RHUL Mathematics Seminar

Jahan Zahid: Forms in many variables and recent progress on Artin's conjecture
After a brief survey of the charted and uncharted landscape of large dimensional projective varieties, we shall focus our attention to such objects defined over local fields. In particular we shall discuss recent progress on a conjecture due to Emil Artin.

Jan Hladký: Hamilton cycles in dense vertex-transitive graphs
We prove that every large dense connected vertex-transitive graph G contains a Hamilton cycle, that is a cycle through all the vertices of G. This answers partially a question of Babai and Lovasz. The proof is based on the Regularity Lemma. This is a joint work with Demetres Christofides and Andras Mathe (both Warwick).

Mark Wildon: Commuting conjugacy classes: an overview
Let us say that two conjugacy classes of a group commute if they contain representatives that commute. When G is a finite group with a normal subgroup N such that G/N is cyclic, one can use this definition, together with Hall's Marriage Theorem, to describe the distribution of the conjugacy classes of G across the cosets of N. I will give an overview of this result, and then talk about some more recent work on commuting conjugacy classes in symmetric and general linear groups. This talk is on joint work with John Britnell.

Graeme Taylor: Cyclotomic matrices and graphs
Lehmer's problem on the Mahler measure of integer polynomials motivates the study of matrices satsifying certain eigenvalue constraints. For integer symmetric matrices, a complete classification of cyclotomic matrices - those with all eigenvalues in [-2,2] - and minimal noncyclotomic matrices was obtained by McKee and Smyth. These results confirm Lehmer's conjecture for a broad class of polynomials, but the general problem remains open. I'll give an overview of the rational integer case, as well as some recent work generalising their approach - based on charged signed graphs - to matrices / graphs over the rings of integers of some imaginary quadratic fields, where Lehmer's conjecture also holds.

Graham Niblo: Topological superrigidity
There is a recurring theme in topology of starting with a map satisfying some control condition, and deriving the existence of a map with much more control. Examples include Whitney's embedding theorem, the sphere theorem, Waldhausen's torus theorem and the geometric superrigidity theorem. I will outline a new result in this spirit concerning the existence of codimension-1 embeddings in aspherical manifolds of high dimension and suggest some applications and possible generalisations. The results use a mixture of ideas from geometric group theory, surgery theory, Poincaré duality and rigidity. This is joint work with Aditi Kar.

Jozef Skokan: Ramsey-goodness
Given two graphs G and H, the Ramsey number R(G,H) is the smallest N such that, however the edges of the complete graph K_N are coloured with red and blue, there exists either a red copy of G or a blue copy of H. Burr gave a simple general lower bound on the Ramsey number R(G,H), valid for all connected graphs G: defining σ(H) to be the smallest size of any colour class in any colouring of H with χ(H) colours, we have R(G,H) ≥ (χ(H)-1)(|G|-1) + σ(H). For a given graph H, it is natural to ask which connected graphs G attain this bound. A class of graphs is called Ramsey-good if, for each fixed H, Burr's bound is attained for all sufficiently large graphs in the class. In this talk we will give an overview of some known results about Ramsey-goodness, and offer some new results. In particular, we shall explore connections between Ramsey-goodness and the bandwidth. Joint work with Peter Allen and Graham Brightwell.

Keith Martin: The rise and fall and rise of combinatorial key predistribution
There are many applications of symmetric cryptography where the only realistic option is to predistribute key material in advance of deployment, rather than provide online key distribution. The problem of how most effectively to predistribute keys is inherently combinatorial. We revisit some early combinatorial key predistribution shemes and discuss their limitations. We then explain why this problem is back 'in fashion' after a period of limited attention by the research community. We consider the appropriateness of combinatorial techniques for key distribution and identify potential areas of further research.

Cecelia Busuioc: Eisenstein congruences and K-theory
In this talk, we will investigate congruences of periods of parabolic modular forms at Eisenstein primes and their connection to the arithmetic of the K2-groups of rings of integers. Our approach to this study deals with an explicit construction of Eisenstein cohomology classes in the parabolic cohomology of the arithmetic group Γ₀(N).

