RHUL Mathematics Seminar

Ralf Rueckriemen: Trace formulae and heat kernel asymptotics on quantum graphs
I will introduce quantum graphs and talk about the tools used to study them. I will first show a trace formula and explain where it comes from and then present a method to study the heat kernel asymptotics of a quantum graph.

Ben Fairbairn: Beauville surface, structures and groups
Beauville surfaces are a class of complex surfaces that were first defined by Catanese about ten years ago and have numerous nice geometric properties. What makes them particularly good to work with is the fact that their definition can be translated into entirely group theoretic language. This raises all sort of questions. Which groups can be used? If a given group can be used what sort of surfaces can it define? Can other nice geometric properties can be translated across? In this overview we shall discuss these matters and more. En route we will meet numerous open questions, problems and conjectures.

Eugenio Giannelli: On the decomposition matrix of the symmetric group
In this talk I will give an introduction to the main ideas in the representation theory of the symmetric groups. Then I will describe the long standing open problem of finding the so called decomposition matrix of S_n. In the last part of the talk I will present a new result describing a number of columns of the decomposition matrix in odd prime characteristic.

Tim Browning: How frequently does the Hasse principle fail?
The Hasse principle, when it holds, gives an algorithm for checking whether a Diophantine equation has a solution in rationals. It does not hold in general, however, and counter-examples to the Hasse principle have been known since the 1970s. We give a snapshot of what is known and discuss how often such failures arise for some special families. This is joint work with Regis de la Breteche.

Jens Bolte: Quantum graphs
Quantum graphs are differential operators acting on functions defined on the edges of a metric graph. I will explain the motivation why one is interested in these objects (avoiding physics jargon as much as possible), introduce basic concepts and survey the main results. In the second part I will describe more recent work on many-particle quantum systems on graphs.

Peter Kropholler: Cohomology of finite rank soluble groups with group theoretic and homological applications
Soluble groups entered mathematics at the time of Galois because of the connection between group theory and the problem of solving polynomials in one equation by radicals. Thus, from a historical perspective, one could say that soluble groups go right back to the entrance of group theory onto the mathematical stage. They remain a great testing ground for theoretical questions throughout group theory and in this talk I propose to look at some questions about finiteness conditions motivated not by solving polynomial equations but by certain kinds of geometric construction, especially spaces like the fundamental group which provide a connection with topology and homotopy theory. Homology and cohomology are the natural algebraic theories associated with this territory and in this talk we will look at some of the features of soluble groups in this context.

Aditi Kar: Rank and deficiency gradient of groups
I will survey recent developments in the study of rank gradient and deficiency gradient in group theory. Rank gradient arose in the study of 3-manifold groups and later, was found to have deep connections with areas of mathematics like group cohomology and topological dynamics. Deficiency gradient is a more recent notion but is already at the centre of some frantic research. I will attempt to give an overview of the current state of the research and list some interesting questions.

Ben Martin: Complete reducibility for reductive algebraic groups.
Let G be a reductive algebraic group over a field k of positive characteristic. The notion of a completely reducible subgroup of G generalises the notion of a completely reducible representation (which is the special case when G = GL_n(k)). I will describe a geometric approach to the theory of complete reducibility, based on ideas of R.W. Richardson, and I will discuss some recent work involving non-algebraically closed fields.

Andrei Gagarin: The bondage number of graphs on topological surfaces and Teschner's conjecture
The domination number of a graph is the smallest number of its vertices adjacent to all the other vertices. The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. In a sense, the bondage number measures integrity and reliability of the smallest dominating sets with respect to edge removals, which may correspond, e.g., to link failures in communication networks. The decision problem for the bondage number is known to be NP-hard. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable graph genera. Also, we present stronger upper bounds for graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner's Conjecture in affirmative for almost all graphs. This is joint work with Vadim Zverovich, University of the West of England, Bristol, UK.

Sebastian Egger: The small-t asymptotics of the trace of the heat-kernel for arbitrary quantum graphs
The small t-asymptotics of heat-kernel is a widely studied object in mathematical physics. Its asymptotic expansion contains information about the spectrum of the underlying Schrödinger operator and is an important approach for its spectral analysis. I will generalize the known results for the small-t heat-kernel asymptotics of compact quantum graphs with non-Robin boundary conditions to non-compact quantum graphs with Robin boundary conditions.

Christopher Daw: The Andre–Oort conjecture for a product of modular curves
In this talk, we will explain the Andre–Oort conjecture for a product of two modular curves. We will give the unconditional proof of Pila, relying on the Pila–Zannier strategy and, in particular, the Pila–Wilkie counting theorem from o-minimality.

