21 May 2024:
An introduction to Fourier decoupling theory,
Zane Li (North Carolina State University)
Abstract: Fourier decoupling theory was first introduced by Thomas Wolff in 2000. Since the proof of decoupling for the paraboloid by Bourgain and Demeter in 2014, decoupling has had a wide range of applications in analytic number theory, geometric measure theory, and PDE. For example, in 2015, the long standing main conjecture in Vinogradov's Mean Value Theorem, a conjecture about the number of integer solutions to a particular system of equations, was proven by Bourgain, Demeter, and Guth as a corollary of Fourier decoupling for the moment curve. In this talk, I will try to explain important features that make decoupling effective and explain various tools and techniques used when trying to prove and think about decoupling estimates.
28 May 2024:
Higher-dimensional analogues of stable curves and of MMM invariants,
Valery Alexeev (University of Georgia)
Abstract: The enumerative geometry of the moduli spaces of curves, which began in the 1980s with a famous paper of Mumford, is now an extremely developed field, with perhaps thousands of papers dedicated to it. Some highlights: MMM (Morita-Miller-Mumford) classes, Mumford's conjecture (Madsen-Weiss theorem), Witten's conjecture (Kontsevich's theorem), Gromov-Witten invariants.
After reviewing this "one-dimensional" case, I will explain some "higher-dimensional" generalizations: KSBA spaces of stable surfaces and higher-dimensional varieties, kappa classes on them, "higher" Gromov-Witen invariants, etc.
4 June 2024:
Formalization of p-adic L-functions in Lean 3,
Ashvni Narayanan (University of Sydney)
Abstract:
The Kubota-Leopoldt p-adic L-function is an important concept in number theory. It takes special values in terms of generalized Bernoulli numbers and helps solve Kummer congruences. It is also used in Iwasawa theory. Formalization of p-adic L-functions has been done for the first time in a theorem prover called Lean 3. In this talk, we shall briefly introduce the concept of formalization of mathematics in Lean, the theory behind p-adic L-functions, and its formalization.
11 June:
A computational perspective on the Burau representation,
Oded Yacobi (University of Sydney)
Abstract: The Burau representation of the braid group has generated a lot of interest since its appearance almost a hundred years ago. One reason for this is its connection to low-dimensional topology. I'll survey these foundations and discuss some recent computational approaches to study the case of the four-strand braid group, where famously the faithfulness of the Burau representation is still open. This will include joint work with Joel Gibson and Geordie Williamson, and ongoing work with Francois Charton, Ashvni Narayanan and Williamson.
18 June 2024:
Leavitt algebras,
Roozbeh Hazrat (University of Western Sydney)
Abstract: We give an overview of these fascinating algebras, the current lines of research, open questions and conjectures. These algebras appear in non-commutative ring theory, symbolic dynamics, operator algebras, group theory and even chips firing. We will give a tour...
19 June 2024, 12:00-1:00pm:
[Note unusual day] Forest-skein groups,
Arnaud Brother (UNSW)
Abstract: Motivated by Vaughan Jones' reconstruction program of conformal field theories we have defined a class of groups constructed with tree-diagrams. They form a vast class of groups, called "forest-skein groups", that are interesting on their own, satisfy exceptional properties and have powerful extra-structures. We will present explicit examples and explain how the theory of forest-skein groups is surprisingly well-behaved. Some results are joint work with Ryan Seelig.
25 June 2024:
Algorithms for 3-manifold homeomorphism,
Adele Jackson (University of Sydney)
Abstract: Given two mathematical objects, one basic question we can ask is: are they the same? For groups, for example, there is no algorithm to decide this. In practice there is fast software to answer this question for 3-manifolds, and theoretically the 3-manifold homeomorphism problem is known to be decidable. However, our understanding is limited, and known theoretical algorithms could have extremely long run-times. I will discuss some attempts to better classify the complexity of the 3-manifold homeomorphism problem -- in particular, whether it is in NP -- and discuss the important sub-case of Seifert fibered spaces.
26 June 2024, 2:00-3:00 (Mahler Lecture 2024),
[Note unusual day and time] The Langlands program: a synthesis of
number theory and representation theory,
Matthew Emerton (University of Chicago)
Abstract: The Langlands program posits the existence of
fundamental connections between symmetries of algebraic equations
(Galois groups of algebraic number fields) and representation of Lie
groups arising from harmonic analysis on symmetric spaces. I will try
to explain some of the key ideas underlying these connections, with a
focus on illustrative examples that are also related to important
contemporary developments.
