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 Richard Harland. 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 me.
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.
Summer Seminar Series 2009

This series was jointly organised with Ali Everett.

17^{th} Jul 2009A Short Advert for Homology (Time change: 2pm)
Stephen MillerAbstract (click to view)If you've spent any time at all with topologists, you'll have heard them chatting about homology: singular homology, Cech homology, cellular homology... but the foundations of homology are purely algebraic. In this talk I'll introduce the concept of a chain complex and the homology of a chain complex, and show how a short exact sequence of chain complexes gives rise to a long exact sequence in homology. Topological homology theories will arise as examples of the general theory, and I'll show how homology can be used to prove the famous Brouwer fixed point theorem: any continuous map from the closed ndisc to itself has at least one fixed point.
In case this isn't motivation enough, I'll try to say something about group homology for all you algebraists out there, and if time allows I'll talk about cohomology, explain why it's better than homology, and maybe even explain what a "generalised (co)homology theory" might be. 
31^{st} Jul 2009Finite Groups, and the Atlas Thereof
Paul TaylorAbstract (click to view)In 1985, the pinnacle of human intellectual endeavour was attained with the publication of the Atlas of Finite Groups. In this talk, we will attempt to explain enough group theory to allow one to read a page of this remarkable book, or at least of the associated website, which is a bit easier. A few related theorems will be stated, and a couple might even be proved.

14^{th} Aug 2009Voting Paradoxes
Jerry HopkinsonAbstract (click to view)When people cast their vote, the outcome can sometimes be counterintuitive, or questionably represents the "will of the people". Many alternative systems have been proposed. The important question is then what criteria can be used to decide between them.
I will introduce the topic with examples and background, and discuss some of the ongoing disagreements. 
28^{th} Aug 2009Decidability and the Incompleteness Theorems (Room Change: Sydney Goldstein Room (MAGIC), 1.213)
James DixonAbstract (click to view)At the start of the 20th century, mathematicians wanted a complete, consistent, axiomatized mathematics  everything is either true or false and we have a nice, easy to understand list of axioms to prove everything. I'll be looking at the theorems that proved that such an endeavour is doomed to fail and some nice consequences thereof.

11^{th} Sept 2009Projective Connections and Odd Poisson Structures
Jacob GeorgeAbstract (click to view)In my previous talk at the Postgrad Seminar, I gave definitions of projective connection and gave some indication of how they were related to densities in interesting ways. The first part of this talk will reiterate some of the essential ingredients of this relation. It turns out that on a projectively connected manifold, bilinear symmetric biderivations ("brackets" for short) on the algebra of functions can canonically be extended to the algebra of densities.
Requiring that the brackets are symmetric immediately precludes Poisson brackets and a great number of other interesting structures. A halfway house is possible if instead of considering brackets on manifolds, we consider odd brackets on supermanifolds. I'll round the talk off with conditions for a projective connection on a supermanifold to extend odd Poisson/symplectic structures to odd Poisson/symplectic structures on densities.
Autumn Semester 2009
These seminars are still being filled. If you would like to give a talk on any of the empty slots below, please email me.

25^{th} Sept 2009Rings Without Unique Factorisation
Nicholas GreerAbstract (click to view)It is well known that factorization in integers is unique. It is very easy to assume that extensions of the integers also have this property, but this is not necessarily the case, and problems can occur when consequences of unique factorisation are implicitly assumed. I will construct a ring in which factorization into irreducibles is impossible, and some in which it is nonunique and prove these properties.

2^{nd} Oct 2009Stability Theory in Modules
Simon PereraAbstract (click to view)Linear independence in vector spaces and transcendence in A.C.F.s are examples of the more general model theoretic notion of independence or nonforking. Stability theory involves studying theories via ranks and properties of types, to determine how "well behaved" they are. We'll cover examples ranging from the wild tearaways such as the rational numbers as an ordered set, to the goody goody twoshoes like totally transcendental modules, and I'll attempt to convey a sense of why we like the well behaved theories.

9^{th} Oct 2009Local Cohomology
Rob McKemeyAbstract (click to view)Topologists used the simple and insightful ideas of homology to classify spaces, geometers took these ideas and converted them to more algebraic arguments, homological algebra is what you get when you take these algebraic arguments and take out what's left of the topology. On Friday I'll talk about right derived functors, the key players in homological algebra, and how local cohomology can be calculated and applied using Cech Cohomology.

