# Honours Conference

**Satisfaction Guaranteed: **The talks will be colloquium style. In particular, the first 20 minutes will be accessible to anybody who vaguely remembers the content of Algebra 2 and Analysis 2. They will run precisely on time.

**Talk 1 (Monday 9:30): Ladder Categories (Mitchell Rowett)**

Given two modules over a ring, we can form their tensor product over that ring. Moving up to the world of categories, our analogies for rings are given by (linear) monoidal categories. Indeed, we have a notion of a module over a monoidal category C; given two such modules we can define a tensor product over C, which we will call a ladder category (for reasons which will hopefully become obvious!).

**Talk 2 (Monday 10:30): Beyond Expectations, But Within Limits — The Theory of Coherent Risk Measures (Yuxi Liu).**

Expectation lies at the foundation of probability, but its centrality belies its contingency. Abstractly, the expectation of a random variable is a real number that measures some information about the variable, and one may well explore the consequences of replacing expectation by some alternative.Certain alternatives to expectation, termed "coherent risk measures", have been well-investigated in financial engineering, but they are relatively unknown in the field of machine learning. Here, we collect and prove some fundamental properties of coherent risk measures that we believe would be applicable to machine learning.

In chapter 1, we review the concept of risk measures, point out possible deficiencies of the expectation as a risk measure, then provide a historical overview of the study of risk measures in finance and other areas.

In chapter 2, we review basic probability concepts, then define the concept of coherent risk measures and study the geometric properties of their envelope representations. Armed with geometric insight, we prove a Kusuoka representation theorem when the underlying sample space is finite and uniform, and construct counterexamples when it is finite but nonuniform.

In chapter 3, we generalize some basic probability inequalities and concentration inequalities from expectation to conditional value at risk. Then we review Statistical Learning Theory and generalize its fundamental theorem by replacing expectation with spectral risk measures.

In chapter 4, we review limit theorems in probability, then give a new and intuitive proof of the Central Limit Theorem for the empirical estimator of conditional value at risk. We provide numerical evidence to support our results and generate conjectures.

*Supervisor: Robert Williamson
Cosupervisor: Markus Hegland*

**Talk 3 (Monday 1pm): The direct method in the calculus of variations (Douglas Coulter).**

Often, we'll want to find a function $u$ that minimises an integral like

$$ E(u) = \int f(x,u(x),Du(x)) \, dx.$$ But sometimes $E$ lacks a minimiser.

So what conditions on $f$ and the space of functions that $u$ can belong to are necessary and/or sufficient for a minimiser to exist?

*Supervisor: John Urbas*

**Talk 4 (Monday 2pm): Transport Methods for study of Pdes (James King)**

When one considers the dynamics of particles, certain partial differential equations naturally arise. For example, if one to consider the dynamics of an inert gas, one would arrive at the heat equation.

Traditionally, the existence of solutions to the equation above equation are proven usng analytical techniques -- such as the Fourier transform. This thesis outlines an alternative approach using concepts of transport and gradient flows in metric spaces to not only show the existence of solutions to the above equation and many others like it. Examples of partial differential equations that can be analysed using this technique are the porous medium equation, and the quantum drift diffusion equation.

*Supervisor: John Urbas*

**Talk 5 (Monday 3pm): Proving the Riemann Roch theorem using a very big ring (Christopher Hone)**

The Riemann Roch theorem is a beautiful theorem in both complex analysis and algebraic geometry. Using the ring of adeles associated to an algebraic curve, one may prove this theorem cleanly and rapidly. We will introduce this ring, use it to prove the theorem in the case of the Riemann sphere. This example will only use basic properties of complex numbers, rings, power series, and linear algebra. Time permitting, we will then indicate how to bootstrap this example to a proof of the general theorem.** **

*Supervisor: James Borger*

**Talk 6 (Tuesday 9:30): Drinfeld centers and representations of the annular category (Keeley Hoek)**

We explain a folklore result relating a certain algebraic gadget (the Drinfeld center) to a category of diagrams drawn in the annulus. There are applications to the study of topological phases of matter, as well as purely algebraic motivations. We begin with zero prerequisite knowledge on category theory, and explain the string diagram calculus which we use to make sense of everything on the blackboard. There are some (more colourful!) generalisations left for the end.

*Supervisor: Scott Morrison.*

**Talk 7 (Tuesday 10:30): Modal Logic formalization in HOL (Yiming Xu)**

We will start by introducing what is a theorem proving and what is a theorem prover. Then we will talk about what is modal logic and what we did to formalize it in HOL. By the end of the talk, we will sketch a proof of a theorem called 'finite model property', and talk about its formalization in HOL.

*Supervisor: Michael Norrish.
Co-supervisor: Scott Morrison.*

**Talk 8 (Tuesday 1pm): Fourier Phase Retrieval (James Martini)**

Fourier phase retrieval is the problem of recovering a signal $x^{\ast}\in {\mathbb R}^N$ from the modulus, or modulus square of its Fourier transform

$$

c = |{\mathcal F}(x^{\ast})|, \text{ or } |{\mathcal F}(x^{\ast})|^2.

$$

This problem is incredibly illposed, given that there are many transformations that one may apply to $x^{\ast}$ that leave $c$ invariant. As such, some additional constraints must be imposed to guarantee unique reconstruction. Phase reconstruction is incredibly important in signal processing, with applications ranging across most practical sciences where Fourier magnitude measurements of a signal are measured (astronomy, physics, chemistry and biology).

In this talk, I will be going through the development of two methods to solve Fourier phase retrieval for a sparse signal. One method is to distinguish an appropriate regularisation function $g: \mathbb{R}^N \rightarrow \mathbb{R}$ and formulate the problem as a regularised least squares

\begin{align*}

\min_{x\in {\mathbb R}^N} \left\{\frac{1}{2N} \left\| |{\mathcal F}(x)| - c\right\|_2^2 + g(x)\right\}.

\end{align*}

Given conditions on $g$, one may formulate an iterative method called alternating minimisation (AM) with prior knowledge $g$. Different methods can be derived by distinguishing square and linear measurement vectors. In this case, linear measurements are used and the AM iterative method is realised in terms of the Moreau proximal mapping of the prior knowledge $g$. There are convergence theorems for this method with quite technical proofs requiring tools from non-smooth and non-convex analysis.

On the otherhand, given square measurements another method called GESPAR can be developed by directly imposing a L0 sparsity constraint on the signal. The development of GESPAR involves moving imposed constraints into the least squares objective function, and utilising methods to find the support of $x^{\ast}$. The reconstruction capability of this method can be compared with AM when $g$ is chosen to incorporate L1 data (i.e. to favour sparse signals).

*Supervisor: Qinian Jin*

**Talk 9 (Tuesday 2pm): Contact Surgery (Joshua Tomlin)**

Surgery is a fundamental tool in 3-manifold topology for studying and and creating new 3-manifolds. Contact geometry is also a useful and more recently developed tool for studying 3-manifolds. We will explore how to apply surgery techniques to contact 3-manifolds, particularly focusing on their effect on tight 3-manifolds.

*Supervisor: Joan Licata*