Pure Maths Seminar Archive - Abstracts 2010
Many interesting quantities in mathematics can only be estimated, rather than presented. In some particular cases, there is a dynamical approach based on periodic points which is quite efficient. We illustrate this viewpoint with a number of examples.
Exponential fields are fields with an operation of exponentiation as well as addition and multiplication. I will explain how they can be described by finite sets of generators and relations, in analogue to finitely generated extensions of fields, or finite presentations of groups. I will then discuss what this means for complex exponentiation and give some open problems.
In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures in the appropriate language: pure sets, vector spaces over a fixed division ring and algebraically closed fields.
In 1993 Hrushovski produced a family of counter examples to the conjecture. Each one of these counter examples carry a geometry. We answer a question of Hrushovski about comparing these geometries and their localization to finite sets. We show that the geometries obtained are isomorphic to each other, in particular this proves that the counter-examples arising from Hrushovski's construction are geometrically equivalent to each other.
In this lecture I consider some observations about automorphisms of symmetric designs. In recent years there has been some new work on on symmetric designs which have flag-transitive flag-transitive. The idea is to review some of these results and to make some comments on a way forward.
In this talk we consider counting problems in groups, motivated by representation theory. Let G be an algebraic group. We consider such a group over the p-adic integers Zp and the power series ring Fp[[t]]. As Ax-Kochen philosophy goes, these two rings are model theoretically very similar. We use their model theory to show that counting problems in the groups G(Zp) and G(Fp((t))) do not depend on the ring, even if the groups themselves are not isomorphic. The main tools in this analysis are p-adic integrals. This talk is based on joint work with Berman and Onn.
A hierarchical structure describing the inter-relationships of species has long been a fundamental concept in systematic biology, from Linnean classification through to the more recent quest for a Tree of Life. In this talk we introduce an approach based on discrete mathematics to address a basic question: Could one delineate this hierarchical structure in nature purely by reference to the genealogy of present-day individuals, which describes how they are related with one another by ancestry through a continuous line of descent? We describe several mathematically precise ways by which one can naturally define collections of subsets of present day individuals so that these subsets are nested (and so form a tree) based purely on the directed graph that describes the ancestry of these individuals. We also explore the relationship between these and related clustering constructions.
Siegel proved that a non-singular plane cubic curve has only finitely many integral points. Well, how many? And can we find them all? Attempts to answer these questions have influenced theoretical and practical approaches to number theory for two generations. I will give an example driven survey and report on recent joint work with Valéry Mahé attempting to extend current knowledge.
A knot is a circle embedded into 3-space. Every knot K bounds an embedded orientable surface, called a Seifert surface. The minimum of the genera of all Seifert surfaces bounded by K is a fundamental invariant of the knot K. I will talk about the classification problem of minimal genus Seifert surfaces for a given knot.
I will provide an overview of key results and longstanding open questions concerning models of arithmetic that are in one way or another inextricably intertwined with higher set theory. This will include a discussion of recent work of Saharon Shelah in this area.
This talk presents my and my colleagues' motivation to turn on using the modern computing technologies, in partial grid computing, in our research and educational activities. The talk is rather a collection of examples than a detail discussion on a research problem. It describes how we use the grid computing to simulate communication systems in our research on coded modulation as well as our experience in involving computing in courses on number theory, algebra, coding theory, etc.
Let X be a set, κ be a cardinal number and let H be a family of subsets of X which covers each element x in X at least κ times. What assumptions can ensure that H can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover H: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of Rn by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite κ. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.
Ariki-Koike algebras are generalizations of Hecke algebras associated to Weyl groups of type A and B. When they are not semisimple, their representation theory is fairly well understood thanks to Kashiwara-Lusztig canonical bases theory in affine type A. The purpose of the talk will be to recall some key definitions and results about this theory and to present a natural quantization of a theorem due to Geck and Rouquier on factorization of decomposition maps. This is a joint work in collaboration with S. Ariki and N. Jacon.