16^{th} Oct 2009Pure injectives over String Algebras
Richard HarlandAbstract (click to view)String algebras are a specific band of quiver algebras. I shall present work from a paper by C.M. Ringel, where he explains how to construct a set of pureinjective indecomposabes over any given string algebra. I shall then show how we are trying to extend this to give a complete classification of such modules.

23^{rd} Oct 2009Groups corresponding to geometric structures and viceversa.
Steve CleggAbstract (click to view)As group theory can be considered as the study of the symmetries in geometric objects, we would like to be able to associate an (abstract) group with something geometrical. I will talk about some methods to this aim. In particular, if time permits, I will discuss the notion of BeukenhoutTits geometries.

30^{th} Oct 2009Loops, Groups and Disconnected Graphs
Katie StecklesAbstract (click to view)In my work, I have been studying loop spaces as a way to find periodic orbits in dynamical systems. Many of the spaces I have looked at are examples of EilenbergMac Lane spaces. In this seminar I will attempt to calculate the homology of such spaces, and will relate this to homology of groups, as well as giving some illuminating examples.

6^{th} Nov 2009Reading Week
No Seminars Scheduled 
13^{th} Nov 2009Syntactic Semigroups and Eilenberg's Variety Theorem
Elaine RenderAbstract (click to view)In this talk I intend to give an introduction to the algebraic theory of finite automata. For a given finite automaton accepting a language L (taken as a subset of a finitely generated free semigroup), there exists a semigroup canonically attached known as the syntactic semigroup of L. An extension of a key result of Kleene tells us that a language is accepted by a finite automaton precisely if this semigroup is finite. Which raises the question, what other algebraic properties of a syntactic semigroup can tell us things about the associated language? Eilenberg's variety theorem states that varieties of finite semigroups (so called pseudovarieties) are in one to one correspondence with certain classes of recognisable languages. However, in order to formulate these ideas with finite semigroups we must move to the profinite completion of the free semigroup. I will go through all of the necessary definitions and background, as well as giving lots of examples.

20^{th} Nov 2009Injective modules
Lorna GregoryAbstract (click to view)Injective modules were first introduced by Baer in 1940. They are a generalisation of the concept of divisible abelian group and are dual (in the reversing arrow sense) to the notion of a projective module. In 1953 Eckmann and Schopf showed that every module M (over a ring R) is contained in a unique minimal injective module E(M), the injective hull of M. In 1958 Matlis showed that for R a commutative noetherian ring there is a bijective correspondance between the prime ideals of R and the indecomposable injective modules. In the same paper he also gives a structure theorem for indecomposable injectives over a commutative Noetherian ring. During this talk I will discuss the above results and give some examples of injective modules.

27^{th} Nov 2009The 'Thirty Years War', and related battles in group theory
John BallantyneAbstract (click to view)The JordanHolder Theorem tells us that every finite group can, in some sense, be broken down into a product of finite simple groups. Therefore, if we hope to understand groups in general, it is necessary to understand these 'building blocks'. This was the aim of the Classification of Finite Simple Groups, the pursuit of which occupied many of the great group theorists of the twentieth century. Finally, in the 1980s, after thousands of pages of journal articles and contributions from over 100 authors, the proof was 'complete'. In this talk I will present a brief history of this monumental achievement, with emphasis on grand strategy, rather than gory details.

4^{th} Dec 2009Complex noncommutative 2tori
Chen HarlandAbstract (click to view)I will introduce some background on noncommutative geometry invented by Alain Connes in the 80's, in particular on how compact Riemannian spin manifolds can be generalised through spectral triples. One may try to write down noncommutative version of familiar structures from classical geometry, for example, complex structures. I will talk about some work in progress along this line by inspecting the simple examples of the noncommutative 2tori.

11^{th} Dec 2009Mathematicians: The Past, The Future and the Just Plain Wrong
Ali EverettAbstract (click to view)In the final seminar of the gregorian year, I will attempt to cobble together a light and entertaining, yet mathematically informative, talk. I shall reveal to you some of history's cranks and crazies of mathematics (and some arguments between them), the future mathematician (or McDonalds worker) and some mathematics which, although people have tried to claim true, is just plain wrong.
Spring Semester 2010

5^{th} Feb 2010Models in general categories
Philip BridgeAbstract (click to view)By showing how to interpret formulae in categories without assuming an underlying set structure, I shall provide a concrete example of a model of a theory in which the law of the excluded middle fails.

