RHUL Mathematics Seminar

Andrew Pollington: Diophantine approximation, a conjecture of Wolfgang Schmidt and irregularities of distribution
We describe Littlewood's conjecture in Diophantine approximation and some related questions and also present some connections to work on irregularities of distribution. Some of the work I will describe is Joint with Dmitry Bhadzian, Sanju Velani and some with William Moran.

Jozef Siran: Orientably-regular and regular maps of a given type
A map is a cellular embedding of a connected graph on a surface. If the surface is orientable and the group of all orientation-preserving map automorphisms is transitive on arcs, the map is said to be orientably-regular. A map on a general surface (orientable or not) is said to be regular if its map automorphism group is transitive on flags.

Intuitively, (orientably-) regular maps exhibit the highest level of (orientation-preserving) symmetry a map can have. All vertices in such maps have the same valency and all faces are bounded by walks of the same length; the pair (valency, length) is known as the type of a map.

In the talk I will discuss constructions of orientably-regular, chiral (orientably-regular but not regular) and non-orientable regular maps of a given type, with the help of a variety of algebraic tools.

Mark Jerrum: Counting small substructures in a large structure
For definiteness, take the large structure of the title to be a graph with n vertices and the small substructures to be induced k-vertex subgraphs satisfying a certain specified property Φ. Many counting problems fit into this general framework. If we fix k, we can certainly count subgraphs satisfying Φ in time O(n^k) by brute force. The question prompted by parameterised complexity is: is there universal constant c, and an algorithm A for counting these subgraphs, such that for all k the running time of A is O(n^c)? If k is small, and n large, such an algorithm might be acceptable, even if the dependence on k implicit in the O-notation is rather strong. There has been substantial progress on this topic recently, and I'll try to describe it using a couple of illustrative examples. The area is characterised by interesting use of combinatorial ideas, so I hope there will be something here even for those with no interest in computational complexity and algorithms. Later in the talk I'll reach some joint work with Kitty Meeks (Glasgow).

John Britnell: Minimum degree pathologies in p-groups
The minimum degree of a finite group G is the smallest degree of a faithful permutation representation of G. It is a well known observation of Neumann that some groups possess quotients with a higher minimum degree, and that this poses a problem for computing in permutation groups. In the case of p-groups it is known that the smallest examples occur when |G| = p⁵.

Eugenio Giannelli: Modular representation theory of symmetric groups
In this talk I will discuss projectivity of simple and Specht modules of the symmetric group. The first part of the talk will be a general introduction to the basic ideas and definitions in the modular representation theory of finite groups.

Victor Beresnevich: Some multiplicative problems in metric number theory
I will first give a 'gentle' introduction to metric number theory describing some basic results and techniques and open problems including Khintchine's theorem and the Duffin–Schaeffer conjecture. I will then explain what I mean by multiplicative Diophantine approximation and give an account of metric results and problems in this setting including some recent development. I will show a link between some of these multiplicative problems and estimates for the classical sums of reciprocals of fractional parts of arithmetical progressions. The latter is to be discussed in more details in my satellite number theory talk later on the day.

Maura Paterson: Combinatorial properties and applications of algebraic manipulation detection codes
Algebraic manipulation detection (AMD) codes are algebraic/combinatorial structures that are closely related to difference sets. They were defined for use in providing robustness against certain active attacks in a range of cryptographic applications. In this talk we will examine bounds on the parameters of AMD codes, and consider the problem of finding a combinatorial characterisation of AMD codes that meet the bounds. We will also discuss an application to the construction of optimal robust secret sharing schemes.

Simone Severini: Graph isomorphism: Quantum ideas
I will review methods for approaching the graph isomorphism problem based on quantum ideas.

Roger Behrend: The combinatorics of alternating sign matrices
An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. In the first half of this talk, the history and background of ASMs will be discussed, and some of the main combinatorial results will be reviewed. In the second half of the talk, the proof of a recent result involving ASMs and certain plane partitions will be outlined.

Nansen Petrosyan: Groups with cocompact classifying spaces for proper actions and a question of K. S. Brown
Many discrete groups we like have finite virtual cohomological dimension. By a well-known construction due to Serre, such groups admit finite dimensional classifying spaces for proper actions. One can think of this construction as a generalisation of a classical result of Eilenberg and Ganea that a group G with finite cohomological dimension cd(G) has a finite dimensional classifying space EG of dimension equal to cd(G) provided cd(G) is not 2 in which case it is equal to 3. The difference here is that if the group G has torsion then Serre's construction produces a space of dimension at least twice as big as the vcd(G).

In 1977, Ken Brown asked given a group G with finite vcd(G) whether one could always construct a model for the classifying space for proper actions of dimension equal to the vcd(G) and under which conditions on the group there exists such a cocompact model.

In this talk I will discuss how one can construct families of finite extensions of right-angled Coxeter groups that have a cocompact models for the classifying space for proper actions of minimal dimension but their virtual cohomological dimension is strictly less than this dimension. In fact, for these groups the gap between the two dimensions can be arbitrarily large.

This is joint work with Ian Leary.

Ged Corob: Soluble profinite groups
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 groups from simpler ones, and how these interact with properties of abstract and profinite groups, particularly cohomology properties. I will show how to use these techniques to prove results about soluble profinite groups.

