Academic Year 2024-25
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Talks are on Mondays at 4pm in the MALL, unless otherwise stated. By custom, there is an offering of coffee and cakes after!
To give a talk, or if you have a preference for post-seminar treats, contact one of the organisers:
Previous incarnations of this seminar can be found here.
Abstract: Vortices are topological solitons arising in the Ginzburg-Landau theory of superconductivity. They have been observed experimentally and explain some of the magnetic properties of type-II superconductors. A remarkable feature of low-energy vortices is that they can be completely described by algebro-geometric data. Moreover low-energy vortex dynamics can be described by the path of a point in a high dimensional moduli space which parametrises all possible data.
In this talk, I will introduce moduli spaces in the context of 2-vortices on elliptic curves. After a brief review of some algebraic geometry, we will construct the moduli space of 2-vortices on an elliptic curve and show that it is a fibre bundle. Time permitting, we will discuss some recent results on approximations to low-energy vortex dynamics and construct exact solutions for approximate 2-vortex motion on an elliptic curve.
NOTE THE CHANGE OF ROOM: ROGER STEVENS LT 15
Abstract: We go over some basic results of recursion theory and the undecidability of the halting problem. We then lay out a standard proceedure used to show that a problem is undecidable and prove a few classic examples.
Abstract: A cellular algebra is an (associative and unital) algebra with a specified basis, defined by certain multiplicative properties. This notion was crystallised by Graham and Lehrer in the late 90’s in what has proven to be a popular work in modern representation theory. In this talk we will present this notion and its representation theoretic consequences, all while following a friendly example for concreteness.
Abstract: Permutation models are a technique developed in the 1920s-30s to begin trying to answer questions about the independence of AC from ZF. Instead of working in the usual ZF set theory, we instead construct a model of ZFA (ZF with “atoms”: distinct, non-set objects). By then shaving down our model to only a collection of “symmetric enough” objects, we can obtain models of ZFA that violate choice in some strange ways. I shall introduce this concept and show off some models that give us independence results.
Abstract: Lie groups are used to study continuous symmetries and have applications in numerous areas of maths, from mathematical physics to number theory. The tangent space to the identity of a Lie group is a Lie algebra, hence Lie algebras are a useful tool in order to understand Lie groups. In this talk we will look at Lie algebras and Lie groups from an algebraic perspective, firstly reviewing basic notions of Lie algebras, before discussing at the Lie algebra of a matrix Lie group, and some of the properties preserved between these two objects.
Abstract: Kolmogorov complexity is a way of showing how compressible a finite binary string is. This has useful applications when looking at algorithmic randomness of infinite strings by looking at the compressibility of initial segments of the sequence, this can be shown to be equivalent to other notions of randomness such as Martin-Löf randomness. We begin by introducing some basic computability theory, then we go on to define plain and prefix-free Kolmogorov complexity with some proofs relating these notions. At the end we will briefly discuss the application to randomness.
Abstract: A first-order structure is said to be homogeneous if any isomorphism between two finitely generated structures of extends to an automorphism of . In this talk we will consider examples of such structures as well as introducing Fraïssé’s theorem, a powerful result determining when such structures arise. We will also look at other results in the study of homogeneous structures.
Abstract: TBA.
Abstract: TBA.