# Honours Seminar

**Talk 1 (Dean Muir): Krylov subspace methods.**

Supervisors: Prof Matthew Hole, Prof Markus Hegland.

Krylov subspace methods are a class of projection techniques, which find approximate solutions a linear system $\mathbf{A}\vec{x}=\vec{b}$. They are used to construct a subspace of $\mathbb{R}^n$, the subspace is defined by:

$$

\mathcal{K}_m(\mathbf{A},\vec{r})=span\{ \vec{r},\mathbf{A}\vec{r},\mathbf{A}^2\vec{r},...,\mathbf{A}^{m-1}\vec{r} \},

$$

where $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\vec{r}\in\mathbb{R}^n$, $m\leq n$. Computational methods that utilise Krylov subspaces are amongst some of the most popular for solving systems of linear equations, due to their ability to represent a large, intractable problem in a much smaller space, thereby making it solvable.

In this talk I will introduce Krylov subspace methods. Additionally, I will provide an example of how to construct and utilise a Krylov subspace method to solve a time dependant 2-dimensional nonlinear PDE, describing flow though a non-homogeneous porous medium.

**Talk 2 (Keeley Hoek): Noether's Theorem and the Variational Bicomplex**

A modern way to formulate classical mechanics and classical field theory is in terms of the variational bicomplex. Given a manifold $M$ which represents points of space amd time and a fiber bundle $E \to M$, a section of this bundle can be thought of as the data of the configuration of the (e.g. electromagnetic, gravitational) fields at every point of spacetime. In this situation the jet bundle of $E \to M$ has a de Rham complex which splits into bigraded pieces; the two differentials correspond to changes in space and time, and changes in the field configuration. The field configuration which occurs physically is governed by a functional on local sections of the jet bundle called the Lagrangian, and every generalised symmetry of the Lagrangian gives rise to a conserved quantity (e.g. energy or momentum)---this is called Noether's theorem.

**Talk 3 (James Martini): Phase retrieval problem.**

The phase retrieval problem in it's most general form is the problem of recovering a signal from measured data (usually the measured data is the magnitude of the Fourier coefficients, that act as constraints on the function in the Fourier domain) and a priori suppositions on the structure of the signal itself (which act as constraints on the function in the object domain, usually this can be positivity or sparsity). Using these constraints, algorithms can be developed and convergence analyses can be performed. In my talk, I will mention how some applications map onto constraint types, and how the current and past literature has dealt with such constraint types, i.e. different algorithms and their advantages and disadvantages. Essentially, the presentation will be a problem statement and overview of the papers I have read so far.

**Talk 4 (Mitchell Rowett): A category theoretic approach to the distributivity of the natural numbers**

This talk will serve as an introduction to basic category theory, using the category of finite sets as an example. In particular, it will cover two of the more fundamental results of category theory - the Yoneda lemma and the theorem that right adjoints preserve limits. This second theorem, when specialised to the category of finite sets, can be used as a proof of the distributivity law of the natural numbers.