Pure Postgraduate Seminars

The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. If you would like to give a talk or have any comments or suggestions as to the organisation of the seminars please contact Simon Baker or David Naughton. Every week, a reminder will be sent to all Pure Postgrads. If you are not a pure postgrad and would also like to be sent a reminder then please e-mail us.

The seminars are held in the Alan Turing Building, Frank Adams Room 2 (1.212), Fridays from 4pm to 5pm. We will have tea, coffee and biscuits before the seminar at 3:45pm on the Atrium bridge. In the evening, we often go to a pub.

For the Spring/Summer 2012 seminar timetable, please click here.

Autumn Semester 2011

23^rd Sept 2011
A proof of Gabrielov's theorem
Mohsen Khani

Abstract (click to view)

Gabrielov proved that the complement of a subanalytic set is subanalytic. In this talk, we will see how model theory provides us with a shorter and more explicit proof, while the foundational analytic one is long and complicated."
30^th Sept 2011
MRSC 2011

Abstract (click to view)

A day of talks aimed to welcome the new postgraduate students and promote interdisciplinary research. The day begins at 9 o'clock at Hulme Hall.
7^th Oct 2011
A Short Overview of Lie Correspondence
Lewis Topley

Abstract (click to view)

Large classes of groups have some form of geometric structure (manifold, variety, scheme, etc). Nearly 100 years ago some bright spark realised that the algebraic properties of such connected groups could be encapsulated by the infinitesimal behaviour around the identity element, and the concept of a Lie algebra was introduced. In my talk I intend to roughly explain how to attach a Lie algebra to an affine algebraic group, and state some theorems which allow us to learn information about the one from the other. I intend to describe everything in an intuitive manner and not worry about the technicalities of algebraic geometry. If I have time at the end I'll explain why Lie algebras are much better than groups.
14^th Oct 2011
title
Dave Naughton

Abstract (click to view)

So, Szemeredi's theorem says that any subset of the natural numbers which possesses some upper density property, contains arbitrarily long arithmetic progressions. This was proved in 1975 in a lengthy and difficult argument. However in 1979 Ergodic Theorist and absolute legend Hillel Furstenburg found that the theorem was in fact implied by a then unsolved problem in Ergodic Theory, known as multiple recurrence. I'll introduce all the notions from number theory needed, then proceed to give a crash course in ergodic theory, and then hopefully talk about the correspondence between these results. Examples will be included.
21^st Oct 2011
A few examples in algebraic geometry
Andrew Davies

Abstract (click to view)

When first encountering algebraic geometry there is a tendency for the geometry to be obscured. In this seminar I will first give some intuition behind the concepts involved, and following this I'll describe a couple of interesting pictures which illustrate the geometry behind the (commutative) algebra. I'll also associate some geometry to a noncommutative ring via elliptic curves.
28^th Oct 2011
Elliptic Integrals and Schneider's Theorem
Adam Biggs

Abstract (click to view)

Schneider's theorem is an amazing statement that encompasses a wide variety of mathematics. Briefly put it says that the value of an algebraically defined elliptic integral is transcendental. We shall introduce the necessary tools to understand the above sentence.
4^th Nov 2011
Twisted Products in Homotopy Theory
Alexander Longdon

Abstract (click to view)

In algebraic topology we find ways of associating various algebraic objects to topological spaces, and study the spaces using these invariants. Particularly useful such invariants are the homotopy groups of a space, but whilst these are easy to define, they are notoriously difficult to compute. In this seminar, we shall look at these homotopy groups and the problems faced in calculating them. We shall then investigate a geometrically interesting type of topological space and see how the geometry – intuitively, that of a "twisted product", like a Möbius band – allows us to get a handle on computing the homotopy groups of these spaces, by way of a marvellous exact sequence.
11^th Nov 2011
Open dynamical systems and binary expansions
Rafael Alcaraz Barrera

Abstract (click to view)

I'll introduce the notion of open dynamical systems, and state some problems related to them. Using a "easy" and well know example I'll try to show some relationship between a open system and symbolic dynamics.
18^th Nov 2011
A graph theoretic proof of Roth’s theorem
Amit Kuber

Abstract (click to view)

Roth’s theorem is a special case of Szemeredi’s theorem. It states that any subset of natural numbers with positive upper density has a 3-term arithmetic progression. I will start with the definition of a graph and introduce some ideas required to state Szemeredi’s regularity lemma which is one of the most powerful tools in extremal graph theory. We shall use it to prove triangle removal lemma which then gives the required result in the special case combined with the triangle counting lemma. The talk will be self contained and I won’t assume any prior knowledge of graph theory. Also the talk will be full of sketches to motivate geometric intuition behind the proof.
25^th Nov 2011
Groups and their Geometries
Stephen Clegg

Abstract (click to view)

We'll consider well-known geometric objects and their automorphism groups, then extend the notions to obtain geometries for other groups.
2^nd Dec 2011
ODEs the pure way: The Differential Nullstellensatz
Nikesh Solanki

Abstract (click to view)

What? ODes stuff? In a pure postgrad seminar? Indeed it is true and in fact it is very pure in style. Differential algebra is the study of everyday algebraic structures such as rings, fields, modules etc. equipped with a map called a formal derivative. Within this world there is an equivalent of Hilbert’s Nullstellensatz, which says there is one-to-one correspondence between the zero sets of systems of differential polynomial equations over a differentially closed field (which is the differential equivalent of an algebraically closed field) and radical differential ideals over that field. These differential polynomials can be viewed as ODEs. In this talk I will give a brief introduction to differential algebra, defining what is meant by a formal derivative, differential polynomial/ideal and differentially closed field etc, with the aims to lead up to the differential nullstellensatz which we shall prove at the end. Even applied kids are welcome to come along.
9^th Dec 2011
Cohomology and Structure Theorems
Rob Mckemey

Abstract (click to view)

Given a vector space, < x_1,x_2,...,x_n >, one may construct the symmetric algebra on its basis, S=k[x_1, ... , x_n]. If we are given a group action on this vector space, we may extend this action linearly to the whole of S. A structure theorem for S, is an attempt to understand the group action on S. The question I will be addressing in this talk is how to find the best one
16^th Dec 2011
Title
speaker

Abstract (click to view)

ABSTRACT

For the Spring/Summer 2012 seminar timetable, please click here.

Previous Seminars

List of 2010/2011 seminars (Philip Bridge)
List of 2009/2010 seminars (Richard Harland)
List of 2008/2009 seminars (Ali Everett)
List of 2007/2008 seminars (Jacob George)
List of 2006/2007 seminars (Stephen Clegg)
List of 2005/2006 seminars (Marianne Johnson)
List of 2004/2005 seminars (Matt Horsham)
List of 2003 seminars (Matthew Craven/Sara Santos)
List of 2002 seminars (Sarah Perkins/Sara Santos)