Eugen Keil: Non-trivial pigeonhole principles
The pigeonhole principle is one of the simplest and at the same time probably the most versatile tool in combinatorics. In this talk I want to show how a more general view can help us to see some of the great results and proofs in additive combinatorics as non-trivial instances of this wide-ranging idea.

Chris Smyth: Polynomials with roots on the unit circle
For a single polynomial, we can readily find its roots. For infinite families of polynomials, however, it can be difficult even to give the general location of their roots. Here I will describe techniques for proving that families of polynomials have their roots on the unit circle, and also for telling which of these roots are roots of unity. If time permits I will also discuss the corresponding problem for hypersurfaces.

Efthymios Sofos: Distribution of rational points on surfaces
Manin's conjecture regards the distribution of rational points on varieties and can be viewed as an attempt to describe the analytic behaviour of the height zeta function via that of the Hasse-Weil zeta function. We will provide a brief account of previous results in the area and report on recent progress regarding the development of a counting method based on fibrations. Our new applications include proving the lower bounds predicted by Manin's conjecture for families of smooth cubic surfaces with a rational line.

Giovanni Gandini: Homological dimension via L² Betti numbers
I will report on work in progress which aims to determine the rational homological dimension of S-arithmetic groups over function fields. I will explain how in certain situations it's possible to use L² Betti numbers to show that the homological and the cohomological dimension coincide. I should conclude discussing present and future directions.This is joint work with Peter Kropholler, Ian Leary, and Andreas Thom.

Francesco Amoroso: Small points on varieties of algebraic tori
In this talk we present an overview on lower bounds for the height and on the distribution of small points in a power G_mⁿ of the multiplicative group. In the first part of the talk we discuss several one-dimensional lower bounds, starting with the still open 1933 Lehmer's problem. We then turn to multivariate problems, the so-called generalized Lehmer and Bogomolov lower bounds, concerning the distribution of points of small height in a subvariety of G_mⁿ.

Alexey Koloydenko: Positive definite matrices, Procrustes analysis, and other non-Euclidean approaches to diffusion weighted MRI
Symmetric positive semi-definite (SPD) matrices have recently seen several new applications, including Diffusion Tensor Imaging (DTI) in MRI, covariance descriptors and structure tensors in computer vision, and kernels in machine learning. Depending on the application, various geometries have been explored for statistical analysis of SPD-valued data. We will focus on DTI, where the Euclidean approach was generally criticised for its 'swelling' effect on interpolation, and for its violation of positive definiteness in extrapolation and other tasks. The affine invariant and log-Euclidean Riemannian metrics were subsequently proposed to remedy the above deficiencies. However, practitioners have recently argued that these geometric approaches are an overkill for some relevant noise models. We will examine a couple of related alternative approaches that in a sense reside between the two aforementioned extremes. These alternatives are based on the square root Euclidean and Procrustes size-and-shape metrics. Unlike the Riemannian approach, our approaches, we think, operate more naturally with respect to the boundary of the cone of SPD matrices. In particular, we prove that the Procrustes metric, when used to compute weighted Frechet averages, preserves ranks. We also establish and prove a key relationship between these two metrics, as well as inequalities ranking traces and determinants of the interpolants based on the Riemannian, Euclidean, and our alternative metrics. Remarkably, traces and determinants of our alternative interpolants compare differently. A general proof of the determinant comparison was developed (8th February this year!) by Koenraad Audenaert, whose help was also crucial in proving the key relationship between these two metrics.

Several experimental illustrations will be shown based on synthetic and real human brain DT MRI data. No special background in statistical analysis on non-Euclidean manifolds is assumed.

This is a joint work with Prof. Ian Dryden (University of Nottingham) and Dr Diwei Zhou (Loughborough University), with a recent participation of Dr Koenraad Audenaert (RHUL).

Ian Leary: Right-angled Coxeter groups as a source of examples
Coxeter groups are groups generated by reflections; right-angled Coxeter groups are the simplest ones in which any two reflection planes are either parallel or perpendicular. I shall explain some of the ways in which these groups give rise to interesting examples in a range of areas, following the seminal work of Mike Davis.

Sanju Velani: Metric Diophantine approximation: Lebesgue versus Hausdorff
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We discuss these theorems and show that Lebesgue statement implies the general Hausdorff statement. The key is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

Matteo Vannacci: On Hereditarily just infinite profinite groups
A profinite group G is said just infinite if every non-trivial continuous quotient of G is finite. Just infinite groups have been studied widely in the past as they yield interesting counterexamples to many problems in group theory (i.e. Burnside and growth problems). By a theorem of J. Wilson, a non-virtually abelian just infinite profinite group G is either branch (special subgroup of the automorphism group of a rooted tree) or hereditarily just infinite (every open subgroup of G is just infinite). While a lot of work on branch groups has been done, the family of hereditarily just infinite groups still remains very mysterious. In this talk I will give a short overview on just infinite profinite groups and then I'll present some of my results on hereditarily just infinite profinite groups.

Federico Passini: The classifying spaces of knot groups for the family of meridians
Classifying spaces for families are CW-complexes on which a group acts with prescribed fixed-point sets. They attracted strong interest as they appear in two crucial conjectures in K-theory; as they are also invariants of the group, their topology is expected to reflect some structure properties of the group itself. Therefore it is important to have nice concrete models for these spaces on which to perform explicit computations.

In this seminar we present the construction of a classifying space for the family of the meridians of a knot group.