Pure Maths Seminar Archive - Abstracts 2005
A. Pillay adapted Hrushovski's proof of a special case of Weil's 'group chunk theorem' to show that a group definable in an o-minimal structure can be definably equipped with a topology making it a topological group and a definable manifold. The topology in question on a large definable subset of the considered definable group coincides with the usual topology induced by the ordering of the structure. It follows that a field definable in an o-minimal structure can be definably made into a topological field.
Pilllay's proofs were generalised by A. Mosley to the context of groups definable in first order topological structures satisfying certain natural coditions. In particular, he obtained analogues of the mentioned Pillay's results for groups and fields definable in models of weakly o-minimal theories satisfying the exchange principle. Both Pillay's and Mosley's proofs make an extensive use of the theory of generics, which depends on the existence of a well-behaving notion of an algebraic dimension.
In the talk I am going to present an alternative method of investigation of groups and fields definable in weakly o-minimal structures, a class which is not closed under elementary equivalence, and whose members in general do not enjoy the exchange principle. This approach is based on the theory of topological dimension.
Error-correcting codes in the Hamming space. Universal bounds for codes with a given code distance. Codes in deletion/insertion metric. Combinatorics of subsequences and supersequences. Reconstruction of sequences from the minimum number of their subsequences and supersequences. Ordered Steiner systems as perfect codes capable of correcting deletions of symbols.
If P is a polynomial in n variables, its Mahler measure, m(P) is defined to be the average of log|P| integrated over the product of n circles. This quantity appears naturally as an entropy in certain discrete dynamical systems and as a rate of growth in many other situations. When n = 1, there is a classical formula of Jensen that expresses m(P) in terms of the zeros of P, but for n > 1 there is no such general formula. In the late 1970's, Smyth proved some intriguing formulas for a few polynomials of 2 and 3 variables that showed that m(P) can sometimes be related to special values of Dirichlet L-functions. Recently, starting from an insight of Deninger, formulae have been proved and conjectured for infinite families of polynomials in 2, 3 and more variables relating the value of m(P) to special values of L-functions of various kinds including L-functions of elliptic curves, Hecke L-functions and L-functions defined by modular forms. We will present a varied selection of some of these formulas and explain how some of them are proved.
In this talk we discuss Alvis-Curtis duality of finite groups of Lie type, which is induced by a derived equivalence of categories. We show in detail how this duality operates on general linear groups and we associate it to dualities for Hecke and q-Schur algebras. Furthermore, we describe how these dualities are related to the decomposition matrices of the corresponding algebras.
We give an overview of various forms of rigidity in dynamics determined by topology. We begin with a form of rigidity for orbit equivalence of translational flows on solenoids. Then presenting joint work with L.Sadun, we consider tiling space flows and discuss which tiling spaces are rigid and which are not. Then we draw a connection between certain tiling spaces and the previously considered translational flows and automorphisms of solenoids. Finally, we introduce a new form of rigidity for automorphisms of solenoids developed with R.J.Fokkink and discuss which automorphisms are rigid and which are not.
Let F be a local field with finite residue field, ring of integers o, and maximal ideal ℘. Let G be a reductive group scheme over o (for example, G = GLn). We present an approach to the study of representations of the finite groups Gr := G(o/&weierpr), which ,for r = 1, coincides with the theory of Deligne and Lusztig. One reason why such a study is of interest is the close connection between the representation theory of the groups Gr, and the representation theory of the group G(F). One of the few cases where the representations of Gr are known for all r ≥ 1, is when G = GL2. This is due to several people, including Kutzko, and the method used is purely algebraic, and quite different from our geometric approach. We show how the two methods can be linked, and in particular how the algebraic method can be used to analyse representations constructed geometrically.
The Monster group is probably the most famous among the finite simple groups. It can be approached through the so-called Monster amalgam which is formed by a triple of subgroups C ≅ 2+1+24⋅Co1, N ≅ 22+11+22⋅(M24 × S3), L ≅ 23+6+12+16⋅(3.S6 × L3(2)), such that [N:C∩N] = 3 and [L:C∩N] = [L:L∩N] = 7. My intention is to construct a uniform theory of the Monster adopting the Monster amalgam as the first principle. I am going to report on the current status of this project.
We discuss several problems that stem from Banach space theory, but in fact boil down to constructing boolean algebras with various properties. Some such constructions using set-theoretic, algebraic and model-theoretic methods will be shown.
We give examples of a variety of combinatorial and geometric reconstruction problems and show how these can be viewed as a general reconstruction problem for groups. This involves quite elementary methods from permutation group theory and model theory. We shall discuss some recent results on a geometric reconstruction problem.
Conway's Angel and Devil game is played on an infinite two-dimensional grid. At each turn, the angel moves to a square at distance at most 100 (say) from his current position, and then the devil kills any square. The devil wins if the angel is forced to land on a dead square. Does the devil win, or can the angel survive forever?
In this talk we will look at some higher-dimensional versions, as well as their curious connections with the two-dimensional game.