million-dollar prize for a proof. I shall outline some of the many roadblocks to a proof and hope to convince the audience that the prize offered is off by three orders of magnitude.

7 April 2022: Linking vertex algebras and Wightman QFT, Chris Raymond (Australian National University)

Abstract: Conformal field theory (CFT) has been an active area of mathematical and physical research for the last 30 years. Several different axiomatic approaches have been developed to describe CFT, each with its own strengths and weaknesses. It is then natural to ask, under which assumptions are these different approaches equivalent? In this talk, I will discuss some recent results linking the algebraic approach to CFT given by vertex algebras with the approach to axiomatic quantum field theory given by Wightman. Based on joint work with James Tener and Yoh Tanimoto.

14 April 2022: Cohomology theories for algebraic varieties, Lance Gurney (University of Melbourne)

Abstract: In 1949 Andre Weil proposed a series of conjectures about varieties over finite fields and showed that they would follow from the existence of a 'good' cohomology theory. Twenty years later Alexander Grothendieck constructed such a 'good' cohomology theory and proved all but one of Weil's conjectures. However, in doing so he constructed infinitely many and so began our embarrassment of riches. Today we have even more cohomology theories and for fifty years one of the central problems of arithmetic geometry has been to try to understand how they are related and, just maybe, to find one to rule them all. In this talk I will discuss these cohomology theories, how they are related and recent progress towards their unification.

21 April 2022: Explicit bounds on Exponential Sums of Hecke Eigenvalues, on the modular group SL(2, Z) and congruence subgroups, Gal Aharon (UNSW Sydney honours presentation)

Abstract: Bounds on exponential sums have been an important topic of study in Analytic Number Theory, offering solutions to many problems, including the famous Ternary Goldbach Conjecture and Fermat's Last Theorem. In 2006, L. Zhao published a paper finding bounds on a particular exponential sum with square root amplitude, twisted with Hecke-eigenvalues over the full modular group. In this talk I will present some history of these sums, as well as demonstrating more explicit constants for the bounds found by Zhao, and finally an extension of these bounds to the congruence subgroups.

Tasman Fell. Singular Soergel bimodules. (research masters, UNSW)

Simon Bohun. Beilinson-Bernstein Localization on base affine space. (honours, UNSW)

Victon Zhang. Diagrammatic categories in representation theory. (honours, UNSW)

Amy Bradford. Whittaker modules for affine Lie algebras. (REU, University of Utah 2021)

Dylan Soller. Cracking points of finite Gelfand pairs. (REU, University of Utah 2019)

Faith Pearson. Cracking points of finite Gelfand pairs. (REU, University of Utah 2019)