Pure Maths Seminar Archive - Abstracts 2009
Algebraicity and its counterpart, transcendence, are basic notions in field theory. The algebraic closure operator is central to the model-theoretic study of fields. Similar closure operations - pregeometries - control all "tame" theories. The model theory of complex exponentiation is far from tame by most measures, since it interprets arithmetic and Goedel's theorem applies. I will explain the notions of exponential algebraicity and exponential algebraic closure, and show why there is hope that much of the theory of exponentiation may yet be tamed. I will also give some applications to transcendental number theory.
Hilbert's 10th problem asks for an algorithm to decide whether a polynomial equation over the integers has an integer solution. Due to a deep Theorem proved by Matyiasevich in 1970, no such algorithm exists. The corresponding problem with the integers replaced by rationals is still open. One way to reduce this problem to the original one would be to define Z in Q by an existential formula - again, it is open whether that can be done. Instead we will show that Z can be defined in Q by a universal formula, and give arguments for why Z should not be definable in Q by an existential formula.
Distance-regular graphs are a combinatorial generalization of distance-transitive graphs, and were introduced by Biggs in the 1970s. They have many combinatorial properties, for example, they are regular graphs. In 1984 Bannai and Ito conjectured in their book "Algebraic Combinatorics" that there are finitely many distance-regular graphs with given valency k≥3. They showed that their conjecture is true for k=3 and also for the class of bipartite distance-regular graphs. In this talk I will present the main idea behind a proof for this conjecture. This is joint work with Sejeong Bang (Busan) and Vincent Moulton (UEA).
A super real closed ring is a commutative ring equipped with the operation of all real valued continuous functions defined on euclidean spaces. Examples are rings of continuous functions and super real fields attached to z-prime ideals in the sense of Dales and Woodin. I will use a Nullstellensatz for rings of continuous functions to show that super real closed rings provide a flexible category in which commutative algebra may be used to study topological spaces (in terms of their rings of continuous functions). Moreover I will show that the elementary class (in the sense of first-order logic) of super real closed rings which are fields, act as "saturated o-minimal structures".
The facetious title is hopefully not entirely dumb - it's meant to suggest the "correct" point of view. A more fancy version of the same title might be "Classifying affine Hopf algebras of finite Gelfand-Kirillov dimension". I will explain the concepts involved in (both) titles, motivate the project, and describe progress.
Several mathematics educators have argued that students should be encouraged to behave in mathematical situations in a similar manner to that deployed by expert mathematicians. However, in order for this belief to be useful for curriculum design, it is crucial to have a detailed understanding of how both expert mathematicians and students actually do behave in mathematical situations. In this talk I will report on a research study which looked at one factor which may influence how persuasive mathematicians and students find various types of mathematical arguments: the authority of the argument's source. In doing so, I will raise issues surrounding the role of visual arguments in mathematics (and mathematics education), and will discuss different ways of interpreting a request to rate the persuasiveness of a mathematical argument.
An abstract group H is finitely generated if and only if the augmentation ideal of Z[H] is a finitely generated Z[H]-module. We show that in the profinite case, namely with H a profinite group and with the profinite completion of Z instead of Z, this is not true. We find a formula for computing the minimum number of generators for the augmentation ideal in the profinite setting; this is a generalisation of a result obtained in the finite case by Cossey, Gruenberg and Kovacs. We then consider probabilistic questions connected with the generation of the augmentation ideal and we define the class of APFG groups as those profinite groups for which the probability of generating the augmentation ideal with t random elements is non zero for some positive integer t. We give a characterisation of these groups which allows us easily to compare APFG groups with PFG groups and we show that PFG groups are APFG but we give examples showing that this is a strict inclusion.
An isometry R of Rd is called a coincidence isometry of a lattice Γ ⊂ Rd, if Γ and RΓ have a common sublattice. The intersection Γ ∩ RΓ is then called the coincidence site lattice (CSL) for the isometry R. In 3 dimensions CSLs have been widely used to describe grain boundaries in crystals. CSLs in dimension d>3 may be of interest in the investigation of quasicrystals.
In this talk I will consider the CSLs of the root lattice A4 ⊂ R4 and indicate how an embedding of A4 into the icosian ring, with its rich arithmetic structure, leads to a Dirichlet series generating function for the number of CSLs.
This is joint work with Michael Baake, Uwe Grimm and Peter Zeiner.