12^{th} Feb 2010Odd Symplectic Connections and BatalinVilkovisky Algebras
Oliver LittleAbstract (click to view)I'll talk about what it means for a connection on a Z_{2} graded manifold to be compatible with an odd symplectic structure and describe completely the space of such compatible connections. I'll then investigate what extra structure is necessary to turn the algebra of functions on an odd symplectic manifold into a BatalinVilkovisky algebra. This extra structure can be interpreted either as a connection in the bundle of volume forms or as a generalised divergence operator and it also possesses a natural notion of compatibility with a connection.

19^{th} Feb 2010Complex projective space and lens space....with a twist.
Beverley O'NeillAbstract (click to view)Lens space and complex projective space are the orbit spaces of a free cyclic or S^{1} action on the odddimensional sphere respectively and the cohomology ring structure of both is pretty much text book stuff. "So Bev what happens when we consider an nonfree cyclic or S^{1} action?!" I hear you cry....well that's more interesting. In this seminar I will determine the integral cohomology ring structures of weighted projective space and weighted lens space as calculated by Kawasaki.

26^{th} Feb 2010Involutions in the Thompson sporadic simple group
Paul TaylorAbstract (click to view)The detailed suborbit structure of the action of a finite group on a conjugacy class of involutions has been studied for many groups, in particular most of the sporadic simple groups. I shall give some vague justification for this study, describe recent work on one particular sporadic group, the Thompson group of order 90745943887872000, and perhaps give hints of an application of this to the study of a yet larger group.

5^{th} Mar 2010Introduction to Classifying Spaces
Jerry HopkinsonAbstract (click to view)Classifying Spaces arise in the theory of fibre bundles. I shall introduce these spaces and their main properties, and discuss how they can be calculated. I will give an example of their use in calculations on toric manifolds.

12^{th} Mar 2010Classifying Spaces for Quasitoric Manifolds
Steve MillerAbstract (click to view)Quasitoric manifolds are a topological generalisation of the toric manifolds (smooth projective toric varieties) of algebraic geometry. In particular, quasitoric manifolds are smooth 2ndimensional manifolds with T^n action, satisfying certain conditions that ensure we may reconstruct the quasitoric manifold from combinatorial data. These manifolds play a role in cobordism theory: every quasitoric manifold admits a number of T^nequivariant stably complex structures, and every complex cobordism class in dimension >2 contains a quasitoric manifold. I will show how we can construct universal T^nspaces classifying quasitoric manifolds and their stably complex structures. The cohomology of these spaces has a combinatorial description, as do the induced maps in cohomology, and we can make use of the classifying spaces to simplify calculations in cobordism.

19^{th} Mar 2010The MOG and the Mathieu groups
Tim CrinionAbstract (click to view)The aim of the seminar will be draw the MOG and to pin down the 5 Mathieu groups M24, M23, M22, M12 and M11. These are 5 of the 26 "Sporadic simple groups" i.e. very important  much more than they sound! We will start by looking at things called Steiner Systems. In particular, we will construct the Steiner System S(5,8,24). We will draw this thing as the MOG. The Mathieu group M24 turns out to be the group of automorphisms of this Steiner System. We will be able to get M23 and M22 from M24, as M22 is in M23, and M23 is in M24. We'll say a little bit about these "Big Mathieu groups". We will then find the "Little Mathieu groups" M12 and M11. M12 is found inside M24 and M11 is found inside M12. These 5 groups were the first to be discovered of the 26 Sporadic Simple groups, by Emile Mathieu in 1860 and 1873, thirty years after Galois died. If you're not into group theory then that probably doesn't mean much, but We'll go through why this is so amazing at the start of the talk!

Easter BreakNo Seminars Scheduled

23^{rd} Apr 2010Real Closed Fields
James DixonAbstract (click to view)Real closed fields are fields which are firstorder equivalent to the real field. I'll be looking at what this means and using RCFs as a springboard to introduce various model theoretic notions which often lurk in the background of talks given by logicians and are neat generalizations of phenomena found throughout algebra. In particular I'll be looking at quantifier elimination and what this tells us about real closed fields. If all else fails, we can have a group Puzzle Hunt session in the dying embers of this week.