In this talk I will introduce Vopenka's Principle. This is an axiom from set theory that is quite easy to state and yet is very strong, lying near the top of what is known as the large cardinal hierarchy. It has found applications in category theory, leading to the resolution of an important problem in algebraic topology that had been open for 30 years: the question of existence of localisation functors for generalized cohomology theories. After giving an overview of these things, I will talk about some recent results regarding the compatibility of Vopenka's Principle with other statements such as the Generalised Continuum Hypothesis and the existence of a definable well-order on the universe of all sets.
We will discuss recent progress related to the Littlewood and mixed Littlewood conjectures. By using the metric theory of non-monotonic approximation we can show that for any prime p and for almost every real number α,
Furthermore by extending results of Schmidt and others concerning the distribution of the sequence {nα} modulo 1 we are able to prove new metric results about the Littlewood conjecture itself.
By a discrete torus we mean the Cayley graph associated to a finite abelian group with its canonical generators. A natural invariant associated to the graph is the set of eigenvalues of the adjacency matrix. In this talk, we will study results associated to the set of eigenvalues as the orders of the abelian groups tend to infinity. In particular, we will recover spectral theory on real tori as a limiting case of the discrete tori. The research described in this talk is joint work with Jay Jorgenson and Anders Karlsson.
L-functions are fundamental objects in number theory and their central values encode important arithmetic information. In this talk I will present joint work of mine with David Whitehouse where we obtain exact formulas for averages of central L-values using the relative trace formula and relations between periods and L-functions. I will focus on two examples.
A split of a convex polytope P is a (regular) subdivision with exactly two maximal cells. The tight-span of a subdivision of P is the complex dual to its complex of inner faces. Both concepts were introduced in the case of a very special polytope in order to study decompositions of finite metric spaces. However, we will explain how the concept of splits (and possible generalisations of it) can be used to obtain a better understanding of polytopes and their subdivisions in general. We show that each weight function on the vertices of P admits a unique decomposition into splits with a split prime remainder. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with P, the split complex of P. In the second part of the talk our theory we be used to give examples of applications in different parts of mathematics, one in phylogenetics and another in tropical geometry.
The centraliser algebra FGH of a subgroup H of a finite group G, over a field F, consists of those elements of the group algebra FG which are fixed under conjugation by each element of H. The extreme cases are FG1=FG and FGG=Z(FG). Other important cases include when F is a field of characteristic p > 0 and H is a p-group.
We are interested in determining the "block" decomposition of FGH and its simple modules. This may have applications to conjectures in representation theory.
If F is the field of complex numbers, one can define the irreducible characters of FGH, just as one can for FG. An explicit construction was given in a recent note by Alperin. Now suppose that L is a subgroup of G. Then there is a standard method for "inducing" the characters of L to characters of G. We discuss whether a similar method exists for constructing characters of FGH from characters of L, at least for certain groups H,G.
I will talk about primes dividing critical L-values as moduli of congruences between modular forms, or more generally automorphic representations, starting with Ramanujan's mod 691 congruence.
Homological representations often provide the strongest link between the structure of the group and the geometries preserved by it. This is certainly true for Steinberg representations of finite classical groups and related representations constructed by Lusztig in the '70s. Above all, homological representations often yield essential basic information such as degrees and character formulae which are not available otherwise. In this talk I want to report about recent work with Dan Smith on new homological representations of GL(n,q) in positive characteristic p coprime to q. In the process we will revisit Steinberg's construction.
Let Λ=⊕n ≥ 0 Λn be a graded algebra over a field k. We assume dimk Λi < ∞ for all i ≥ 0, but we do not assume that Λ0 is semi-simple. Suppose gldim Λ0 < ∞. Let T be a graded Λ-module concentrated in degree zero. In this talk I propose the following new definition of T-Koszul algebras: Λ is a T-Koszul algebra if both