S2 2025 Honours Conference

Three-Manifolds as Branched Covers of Lens Spaces

• Speaker: Liam Harcombe

The theory of covering spaces is well-understood via algebraic topology, and branched coverings provide a fruitful generalisation. A branched covering between 3-manifolds is a covering map except along a link in the base space, called the branch set. The classical Hilden-Montesinos theorem states that every closed, orientable 3-manifold is a 3-fold branched cover of the 3-sphere with branch set a knot, thereby bridging the theory of 3-manifolds with knot theory. We investigate an analogue of this theorem over lens spaces instead of the 3-sphere using techniques from surgery and open book decompositions.

Topological Hochschild Homology with a "twist"

• Speaker: Tobin Carlton Van Buizen

We introduce a topological version of Hochschild homology for rings, which turns out to be a "stronger" invariant in some sense. We then extend this to a “twisted” version that takes inputs with a G-action.

Chromatic Homotopy Theory and Topological Modular Forms

• Speaker: Brice Gardner

One of the largest open problems in homotopy theory is to understand and compute the stable homotopy groups of spheres. This is incredibly difficult to do, but theoretically splitting up the stable homotopy of a space into ‘pieces’ makes it morally more approachable, provided we have a means to reassemble them. To do so we consider a collection of distinguished cohomology theories; for n=0, it is ordinary cohomology, and for n=1, we recover a version of K-theory. For n=2, we somehow make a miraculous quest into the world of elliptic curves, and the modular forms of complex analysis.  We will spend this talk providing a brief outline of chromatic homotopy theory, why it’s something topologists should care about, and how on earth the words ‘homotopy’, ‘comodule’, ‘etale’ and ‘holomorphic’ ever have business being in the same sentence!

Morning tea

An Idelic View of Number Theory

• Speaker: Michael Guthrie

A common method of solving problems in the integers is to map the problem to the quotient spaces \(\mathbb{Z}/p^{n_p}\mathbb{Z}\) for each prime \(p\). This method of solving a 'global' problem by breaking down into 'local' components motivates the idea of adeles and ideles, structures that allow us to capture the different analytic completions of \(\mathbb{Q}\) (or finite extensions of \(\mathbb{Q}\)) together in one place. This allows us to leverage analytic tools to solve problems whilst keeping number theoretic structure.
 

Elliptic Curves and Fermat’s Last Theorem

• Speaker: Zirui (Harry) Zhang

Elliptic curves are the intersection of algebraic geometry with algebraic number theory. They connect the abstract notion of algebraic varieties with concrete cubic equations known as Weierstrass equations, and its applications ranges from Diophantine equations to cryptography. In this talk, I will introduce the theory of elliptic curves and show many of their interesting properties. I will talk about rational points on an elliptic curve and show how they can be used to solve certain difficult Diophantine equations. Lastly, I will briefly talk about the role elliptic curves play in the proof of Fermat's last theorem, in particular, how the Frey curve would be "too special" if a solution to Fermat's last theorem existed. 
 

Braids and Categories

• Speaker: Yangda Bei

The braid group arises naturally in many areas of group theory and topology. It is generated by the basic operations of crossing strings over and under one another, which explains its name. Beyond this classical setting, braiding can also be interpreted in more abstract mathematical contexts, where objects can be “twisted” much like strands in a braid. In this talk, we will explore such structures in the setting of category O.
 

Geometric Models of Stability Conditions

• Speaker: Micah Sinclair

Bridgeland stability conditions are used to study the objects and structure of triangulated categories, such as the derived category of representations for a quiver, ($$D^b(Q)$$). A fundamental result in Bridgeland's first paper on stability conditions is that the moduli space of stability conditions for a given category form a complex manifold. However, stability conditions are quite abstract, so these moduli spaces are in general poorly understood. A geometric model for the total stability conditions on $$D^b(Q)$$ was recently developed. This model has already been used to prove an outstanding conjecture, and has the potential to be used to approach further conjectures or be extended to a broader class of stability conditions.
 

Classification Theorems in Complexity Theory

• Speaker: Hadyn Tang

In 1971, Cook showed that the Boolean Satisfiability Problem was NP-complete. In 1972, Karp showed that Cook's result implied the NP-completeness of 21 different problems, including some restrictions of SAT: 3-SAT and exactly-1 positive SAT. On the other hand, it was known by Krom in 1967 that a different restriction, 2-SAT, was in P. In 1978, a more general analysis by Schaefer showed that in fact we have a dichotomy: all restrictions of SAT of a certain form can be classified as either being NP-complete or lying in P. In our talk, we will discuss why classification theorems are important in complexity theory, how to prove them, and how we can apply our knowledge to the field of automated planning.
 