Let G be a simply-connected (real) Lie group. A lattice Γ in G is said to be rigid if every automorphism of Γ extends to an automorphism of G. Landmark results in the context of semisimple groups are the Mostow Rigidity Theorem and the Margulis Superrigidity Theorem.
In this talk I will report on onging joint work with Oliver Baues, concerning rigidity and non-rigidity of lattices in soluble Lie groups. I will start by discussing a classical theorem of Malt'sev-Saitô, instructive examples given by Starkov and a result of Witte. Then I will explain our more recent approach towards a "quantative description" of the phenomenon of non-rigid lattices in soluble Lie groups.
I will describe the main ideas behind Carnap's programme for inductive logic and outline some classical results achieved in the unary context, in particular as regards instantial relevance. Then I will explain why allowing non-unary properties introduces fundamental differences and difficulties, and I will consider some recent advances in this field that emerged during the last decade.
Public Key cryptography is a fundamental tool in securing modern communications including the internet. In my talk I show how mathematical concepts have been central to the development of this subject, from its beginnings in the 1970s up to the present day.
In this talk I will discuss a geometric instance of mutation (as defined by Fomin and Zelevinsky) for a bipartite graph embedded in surface and describe quantities connected to the graph's dimer partition function which are conserved under mutation. As application I will compute Laurent expansions — anticipated in theory of cluster algebras — for "twisted" Plücker coordinates of a Grassmannian.
Distinguished representations are those who appear in the harmonic analysis of the symmetric space GL(n,K)/GL(n,F) for a quadratic extension K/F of p-adic fields. We will talk about the link between the classification of "generic" distinguished representations and some meromorphic functions called Asai L-functions.
(Joint work with Michal Rams) Multifractal analysis involves decomposing a set into level sets where some local quantity is equal. This local quantity usually relates to a measure or a function which is unevenly distribution. You then look at properties such as the Hausdorff dimension of these level sets. For self-similar sets (e.g. the Sierpinski gasket) it is possible to get very complete results. However for self-affine sets several difficulties arise. We will go through the case of self-similar sets and then explain some of he difficulties for self-affine sets. Finally we will show how some of these difficulties may be overcome in a special case. The talk will assume very little previous knowledge of the area.
We shall study matrix groups acting as isometries of the hyperbolic plane. Under natural conditions, we shall discuss the statistical behaviour of orbits with respect to a large class of measures on a geometrically defined limit set. In particular, we shall discuss approximation by Brownian motion. (This is joint work with Mark Pollicott.)
In this talk I will introduce the concept of Kazhdan-Lusztig cells in Coxeter groups with unequal parameters. I will then present a conjecture (the semicontinuity conjecture) of Bonnafé which describe how the partition into cells is changing when the parameters are varying. Finally we will discuss the partition of affine Weyl groups of rank 2 into cells for any choices of parameters and show that the "semicontinuity conjecture" holds in those cases.
I will talk about Fontaine's classification of the p-adic representations of the absolute Galois group of Qp. Such representations occur naturally in number theory, for example in the study of the p-torsion points of an elliptic curve defined over Q. I will explain the geometric motivation behind Fontaine's classification, and will will outline important applications in number theory and algebraic geometry.
I will explain the recursive structure of categories of representations of GL2 over a field of positive characteristic and briefly mention some homological consequences. This is joint work with Will Turner.
It follows from the work of Kirchhoff (1847) that the spanning trees of a connected graph satisfy the following negative correlation inequality: if T is a spanning tree of G chosen uniformly at random, then
for all distinct edges e and f of G. In other words, the event T contains e and the event T contains f are negatively correlated. More recently, there has been a growing interest in the following related conjecture: if F is a forest of G chosen uniformly at random, then
for all distinct edges e and f of G. In this talk, we shall survey some related inequalities and describe some recent investigations into the above conjecture. No knowledge of matroids is required for the talk. This is joint work with Dominic Welsh (University of Oxford).
The descendant set desc(α) of a vertex α in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from α. We investigate desc(α) in an infinite highly arc-transitive digraph D of finite out-valency and whose automorphism group is vertex-primitive. We formulate four conditions which the subdigraph desc(α) must satisfy, and show that there are (up to isomorphism) only countably many digraphs Γ satisfying our conditions.
I will present results from the joint paper with Szymon Glab on a certain naturally defined ideal of subsets of the plane. The stress will be put on presenting the basic techniques of the set theory of the reals.