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 email 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.
Autumn Semester 2011

23^{rd} Sept 2011A proof of Gabrielov's theorem
Mohsen KhaniAbstract (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 2011MRSC 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 2011A Short Overview of Lie Correspondence
Lewis TopleyAbstract (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 2011title
Dave NaughtonAbstract (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 2011A few examples in algebraic geometry
Andrew DaviesAbstract (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 2011Elliptic Integrals and Schneider's Theorem
Adam BiggsAbstract (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 2011Twisted Products in Homotopy Theory
Alexander LongdonAbstract (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 2011Open dynamical systems and binary expansions
Rafael Alcaraz BarreraAbstract (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 2011A graph theoretic proof of Roth’s theorem
Amit KuberAbstract (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 3term 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 2011Groups and their Geometries
Stephen CleggAbstract (click to view)We'll consider wellknown geometric objects and their automorphism groups, then extend the notions to obtain geometries for other groups.

2^{nd} Dec 2011ODEs the pure way: The Differential Nullstellensatz
Nikesh SolankiAbstract (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 onetoone 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 2011Cohomology and Structure Theorems
Rob MckemeyAbstract (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

20^{th} Jan 2012An overview of Ergodic theory
Simon BakerAbstract (click to view)Ergodic theory can be described as the statistical and qualitative behaviour of a given system that evolves with time. In this talk I shall give an overview of several key topics arising from this field. We shall see applications to Number Theory and Fractal Geometry (there will be pictures).

27^{th} Jan 2012An introduction to ominimal fields
Javier UtrerasAbstract (click to view)During the 80s, van den Dries, Pillay and Steinhorn introduced the concept of ominimality, which allows to reproduce nice properties of semialgebraic sets in extensions of real closed fields. In this talk, I shall define ominimality and give an overview of some of these properties.

3^{rd} Feb 2012Poisson Geometry (+ Deformation Quantization)
Adam BiggsAbstract (click to view)Poisson geometry is the arena in which mathematicians tackle questions pertaining to the study of dynamicals systems that have similar properties to the standard flat Hamiltonian method. We shall describe what such objects are and lightly explore the mechanics that can be defined on them, mainly through examples. If we have time we shall also define what a deformation of a Poisson algebra and try to explain the philosophy behind this form of quantization. Again this will be example heavy to try and get a feel of what is going on.

10^{th} Feb 2012An Introduction to Complex Dynamics
Dave NaughtonAbstract (click to view)I wanted to call this talk 'Why those pictures look so full on awesome' but I decided to be sensible. Don't worry it won't happen again. So, we've all been asked the question by friends and family pretending to be interested  'so what is it you do your PhD in?'  if maths doesn't scare them off, pure maths likely will. If they stick around long enough to hear 'Ergodic Theory and Dynamical Systems' then they look at me like I've got 10 heads (however...). So when asked what is it you do I usually just mention fractals and the Mandelbrot set. Even the bloke who sells pies down Wigan market has probably seen Th'Mandelbrot Set somewhere down the line. Or maybe not. Anyway i'm going off on a tangent once again. In this talk my main aim is to introduce complex dynamics and Julia sets. These arise from iterations of seemingly simple polynomials with complex coefficients, but the results can be very interesting. I want to talk about how these things are constructed, and why they look like they do in all the computer generated fractal images we see on many mathematics books. I will also relate this to an application of Newton's Method for finding zero's of polynomials, in particular how the choice of a starting point in the Newton iteration determines which (if any) root the method will converge to, and how this leads to fractals. Then I'll talk about the construction and properties of the Mandelbrot Set itself, which is intrinsically related to Julia Sets. There will be plenty pictures, hopefully not too many definitions and it should be pretty straightforward and accessible to all. The only prerequisite is a knowledge of what a complex number is  so yeah, everyone should get something from it, even the pie bloke from Wigan market, but unfortunately he takes his Whippets out for a walk on a Friday afternoon. Also more tangents are inevitable.

17^{th} Feb 2012The Algebraic theory of Quadratic forms
Sian FryerAbstract (click to view)Quadratic forms show up in all areas of maths, from number theory to algebra to Ktheory, but given their basic definition: the diagonal part of a symmetric bilinear form, or a form in n variables that's homogeneous of degree 2, it's not especially clear what they are or what we can do with them. I will present an introduction to the algebraic theory of quadratic forms, which reduces them to a form that's easier to work with and packages them up into a nice ring structure, and if there's time I will show how to compute this ring for a few wellknown fields.

24^{th} Feb 2012An overview of Continuum Theory
Rafael Alcaraz BarreraAbstract (click to view)Continuum theory is an intriguing area of General Topology. Despite the definition of a Continua it is quite simple, some of these spaces have some non intuitive properties. Some interesting examples will be given and I will introduce a useful tool to study them, Inverse Limits. Maybe Dynamical Systems will be mentioned in this talk, or maybe not.

2^{nd} Mar 2012Title
Steve CleggAbstract (click to view)ABSTRACT

9^{th} Mar 2012An Introduction to Topos Theory
Amit KuberAbstract (click to view)Do you insist on exhibiting a construction demonstrating each proof you write? If yes, then there is a world of rich categories where your dreams come true! A category in which all of the constructions (except complements!) in set theory are possible is called as an elementary topos. I will present this viewpoint of looking at topos theory with some examples.