30^{th} Apr 2010Commuting Involution Graphs of some Symplectic Groups
Ali EverettAbstract (click to view)As with most (if not all) group theory talks, I shall begin with: LetG be a group. Let X be a subset of G. We construct a graph structure using the elements of X as the vertices and drawing an edge between two vertices if and only if they commute in G. I will use a ruler if it makes people happy. The aim of this talk is to highlight the motivations, some examples, and past progress of this avenue of study, together with current progress and an overview of one particular example. Prerequisite knowledge: you will probably need to know what a group is.

7^{th} May 2010Capturing Analogical Reasoning in Inductive Logic
Alex HillAbstract (click to view)Since argument by analogy is a major part of everyday inductive reasoning it has long been seen as desirable that it be captured in formal inductive logic. Philosophers have tended to approach this problem by modifying Carnap's inductive methods to make them sensitive to analogy. Our approach is to look at a candidate Analogy Principle and attempt to classify the probability functions that satisfy it, working initially in a language with two predicates.

14^{th} May 2010The Classification and Representation Theory of Semisimple Lie algebras
Lewis TopleyAbstract (click to view)I intend to give an overview of the successful classification of both semisimple Lie algebras and their representations over K algebraically closed, characteristic 0. I shall start with all the basic definitions and introduce the important structural concepts involved: Killing form, Cartan decomposition, roots of a Lie algebra, universal enveloping algebras

21^{th} May 2010Geometric Group Theory
Alistair DarbyAbstract (click to view)In 1872, Klein proposed group theory as a means of formulating and understanding geometrical constructions; the "Erlangen Program." We can think of Geometric Group Theory as this program in reverse. I intend to give a basic introduction to this relatively new area in mathematics, where we look at geometric and topological properties of spaces on which a finitely generated group acts, and then deduce properties of the group from this. The subject draws on ideas from across mathematics, in particular lowdimensional topology (mainly, 3manifold theory) and hyperbolic geometry. Thurston's work has shown that these two subjects are intimately linked. I aim to define hyperbolic (negatively curved) groups and give some of their properties.

28^{th} May 2010The Ziegler spectrum and the tensor embedding
Lorna GregoryAbstract (click to view)The Ziegler spectrum of a ring is a topological space with points being the isomorphism classes of indecomposable pureinjectives and closed sets corresponding to complete theories of modules closed under products. In this talk, I will give various equivalent definitions of the Ziegler spectrum, giving some indication of why they are equivalent. I will state some of the basic properties of the Ziegler spectrum and hopefully show that it is a useful and interesting object attached to a ring.

4^{th} Jun 2010The collection of subsets of 1
Ali LloydAbstract (click to view)P(1) is a central object in set theories. Original formulations of the axioms failed to completely determine what a subset is. Subsets may in fact be reinterpreted in over six ways (maybe even 13). These reinterpretations result in new ways of viewing V (universe of set theory). Partly our queries into what V can be aim to help in the rewriting of the axioms. Algebraic Set Theory, introduced by Moerdijk and Joyal, by tackling things in the language of categories, allows one to examine nonclassical analogues of Top, the categorical generalisation of set theory, where pullback reigns supreme. I'll prove a theorem of Diaconescu which had consequences for nonclassical theories (roughly in CZ you can't have the axiom of choice). Rejecting AC then, I'll look at truth values (not in much detail) and show what one can infer when P(1) is more complicated than the simple unordered pair {0,1}.

11^{th} Jun 2010Positive Quantifier Elimination in Algebraic and Real Geometry
Nikesh SolankiAbstract (click to view)Model theory can provide helpful and elegant approaches to everyday pure mathematics. I aim to show an example of this in the areas of algebraic and real geometry using a modeltheoretic result called the Lyndontype lemma. The Lyndontype lemma gives a criterion for positive quantifier elimination. In this talk I shall define what is meant by (positive) constructible and semialgebraic sets and I will show how the Lyndontype lemma can be used to prove that the projection of certain positive constructible and semialgebraic is again positive. Along the way we shall drop in on some of my good friends basic logic (I bet you didn't see that coming), valuation theory and many others.
Previous Seminars
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)