Pure Postgraduate Seminars

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 2009
  Rings Without Unique Factorisation
  Nicholas Greer
  Abstract (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 non-unique and prove these properties.

• 2^nd Oct 2009
  Stability Theory in Modules
  Simon Perera
  Abstract (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 non-forking. 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 tear-aways such as the rational numbers as an ordered set, to the goody goody two-shoes like totally transcendental modules, and I'll attempt to convey a sense of why we like the well behaved theories.

• 9^th Oct 2009
  Local Cohomology
  Rob McKemey
  Abstract (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 2009
  Pure injectives over String Algebras
  Richard Harland
  Abstract (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 pure-injective 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 2009
  Groups corresponding to geometric structures and vice-versa.
  Steve Clegg
  Abstract (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 Beukenhout-Tits geometries.

• 30^th Oct 2009
  Loops, Groups and Disconnected Graphs
  Katie Steckles
  Abstract (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 Eilenberg-Mac 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 2009
  Reading Week
  No Seminars Scheduled
• 13^th Nov 2009
  Syntactic Semigroups and Eilenberg's Variety Theorem
  Elaine Render
  Abstract (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 2009
  Injective modules
  Lorna Gregory
  Abstract (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 2009
  The 'Thirty Years War', and related battles in group theory
  John Ballantyne
  Abstract (click to view)

  The Jordan-Holder 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 2009
  Complex noncommutative 2-tori
  Chen Harland
  Abstract (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 2-tori.

• 11^th Dec 2009
  Mathematicians: The Past, The Future and the Just Plain Wrong
  Ali Everett
  Abstract (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 2010
  Models in general categories
  Philip Bridge
  Abstract (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 2010
  Odd Symplectic Connections and Batalin-Vilkovisky Algebras
  Oliver Little
  Abstract (click to view)

  I'll talk about what it means for a connection on a Z₂ 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 Batalin-Vilkovisky 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 2010
  Complex projective space and lens space....with a twist.
  Beverley O'Neill
  Abstract (click to view)

  Lens space and complex projective space are the orbit spaces of a free cyclic or S¹ action on the odd-dimensional sphere respectively and the cohomology ring structure of both is pretty much text book stuff. "So Bev what happens when we consider an non-free cyclic or S¹ 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 2010
  Involutions in the Thompson sporadic simple group
  Paul Taylor
  Abstract (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 2010
  Introduction to Classifying Spaces
  Jerry Hopkinson
  Abstract (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 2010
  Classifying Spaces for Quasitoric Manifolds
  Steve Miller
  Abstract (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 2n-dimensional 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^n-equivariant stably complex structures, and every complex cobordism class in dimension >2 contains a quasitoric manifold. I will show how we can construct universal T^n-spaces 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 2010
  The MOG and the Mathieu groups
  Tim Crinion
  Abstract (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 Break
  No Seminars Scheduled
• 23^rd Apr 2010
  Real Closed Fields
  James Dixon
  Abstract (click to view)

  Real closed fields are fields which are first-order 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 2010
  Commuting Involution Graphs of some Symplectic Groups
  Ali Everett
  Abstract (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. Pre-requisite knowledge: you will probably need to know what a group is.

• 7^th May 2010
  Capturing Analogical Reasoning in Inductive Logic
  Alex Hill
  Abstract (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 2010
  The Classification and Representation Theory of Semisimple Lie algebras
  Lewis Topley
  Abstract (click to view)

  I intend to give an overview of the successful classification of both semi-simple 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 2010
  Geometric Group Theory
  Alistair Darby
  Abstract (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 low-dimensional topology (mainly, 3-manifold 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 2010
  The Ziegler spectrum and the tensor embedding
  Lorna Gregory
  Abstract (click to view)

  The Ziegler spectrum of a ring is a topological space with points being the isomorphism classes of indecomposable pure-injectives 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 2010
  The collection of subsets of 1
  Ali Lloyd
  Abstract (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 non-classical analogues of Top, the categorical generalisation of set theory, where pullback reigns supreme. I'll prove a theorem of Diaconescu which had consequences for non-classical 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 2010
  Positive Quantifier Elimination in Algebraic and Real Geometry
  Nikesh Solanki
  Abstract (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 model-theoretic result called the Lyndon-type lemma. The Lyndon-type lemma gives a criterion for positive quantifier elimination. In this talk I shall define what is meant by (positive) constructible and semi-algebraic sets and I will show how the Lyndon-type lemma can be used to prove that the projection of certain positive constructible and semi-algebraic 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.