Forthcoming events in this series


Wed, 09 Nov 2016
11:30
N3.12

Hilbert's Third Problem

Alex Margolis
Abstract

Two polyhedra are said to be scissors congruent if they can be subdivided into the same finite number of polyhedra such that each piece in the first polyhedron is congruent to one in the second. In 1900, Hilbert asked if there exist tetrahedra of the same volume which are not scissors congruent. I will give a history of this problem and its proofs, including an incorrect 'proof' by Bricard from 1896 which was only rectified in 2007.

Wed, 02 Nov 2016
11:30
N3.12

Methods of Galois group computation

Adam Jones
Abstract

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.

Wed, 15 Jun 2016

11:30 - 12:30
N3.12

2x2 Matrices

Giles Gardam
(Oxford)
Abstract

We will explore the many guises under which groups of 2x2 matrices appear, such as isometries of the hyperbolic plane, mapping class groups and the modular group. Along the way we will learn some interesting and perhaps surprising facts.

Wed, 08 Jun 2016

11:30 - 12:30
N3.12

TBA

Alex Betts
(Oxford)
Wed, 25 May 2016

11:00 - 12:30
N3.12

TBA

Philip Dittman
(Oxford)
Wed, 11 May 2016

11:00 - 12:30
N3.12

Wild spheres in R3

Simon Bergant
(Oxford)
Abstract

In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.

Wed, 04 May 2016

11:00 - 12:30
S2.37

Combinatorics in the representation theory of the symmetric group

Kieran Calvert
(Oxford)
Abstract

Since the symmetric group is a finite group it’s representation theory is not too complex, however in this special case we can realise these representations in a particular nice combinatorial way using young tableaux and young symmetrizers. I will introduce these ideas and use them to describe the representation theory of Sn over the complex numbers.

Wed, 24 Feb 2016

11:00 - 12:30
N3.12

Outer Automorphisms of Hyperbolic Groups

Alex Margolis
(Oxford)
Abstract

I will talk about a remarkable theorem by Paulin, which says
that if a one-ended hyperbolic group has infinite outer automorphism
group, then it splits over a two-ended subgroup. In particular, this
gives a condition which ensures a hyperbolic group doesn't have property
(T).

 

Wed, 17 Feb 2016

11:00 - 11:30
N3.12

The Riemann zeta function, quantum chaos and random matrices

Simon Myerson
(Oxford)
Abstract
The Riemann zeta function is linked to quantum chaology by some totally neat results and utterly wacky conjectures concerning random matrices. Join me to see the horrifying extent of these unexpected connections!
Wed, 20 Jan 2016

11:00 - 12:30
S2.37

Bieberbach's Theorems

Robert Kropholler
(Oxford)
Abstract
I will go through a proof of Bieberbach's theorems proving that a group acting cocompactly on Euclidean n-space has a subgroup consisting of n independent translations. Time permitting I will also prove that there is a bound on the number of such groups for each dimension n. I will assume very little requiring only a small amount of group theory and linear algebra for the proofs. 
Wed, 02 Dec 2015

11:30 - 12:30
S2.37

Representation Dimension and Quasihereditary algebras

Teresa Conde
(Oxford)
Abstract


The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

Wed, 21 Oct 2015

11:00 - 12:30
N3.12

Some Theorems of the Greeks

Gareth Wilkes
(Oxford)
Abstract

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

Wed, 14 Oct 2015

11:00 - 12:30
N3.12

Properties of random groups.

Rob Kropholler
(Oxford)
Abstract

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

Tue, 16 Jun 2015

11:00 - 12:30
N3.12

(Spin) Topological Quantum Field Theory

Thomas Wasserman
(Oxford)
Abstract

This'll be a nice and slow paced introduction to topological quantum field theory in general, and 1-2-3 dimensional theories in particular. If time permits I will explain the spin version of these and their connection to physics. There will be lots of pictures. 

Wed, 10 Jun 2015

11:00 - 12:30
N3.12

The arithmetic of K3 surfaces.

Chris Nicholls
(Oxford)
Abstract

In the classification of surfaces, K3 surfaces hold a place not dissimilar to that of elliptic curves within the classification of curves by genus. In recent years there has been a lot of activity on the problem of rational points on K3 surfaces. I will discuss the problem of finding the Picard group of a K3 surface, and how this relates to finding counterexamples to the Hasse principle on K3 surfaces.

Wed, 27 May 2015

11:00 - 12:30
S1.37

Lackenby's Trichotomy

Henry Bradford
(Oxford)
Abstract

Expansion, rank gradient and virtual splitting are all concepts of great interest in asymptotic group theory. We discuss a result of Marc Lackenby which demonstrates a surprising relationship between then, and give examples exhibiting different combinations of asymptotic behaviour.

Wed, 13 May 2015

11:00 - 12:30
N3.12

Prime Decompositions of Manifolds

Gareth Wilkes
(Oxford)
Abstract

The notion of prime decomposition will be defined and illustrated for
manifolds. Two proofs of existence will be given, including Kneser's
classical proof using normal surface theory.

Wed, 06 May 2015

11:00 - 12:30
N3.12

Voting Systems and Arrow's Impossibility Theorem

Robert Kropholler
(Oxford)
Abstract

With the general election looming upon I will discuss the various different kinds of voting system that one could implement in such an election. I will show that these can give very different answers to the same set of voters. I will then discuss Arrow's Impossibility Theorem which shows that no voting system is compatible with 4 simple axioms which may be desireable.

Wed, 11 Mar 2015

11:00 - 12:30
N3.12

Expansion, Random Walks and Sieving in SL_2(F_p[t])

Henry Bradford
(Oxford)
Abstract

Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p[t])$

 

We pose the question of how to characterize "generic" elements of finitely generated groups. We set the scene by discussing recent results for linear groups in characteristic zero. To conclude we describe some new work in positive characteristic.

Wed, 04 Mar 2015

11:00 - 12:30
N3.12

Soluble Profinite Groups

Ged Corob Cook
(Royal Holloway)
Abstract

Soluble groups, and other classes of groups that can be built from simpler groups, are useful test cases for studying group properties. I will talk about techniques for building profinite groups from simpler ones, and how  to use these to investigate the cohomology of such groups and recover information about the group structure.

Wed, 25 Feb 2015

11:00 - 12:30

Derived Categories of Sheaves on Smooth Projective Varieties in S2.37

Jack Kelly
(Oxford)
Abstract

In this talk we will introduce the (bounded) derived category of coherent sheaves on a smooth projective variety X, and explain how the geometry of X endows this category with a very rigid structure. In particular we will give an overview of a theorem of Orlov which states that any sufficiently ‘nice’ functor between such categories must be Fourier-Mukai.

Wed, 18 Feb 2015

11:00 - 12:30
N3.12

Groups acting on R(ooted) Trees

Alejandra Garrido
(Oxford)
Abstract

In particular, some nice things about branch groups, whose subgroup structure  "sees" all actions on rooted trees.