Simon Goodwin: Representations of finite W-algebras
Finite W-algebras are certain associative algebras that can be viewed as the enveloping algebra of the Slodowy slice to a nilpotent orbit in a reductive Lie algebra g. The representation theory of W-algebras has a number of important applications: in particular, to the study of the primitive ideals of the universal enveloping algebra of g. In this talk, I will give an overview of the representation theory of finite W-algebras. All terms above will be explained and motivated.

Kevin Buzzard: Artin's conjecture on L-functions
I will tell the story of what little we know about an old conjecture of Emil Artin from the 1930s. The conjecture is deceptively simple. The Riemann zeta function is defined by a series in s which only converges for Re(s) > 1, but the function is well-known to have a meromorphic continuation to the complex plane and to satisfy a functional equation. Artin constructed some more general types of zeta function, defined by series which converged for Re(s) > 1, and conjectured that they also should have a meromorphic continuation. This simple-sounding conjecture is still wide wide open and recent incremental progress in our knowledge about it has only come via recent breakthroughs in subjects like the theory of modular forms. I will explain what little we know. I will not assume prior knowledge of Galois representations or modular forms or anything like this and there will be no proofs presented! It will be more of a survey of the area.

Heidi Gebauer: Game theoretic Ramsey numbers
The Ramsey Number, R(k), is defined as the minimum N such that every 2-coloring of the edges of K_N (the complete graph on N vertices) yields a monochromatic k-clique. For 60 years it is known that 2^k/2 < R(k) < 4^k, and it is a widely open problem to find significantly better bounds. In this talk we consider a game theoretic variant of the Ramsey Numbers: Two players, called Maker and Breaker, alternately claim an edge of K_N. Maker's goal is to completely occupy a K_k and Breaker's goal is to prevent this. The game theoretic Ramsey Number R'(k) is defined as the minimum N such that Maker has a strategy to build a K_k in the game on K_N. In contrast to the ordinary Ramsey Numbers, R'(k) has been determined precisely — a result of Beck. We will sketch a new, weaker result about R'(k) and use it to solve some related open problems.

Brita Nucinkis: Cohomological finiteness properties of the Brin-Thomspon-Higman groups
This is joint work with D. Kochloukova and C. Martinez-Perez. We show that Brin's generalisations 2V and 3V of the Thompson–Higman group V are of type FP_∞. Our methods also give a new proof that both groups are finitely presented.

Our proof is based on an idea of Ken Brown: In 1985 he showed that Thompson's groups F, T and V as well as some generalisations such as Higman's groups V_n,r are of type FP_∞ and are finitely presented. These groups are viewed as groups of algebra-automorphisms and act combinatorially on the geometric realisation of a poset determined by the algebra. It is then shown that this complex has a filtration yielding the required finiteness properties.

In this talk I will begin by explaining various ways to define the original Thompson groups, especially focusing on their description as certain automorphism groups of tree diagrams, and then move on to explaining how to generalise this to V. This will lead to the definition of the poset mentioned above.

Oscar Marmon: Sums and differences of four kth powers
I will discuss the following problem: in how many different ways can we write a natural number N as the sum of kth powers of integers? After a brief survey of relevant conjectures and results I will present some results of my own, giving new upper bounds for the number of representations of an integer as the sum or difference of four kth powers. I will briefly outline the main ideas in the proof, which uses a version of the Bombieri–Pila determinant method and is a mix of analytic number theory and algebraic geometry.

Christian Elsholtz: Sums of unit fractions (Number Theory Day)
We study the diophantine equation m/n = 1/ x₁ + … + 1/x_k (with positive integers m, n, x_i) and give a survey of several questions in this area.

• (1) For m=4 and k=3 Erdős and Straus conjectured that a solution exists, for each n > 1. More generally, for fixed m and k we study those n ≤ N that do not have a solution of the equation above.

