All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!
Seminars for the autumn term 2011:
- 27th September: Uri Onn (Ben Gurion University of the Negev): Representation zeta functions
Abstract: This is an introductory talk to the subject of representation growth and representation zeta functions
of groups. I will also present some recent results, including joint work with N. Avni, B. Klopsch and C. Voll.
- 4th October: Sinead Lyle (UEA): Homomorphisms between Specht modules
Abstract: The Specht modules are important modules for the symmetric group algebra and the Hecke algebras of type A. A lot of their properties can be encoded combinatorially and partitions, Young diagrams and tableaux are used extensively to study them. In this talk, we show how we may use combinatorics to construct homomorphisms between Specht modules indexed by certain pairs of partitions. This is joint work with Andrew Mathas. - 11th October: Julia Brandes (Bristol): A generalised version of the multi-dimensional Waring's problem
Abstract: A generalisation of Waring's problem, considered first by Arkhipov and Karatsuba, is the question of representing not an integer, but a given polynomial, as a sum of powers of linear polynomials. We investigate a related problem and prove a Hasse principle for the number of identical representations of a set of given forms by homogeneous polynomials of general shape. The result leads to sizeable improvements for estimates of the number of linear spaces on the intersection of hypersurfaces. - 18th October: Gary Greaves (RHUL): Algebraic integers and combinatorial objects
Abstract: One gains a better understanding of certain sets of algebraic integers by associating them to combinatorial objects. In a series of recent papers, McKee and Smyth have used various kinds of graphs and matrices to study certain algebraic integers. I will present results supporting a conjecture of Lehmer that have been obtained by classifying combinatorial objects having a certain property. - 25th October: Glyn Harman (RHUL): Primes whose sums of digits are prime
Abstract: Thanks to a recent extraordinary result given by Drmota, Mauduit and Rivat, we know that every sufficiently large prime is the sum of the digits of another prime. In this talk, which is aimed at a general mathematically literate audience, we explore the question of counting the number of primes whose sum of digits is prime, and show that an asymptotic formula is only possible if we assume a hypothesis somewhat stronger than the Riemann hypothesis. Nevertheless, we are able to obtain a formula for the sum of reciprocals of these primes. In addition we can show that if we look at the “decimal” expansion (in any base) of a “typical” real number, we find infinitely many primes whose sum of digits is prime. Technical details will be kept to a minimum! - 1th November: Chris Dowden: Extremal planar graphs
Abstract: One of the best known results in graph theory is Turan's Theorem, which concerns the maximum number of edges that a graph on n vertices can have without containing a "complete" subgraph of size r (i.e. r vertices with an edge between every pair). The Erdos-Stone Theorem then extends this to the case when other subgraphs are forbidden instead. Recently, I have started to look at the analogous question for planar graphs, which are graphs that can be drawn in the 2-dimensional plane without any edges crossing. In other words, how many edges can a planar graph on n vertices have without containing a specified forbidden subgraph? In this talk, I intend to present results for the case when the forbidden subgraph is a complete graph, which is quite easy to deal with, and also for the case when the forbidden subgraph is a small cycle, which is more difficult. - 8th November: Martin Kassabov (Southampton): Groups of oscillating intermediate growth (joint work with I. Pak)
Abstract: We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of functions. In particular, we can have growth oscillating between exp(n^a) (for 0.8 < a < 1) and any prescribed subexponential function, growing as rapidly as desired. Our construction is built on top of any of the Grigorchuk groups of intermediate growth, and is a variation on the limit of permutational wreath product. - 15th November: Robert Brignall (Open University): Grid Classes
Abstract: The study of permutation classes (sets of permutations closed downwards under the containment partial order) has recently begun to move from the ad-hoc consideration of specific examples to a more general structure theory. Central to this theory are "grid classes" - classes consisting of the permutations which when viewed graphically can be divided into cells using a collection of horizontal and vertical lines, with each cell satisfying some additional conditions. For example, we may specify that the points of a certain cell must form an increasing sequence.
Highlights in the study of grid classes include startling results in the asymptotic growth rates of so-called "small permutation classes", and general techniques to apply Higman's theorem or construct infinite antichains to resolve the question of well-quasi-order for a large number of permutation classes. Meanwhile, other problems such as finding the minimal set of forbidden patterns (the "basis") for a given grid class remain essentially wide open.
In this talk, I will present some of the most pertinent results and open problems in the study of grid classes. Of particular interest are monotone grid classes (where every non-empty cell of the grid must form an increasing or a decreasing sequence), and results relating to their structure, enumeration and well-quasi-order.
- 24th November (Thursday): There will be a statistics seminar
- 29th November: Inna (Korchagina) Capdeboscq (Warwick): Cocompact Lattices in some Kac-Moody groups
Abstract: After introducing Kac-Moody groups and recalling the necessary prerequisites about lattices, I will discuss existence and constructions of cocompact lattices in some topological Kac-Moody groups. - 6th December: Michael Harvey (RHUL) Representations of quadratic forms
Abstract: in this talk, we shall discuss the problem of representing an integral positive-definite quadratic form of rank m by another of rank n. When m = 1, this reduces to the classical case of representing a positive integer.Siegel obtained an exact formula for the number of averaged representations. Here we shall give an asymptotic formula for the number of representations, provided n is large enough in term of m.