# Pure Mathematics Postgraduate Seminar 2013-14

## Schedule 2013-14

• Thu 17th October, 2013 - 4:00 PM, MAGIC Room
Lovkush Agarwal - "Finite Ramsey Theory"
Abstract: Loosely speaking, Ramsey Theory is about determining whether you can find order in a large amount of disorder. The most widely known example is that of people in a party: if there are at least 6 people at a party, then you can find three people who have all met each other before, or, three who have not met each other before.

In this talk, I will prove the Finite Ramsey Theorem, which generalises the result described, and Van der Waerden's Theorem, which concerns coloured beads on a string. I will also discuss corollaries and generalisations of these results, including a non-trivial alternative to Naughts and Crosses (or Tic-Tac-Toe).
• Thu 24th October, 2013 - 4:00 PM, MAGIC Room
Jacob Hilton - "The Infinite Ramsey Theorem, Uncountably Long Decimals, and Large Cardinals"
Abstract: The Infinite Ramsey Theorem says: take a countably infinite set of vertices, and join every pair with an edge. Colour the edges using a finite collection of colours in any way. Then there will always be an infinite subset of the vertices between which all edges have the same colour.

A natural question to ask is: what happens if we replace the countably infinite vertex set with an uncountably infinite one, such as the real numbers? Along the way to answering this question, we will encounter such things as "uncountably long (binary) decimals" (formally, the power set of a well-ordering, ordered lexicographically) and so-called "large cardinals", whose existence would have consequences for the consistency of ZFC.

Familiarity with some basic set theory will be useful but not essential.
• Thu 31st October, 2013 - 4:00 PM, MAGIC Room
Ricardo Bello Aguirre - "Fraisse Limits or Ages of ultrahomogeneous structures"
Abstract: In this introductory talk we will cover the basic construction of a Fraisse limit, and give some examples.
• Thu 14th November, 2013 - 4:00 PM, MAGIC Room
Michael Toppel - "Logical truth by substitution of terms and an appendix from Boolos"
Abstract: W.V.O. Quine adapted a new notion of logical consequence, originally going back to Peter Abelard, to characterise logical truth in contrast to Tarski\'s characterisation. In a recent paper, Günther Eder rediscovered an argument formulated by Boolos, challengin Quine\'s understanding of logical truth on the basis of its implications for logical consequence. In my talk I will present the proof given by Quine to support his notion as well the argument given by Boolos to refute it.
• Thu 28th November, 2013 - 4:00 PM, MAGIC Room
James Gay - "P vs. NP, an Introduction to Computational Complexity"
Abstract: In this talk I will introduce the topic of computational complexity, starting with Turing Machines and working up to the P vs. NP problem.
• Thu 5th December, 2013 - 4:00 PM, MAGIC Room
Anja Komatar - "The 2nd Neighbourhood Conjecture"
Abstract: We define stretched graphs and explore their basic properties. We define their period and consider two examples - a period of the stretched graph of a cycle and of a graph consisting of two cycles. We also consider contributions using stretched graphs to the second neighbourhood conjecture.
• Thu 30th January, 2014 - 4:00 PM, MALL 1
Rodrigo Pires dos Santos - "Gauge Theory"
Abstract: An introduction to Gauge Theory will be given via models in classical mechanics.
• Thu 6th February, 2014 - 4:00 PM, MALL 1
Lovkush Agarwal - "WQOs"
Abstract: An introduction to WQO theory will be given via its use in the solution of a question proposed by Erdős, namely, Problem 4358 (Amer. Math. Monthly, 56 (1949), 480).
• Thu 13th February, 2014 - 4:00 PM, MAGIC Room
Steve Trotter - "Non-commutative Analysis via C*-algebras"
Abstract: We offer an introduction to C*-algebra theory with a leaning towards what I'm referring to as non-commutative analysis. Non-commutativity (sometimes referred to as quantum) has a different meaning here than what some may be used to. It turns out that given a sufficiently reasonable topological space we can describe it entirely in terms of the algebra of continuous functions on this space. This algebra of continuous functions can be made into a commutative C*-algebra and we can then consider what properties remain valid when we consider general non-commutative C*-algebras. I will describe this in the talk.
• Thu 20th February, 2014 - 4:00 PM, MAGIC Room
Chwas Ahmed - "Specht Modules of $\mathbb{C}S_n$"
Abstract: After introducing some basic module theory notions about semisimple algebras, I will try to describe the nature of the Specht modules of $\mathbb{C}S_n$ by means of Young diagrams.
• Thu 6th March, 2014 - 4:00 PM, Mall 1
Andrzej Kucik - "Isometric maps between Zen spaces on the complex half-plane and weighted $L^2$ spaces on the real half-line"
Abstract: I will show that the n-th derivative of the Laplace transform determines an isometric map from weighted $L^2$ spaces to some corresponding Zen spaces. I will then show how it could be used to define a new norm on the latter spaces and also how to use it to calculate reproducing kernels of these spaces.
• Thu 20th March, 2014 - 4:00 PM, Mall 1
Jacob Hilton - "Boolean algebras and Stone duality"
Abstract: A Boolean algebra can be viewed as a ring $R$ such that $x^2 = x$ for all $x \in R$. This simple notion gives rise to a rich theory, at the heart of which is Stone duality, a structure theorem for Boolean algebras in terms of a certain kind of topological space.

In this talk I will try to describe the key ideas behind Stone duality, focusing on examples rather than proofs. The theory of Boolean algebras will be developed from scratch, but it will be helpful to be familiar with the notions of a ring and an ideal.
• Thu 27th March, 2014 - 4:00 PM, Mall 1