• (2) For given m, n and k there are finitely many solutions only. For example, for m=n=1 an upper bound of 2^2k is almost trivial. In joint work with Tim Browning we gave more sophisticated upper bounds for the number of solutions.

• (3) If one restricts the x_i, for example all x_i are odd, or the x_i are composed of certain prime factors only, then one arrives at a different type of problem.

In joint work with Y. G. Chen and L. L. Jiang we solved several open problems, in particular we have a necessary and sufficient condition for the set of prime factors, to allow for a solution of 1 = 1/x₁ + … + 1/x_k, for sufficiently large k.

Rob Noble: Conjugate algebraic numbers on plane curves (Number Theory Day)
If an algebraic number lies with all of its conjugates on a single line in the complex plane, then the number must either be totally real (so that the line is the real axis) or have rational real part and totally real imaginary part (so that the line consists of those complex numbers with a fixed rational real part). Moving to the next level of complexity, we arrive at the conic sections. Roots of unity lie with all of their conjugates on the unit circle, and by a classical result of Kronecker, are the only examples of algebraic integers that satisfy this property. Work of Robinson, Ennola and Smyth completes the picture for circles, and Smyth and Berry have taken care of the other conic sections. It would be natural to next tackle the case of elliptic curves, but it may be easier to first treat curves of higher degree that arise as conic sections with respect to non-euclidean L^p norms. In this talk, an overview of the literature on conjugate algebraic numbers on lines and conics with respect to the euclidean norm, as well as some preliminary results for circles with respect to L^p norms will be given.

Phillipp Sprüssel: Unfriendly partitions of rayless graphs
The Unfriendly Partition Conjecture is one of the most famous problems concerning infinite graphs. It states that every countable graph has a partition into two parts such that every vertex has at least as many neighbors in the opposite class as in its own. Such a partition is called unfriendly. For almost 20 years, the only substantial result in this area has been the proof by Aharoni, Milner, and Prikry that graphs with only finitely many vertices of infinite degree have unfriendly partitions. We show that every rayless graph (a graph without an infinite path) has an unfriendly partition.

John MacQuarrie: Modular representations of profinite groups
Modern modular representation theory of finite groups is a tricky subject, with many strange and difficult unanswered questions. That said, the basic definitions and principles are very natural and are well understood, due in no small part to some excellent definitions by J. A. Green. By extending some of these definitions in a natural way, we begin to develop the subject for the wider class of profinite groups. Everything mentioned here will be defined, and we'll show that several foundational results required for a useful theory hold in this wider context.

Jan-Christoph Schlage-Puchta: Subgroups of virtually free groups
Let Γ be a virtually free group. We consider the following three questions: How many subgroups of index ≤ n does Γ have? Which subgroups occur as finite index subgroups of Γ? How does an 'average' subgroup of index n looks like? For Γ a free product these problems are by now well understood, however, for general virtually free groups surprising new phenomena do arise.

Steven Galbraith: Kangaroos, Birthdays, Card Tricks and Discrete Logarithms
The discrete logarithm problem in a group is: given g and h find x, if it exists, such that h = g^x. This computational problem has applications in public key cryptography. I will survey some algorithms due to John Pollard to solve the discrete logarithm problem and describe how a card trick, the birthday paradox, and the jumping of kangaroos have all influenced research on this problem. I will present some recently developed algorithms for special cases of the discrete logarithm problem, and a generalisation of the birthday paradox. This is joint work with Mark Holmes, John Pollard and Raminder Ruprai.

Dan Hefez: Hitting time results for Maker-Breaker games
We prove that almost surely a random graph process becomes Maker's win in the Maker-Breaker games 'k-vertex-connectivity', 'perfect matching' and 'Hamiltonicity' exactly when its minimum degree first becomes 2k, 2 and 4 respectively. (Based on joint work with Sonny Ben-Shimon, Asaf Ferber and Michael Krivelevich.)