16^{th} Mar 2012Visualisations of highdimensional manifolds
Alastair DarbyAbstract (click to view)Mentally picturing manifolds such as the 3sphere can be tricky when you try to visualize some of its properties. We will start with this simple case and hopefully move on to visualising the difference between some of John Milnor's exotic 7spheres, which are all homeomorphic but not diffeomorphic. Following Dave's lead there will be some cool colourful videos.

23^{th} Mar 2012Skolem's Paradox
Laura PhillipsAbstract (click to view)A fundamental theorem of model theory guarantees that a theory with an infinite model expressed in a countable language has a countable model. Since we can axiomatise set theory using just one binary relation symbol  the membership relation  there's a countable model of set theory. Great. The trouble is, this model contains uncountable sets (and their members). Why? Well, because Cantor said so... This conflict is known as Skolem's Paradox. In this talk I'll explain the historical background, and after revisiting the basics of languages and structures, show why it isn't really a paradox after all.

20^{th} April 2012Automata: a subset of your wishes is their command
David WildingAbstract (click to view)An automaton is a device for computing a property of a string of symbols. They are typically used to compute membership of formal languages, but they can also count features of strings (think Scrabble scores) and decode strings (think binary numbers). I'll explain these examples and I'll give a characterisation of the properties that automata are capable of computing.

27^{th} April 2012Hyperbolic dynamical systems
Anthony ChiuAbstract (click to view)A dynamical system is "hyperbolic" if it contracts points in one direction and stretches points in another direction. In this talk, we will look at some famous examples of discrete time dynamical systems that have such behaviour. We will then learn a bit of hyperbolic geometry to see how geodesic and horocycle flows (dynamical systems that evolve in continuous time) give rise to hyperbolic dynamics. There will be lots of diagrams.

4^{th} May 2012Hall's Marriage Theorem
David WardAbstract (click to view)Philip Hall was a mathematician of great influence in England in the twentieth centre, contributing many improtant concepts and theroems to the area of group theory nad in particular finite groups. One such result was generalised to form what is now known as Hall's Marriage Theorem  a very important theorem in combinatorics. In this talk, we will give a few examples of applications of Hall's theorem in both fun and more mathematical situations. We will then proceed to illustrate the power of the therorem to give a proof about 3dimensional finite projective spaces.

11^{th} May 2012Beta expansions
Simon BakerAbstract (click to view)For any real number x we can associate to it a binary/tertiary/decimal/nary expansion. This expansion is unique except for a countable set of points that have two expansions. In this talk I shall discuss the case where our base is no longer integer. Amongst other things I shall explain how x can have a continuum of expansions and show that this behaviour is in fact typical.

18^{th} May 2012Get to the point
Nikesh SolankiAbstract (click to view)They said to the algebraic geometer and so he/she did. He/she found that one can investigate the properties that are true arbitrarily close, with respect to the Zariski topology, to a point on a variety using the algebraic technique of ‘localisation’ on the coordinate ring of the variety to obtain a local ring. However, the analyst was not happy this complaining, “That is not close enough (to the point)!” So the algebraic geometer went away and found that if he/she takes the socalled ‘completion’ of the local ring, properties in even smaller neighbourhoods could be obtained. In this talk, a model theorist is going to explain roughly what the algebraic geometer did above.

25^{th} May 2012The Petersen graph and the Petersen geometries
Tim CrinionAbstract (click to view)In the first part of this talk we'll define a geometry. Geometries are interesting in Group theory because of their automorphism groups. We will define the Petersen graph and the Petersen geometries. For the second part I will describe my research so far.

1^{st} June 2012Language Invariance in Inductive Logic
Malte KliessAbstract (click to view)Inductive Logic deals with probability functions on sentences of a language L with a finite number of relations. Most of the time, we are concerned with rational principles that these functions should satisfy, but what if we happen to pick too small a language? Is there a way of extending functions to larger languages? How far can we go up? And does this affect the rational principles we would like our functions to satisfy? The key to answering these questions is Language Invariance. I will tell you what we mean by Language Invariance and why it is a nice thing to have.

15^{th} June 2012The Fixed point property and hyperspaces
Rafael Alcaraz BarreraAbstract (click to view)In Dynamical Systems we are always concerned about finding fixed or periodic points and understand their dynamical behavior. Nonetheless we study them as a property of the function rather than a property of our state space. The topology of some spaces can assure the existence of fixed point for every continuous map. We say that these spaces have the fixed point property. In this talk some examples and non examples of continua with the fixed point propety will be given, and the fixed point property on certain hyperspaces (I will define this notion of course) will be studied.

22^{nd} June 2012Infinite Combinatorics and Large Cardinals
Tahel RonelAbstract (click to view)In this talk I will explain what large cardinal hypotheses are and give some combinatorial properties used to construct large cardinals. I will then talk about why they are important to our understanding of the mathematical universe. Time permitting, I will also say something about results involving rainbow colourings.

29^{th} June 2012Hopf Algebras
Colin DoldAbstract (click to view)Hopf algebras are algebraic structures that combine the notion of an algebra over a field with a “dual” notion. They arise naturally in algebraic topology and Lie theory, and have applications in quantum theory. In this talk I will endeavour to explain what a Hopf algebra is, and what the Hopf algebra structure implies for the category of modules over the underlying algebra.
Spring Semester 2012
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)