Discretising the Nash Cascade Model

• Speaker: Georgia McCulloch

How can we predict something as unpredictable as the time it will take for a flood to occur? Well, we can start with a very simple case, with minimal variables, and create a model with an analytical solution for said case. But then how might we build up the complexities in this model? And what happens if, in adding these complexities, we move away from our correct solution? In this talk, we introduce the Nash Cascade, a linear cascade of identical reservoirs, and focus on the problem of discretising the output. Whilst there exist formulations that will offer an exact discrete solution, these are often overlooked in favour of models that are more adaptable to replicating non-linear reservoir systems but give incorrect outputs. We introduce these models and look at ways to bridge the gaps between mathematical accuracy and physical accuracy.
 

Afternoon Tea

Black Hole Entropy in General Relativity

• Speaker: Peter Ilyk

In 1973, mathematical physicists Jacob Bekenstein and Stephen Hawking deduced that there is a bijective correspondence between the equations relating different properties of a black holes in general relativity and equations relating macroscopic thermodynamic variables, encapsulated in an area of study known as 'black hole thermodynamics'. One of the key results here is the relation between the area of a black hole’s horizon and its 'entropy' - a measure of the inaccessibility of the information contained beyond the black hole's horizon. In this talk, I will describe the geometry of black holes in the context of differential geometry and general relativity, as well as introducing the physical intuition behind the celebrated black hole entropy relation. This correspondence assists in bridging our understanding between the unification of general relativity with quantum theory, an active area of research in both theoretical and mathematical physics.

The definition of a Rational Segal Conformal Field Theory

• Speaker: Chenxiao (Kevin) Zhou

This thesis explores the mathematical structures behind a rational 2D functorial conformal field theory. A functorial field theory is a paradigm for axiomatising quantum field theory. It entails a functor from a cobordism category into another category of state spaces. In the case of a ration 2D CFT, this leads to the construction of a modular tensor category. We give rigorous treatment for this construction and outline the difficulties that mathematicians have encountered. Overall this thesis serves as a self contained exposition for anyone seeking to learn this area of maths. 

Persistent Homology and Chaos in Plasma Heating

• Speaker: Gabrielle Evans

Particle orbits in a Tokamak fusion reactor exhibit a variety of topological properties that can be identified using persistent homology. In some configurations, particles orbits can become chaotic in phase space. The onset of this chaotic behaviour can be determined using techniques from persistent homology, namely by calculating the persistent entropy. In this talk, we will briefly introduce persistent homology and then discuss its application to detecting phase space chaos.
 

Morning Tea

Fourier and Schur Multipliers of the Group Z, and Beyond

• Speaker: Noah Gorrell

This is a talk about operator theory. To some extent, it is also a talk about harmonic analysis. Specifically, it is about Fourier multipliers, which abound in the latter subject. In the 1960s, Eymard introduced an operator-theoretic perspective on Fourier analysis. Through Eymard's lens, we view a Fourier multiplier as a certain linear transformation on infinite-dimensional matrices. In this way, a certain natural class of Fourier multipliers can be identified with another class of matrix operations: Schur multipliers. In this talk, we explore a recent development on the relationship between Fourier and Schur multipliers, paying particular attention to the additive group of integers.

Random Matrices and Matrix Concentration Inequalities

• Speaker: Jerry Mahajan

Classical random matrix theory provides estimates for the size of random matrices in the limit as their dimension becomes large and when the matrices have a lot of inherent structure. By utilising the theory of free probability, we provide tighter bounds which are non-asymptotic and require very little structure on the underlying matrices. These bounds look at the extent to which the "intrinsic freeness" of a random matrix can capture its spectral properties. This topic lies at the intersection of probability theory, functional analysis and linear algebra 

Overview of the Poincare Conjecture

• Speaker: Tahn Rainbird

A brief look into what Ricci flow is, and how Ricci Flow was used to crack the only solved millenium problem; the Poincare Conjecture.
 

Ricci Flow on Piecewise-Flat Spaces

• Speaker: Aden Power

I discuss what a piecewise-flat geometry is, how to define a metric and measure curvature on one and what happens when you evolve one of those geometries with the famous Ricci flow equation. Then I'll talk about the alternative approach that can be taken in dimensions higher than 2, and how they compare.
 

Modern Methods in Planar Curve Evolution

• Speaker: Lekh Bhatia

The heat equation is the most fundamental dynamic equation, and curve shortening flow is its natural 1-d geometric analogue. The longtime behaviour of curve shortening flow was first resolved by a long series of results in the 80s due to Gage, Hamilton and Grayson. Since these results, there have been multiple groundbreaking refinements to the argument due to Huisken, Bryan, Andrews, Langford, Zhu, and others. In this talk, we will motivate the study of curve shortening flow, and outline the work done specifically on chord-arc estimates over the past two decades. We will finish by introducing some new methods presented in the thesis.
 

Afternoon Tea and Wrap-Up