2 July 2024:
Magic trees and mystic matrices, Ian Doust (UNSW)
Abstract: Be amazed at Ian's powers of prognostication! Or at least amused at his attempts at mathemagic.
The mystic matrices are those which contain the distances between some finite set of objects X = {x_1,...,x_m}. The most classical question in the area of 'distance geometry' is whether one can find a subset of Euclidean space with matching distances. Answering this involves looking at various properties of this distance matrix; D_X = (d(x_i,x_j))_{i,j=1}^m. In this talk we shall concentrate on some important examples such as metric trees, or sets of bit strings. In these settings the distance matrices have some rather magical properties.
No knowledge beyond second year mathematics will be required, but a small amount of audience participation may be needed. This joint work with Anthony Weston.
9 July 2024:
Hochschild cohomology of flag varieties and Kostant's tensor square conjecture, Sam Jeralds (University of Sydney)
Abstract: Flag varieties occupy a distinguished position in geometry, representation theory, and algebraic combinatorics and offer a rich source of interplay between these three subjects. At its inception in the mid-twentieth century, the goal of geometric representation theory was to understand the interaction between algebraic objects and their representations and geometric or topological invariants of flag (and related) varieties. In this talk, we will use this classical machinery to relate a relatively-unknown geometric invariant of the flag variety—its Hochschild cohomology—to a representation-theoretic problem known as Kostant's tensor square conjecture.
CANCELLED 16 July 2024:
From H-decompositions to transitive path decompositions,
Ajani De Vas Gunasekara (Notre Dame Australia)
Abstract:
MathJax Example
Let \(\Gamma\) and \(H\) be graphs. An \(H\)-decomposition of a graph \(\Gamma\) is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of \(H\)-decomposition that is highly symmetrical in the sense that the subgraphs (copies of \(H\)) are preserved and transitively permuted by a group of automorphisms of \(\Gamma\). In this talk, I will explore \(H\)-decompositions in general, providing historical context for these decompositions. Additionally, I will delve into transitive \(H\)-decompositions and present our recent results on transitive path decompositions of the Cartesian product \(K_n \Box K_n\) when \(n\) is an odd prime. Part of the talk is joint work with Alice Devillers, UWA. Based on \href{arxiv}{https://arxiv.org/abs/2308.07684} . No prior knowledge is required.
23 July 2024:
Hardy-Littlewood maximal operators on graphs with bounded geometry,
Federico Santagati (UNSW)
Abstract:
MathJax Example
In this talk, we discuss the \(L^p\) boundedness of the centred and the uncentred Hardy–Littlewood maximal operators on the class of trees with \((a,b)\)-bounded geometry, i.e., trees such that every vertex has at least \(a+1\) and at most \(b+1\) neighbours. We provide the sharp range of \(p\), depending on \(a\) and \(b\), for which the centred maximal operator is bounded on \(L^p\). We also extend these results to graphs that are roughly isometric, in the sense of Kanai, to trees with bounded geometry. This is based on joint work with M. Levi, S. Meda, and M. Vallarino.
31 July 2024, 1-2pm: [Note unusual day and time]
Honours Talks
MathJax Example
Wentao Xia, Sobolev Space and Existence Theorems of Weak Solution of Elliptic PDEs: Let \(A\) be an \(n\times n\) matrix and \(x\) is an n-dimensional vector, \(c\) be an n-dimensional vector. In the first-year linear algebra course, we already know the equation \(Ax=c\) admits a unique solution if and only if the homogeneous equation \(Ax=0\) only has a trivial solution. Do we have a similar result for Second Order Elliptic Partial Differential Equations? Yes, we have, which is the Second Existence Theorem of Weak Solutions.
Weak solutions are defined as elements in the Sobolev space, in our talk, we will
1. Introduce the Sobolev space and define the weak solutions of Elliptic PDEs.
2. Then use the Lax-Milgram Theorem to prove the First Existence Theorem of Weak Solutions.
3. Finally, we will use the Rellich-Kondrachov Compactness Theorem and Theory of Compact Operators to show the Second Existence Theorem of Weak Solutions.
Matthew Bignell, :