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.