Simon Griffith: The triangle-free process and R(3,k)
A random graph process consists of a sequence of random graphs G₀, G₁, G₂, …, where G_m+1 is obtained from G_m by adding a single randomly selected edge. The most famous such model — in which the selections are independent — was introduced by Erdős and Rényi in 1959. In this talk we discuss a number of random graph processes, with a particular emphasis on the triangle-free process. We shall give a broad overview of the methods used to analyse such processes and say something about the recent result that the final graph G_n,Δ has strong quasi-randomness properties and has (with high probability) e(G_n,Δ) = (1/2^3/2 + o(1))n^{3/2 (log n)1/2.}

Kenny Paterson: Plaintext Recovery Attacks Against WPA/TKIP
We conduct an analysis of the RC4 algorithm as it is used in the IEEE WPA/TKIP standard. This standard is widely used for protecting wireless communications in domestic and commercial network. In the standard, RC4 keys are computed on a per-frame basis, with specific key bytes being set to known values that depend on 2 bytes of the WPA frame counter (called the TSC). We observe large, TSC-dependent biases in the RC4 keystream when the algorithm is keyed according to the WPA specification. These biases permit us to mount a statistical, plaintext-recovering attack in the situation where the same plaintext is encrypted in many different frames (the so-called 'broadcast attack' setting). We assess the practical impact of these attacks on WPA/TKIP. Joint work with Bertram Poettering and Jacob Schuldt.

Alan Haynes: Constructing cut and project nets which are close to lattices
Separated nets can be constructed from R^d actions on the torus, via the cut-and-project method. This method produces examples of aperiodic patterns which occur in nature (e.g. in quasicrystals), and a natural problem is to understand how 'close’ these patterns are to periodic. In this talk we will outline our recent proof (joint with Henna Koivusalo) that, in any irrational cut-and-project setup, it is always possible to choose acceptance domains in a nontrivial way so that the resulting separated nets are bounded distance to lattices.

Karen Gunderson: Random Markov processes on countable alphabets
In 1990, Kalikow introduced 'random Markov processes' to refine classifications of measure-preserving transformations. Looking at the shift transformation on probability spaces of infinite sequences of states, a random Markov process is a stochastic process for which there is a coupling of the sequences of states with doubly infinite sequences of positive integers (m(i)), so that for every i, the distribution on the states at the ith step of the process depends only on the m(i) previous states. Random Markov processes are a generalization of usual Markov chains and have been used to answer questions about uniqueness of certain types of measure spaces and as extensions for non-periodic transformations. In this talk, I will discuss some new results on the random Markov processes on any finite number of states and extensions of these results to certain processes with infinitely many states.

David Gunderson: Extremal questions for graphs forbidding an odd cycle
If G is a simple graph on n vertices, and C is a cycle with odd length, how many edges can G have while not containing a copy of C as a subgraph? What does an 'extremal' (most edges) C-free n-vertex graph look like? In 1907, Mantel answered these questions completely when C is a triangle. In the 1970s, Bondy and Woodall found, for all but finitely many n, the number of edges in an extremal C-free graph, and surprisingly, this number does not depend on the cycle length. However, their proof did not reveal what the extremal graphs are. Simonovits proved that for sufficiently large n, such an extremal graph on n vertices is unique. In recent work with Furedi, all remaining extremal numbers and extremal graphs are found. If time permits, I will mention recent progress on the following conjecture: the one triangle-free graph with the most number of cycles is the Turán graph. (This talk is intended for a general mathematical audience.)

Alexander Gorodnik: Searching for square-free points
In this talk we would be interested in finding solutions of polynomial equations whose coordinates are square-free (or r-power free) integers. Combining number-theoretic and dynamical tools, we show that in a number of cases such soulutions form a positive proportion among all solutions and establish an exact asymptotic counting formula for them. This is a joint work with Tim Browning.

Robert Brignall: From permutations to graphs: well-quasi-ordering and infinite antichains
The celebrated Graph Minor Theorem tells us that every infinite collection of graphs contains one graph which is a minor of another, so graphs are 'well-quasi-ordered with respect to the minor ordering'. If we replace 'minor' with 'induced subgraph' this statement is not always true: for example, the set of all cycles forms an infinite antichain. However, by restricting to some smaller class of graphs, we can recover this property, which has important consequences in the study of computational complexity.

In this talk, I will show how recent structural developments made in the study of permutation patterns can be used to establish the well-quasi-orderability (or not) of some graph classes, and specifically of permutation graphs defined by a forbidding a path and a clique.

Simon St John-Green: Proper Actions and Mackey Functors
A Mackey functor is an algebraic structure with operations like the induction, restriction, and conjugation maps of representation theory. We will examine cohomological finiteness conditions associated to Mackey and cohomological Mackey functors and ask if these provide good invariants to study proper actions on contractible CW complexes.

Shona Yu: Height extensions of the Temperley–Lieb category
The partition, Brauer and Temperley–Lieb categories and various associated diagram algebras are well-studied objects originating from different parts of mathematics and physics. Inspired by geometrical, representation theorectic and physical reasons, we will look into an 'interpolation' of algebras and categories which lie between the Temeperley–Lieb and Brauer. No prior knowledge of these objects required; but a non-disliking of partitions and pictures would be helpful. This is based on joint work with Zoltan Kadar and Paul Martin.

Alejandra Garrido: On the congruence subgroup problem for branch groups
For any infinite group with a distinguished family of normal subgroups of finite index — congruence subgroups — one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for SL(n,Z) where n > 2, by Bass, Lazard and Serre. Groups acting on infinite rooted trees are a natural setting in which to ask this question. Branch groups, which are a particular family of groups acting on these trees, have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.