RHUL Mathematics Seminar

Imre Leader: Higher order tournaments
Given n points in general position in the plane, how many of the triangles formed by them can contain the origin? This problem was solved 25 years ago by Boros and Furedi, who used a beautiful translation of the problem to a non-geometric setting. The talk will start with background, including this result, and will then go on to consider what happens in higher dimensions in the geometric and non-geometric cases.

Anton Evseev: A refinement of the McKay conjecture
Let G be a finite group and N be the normalizer of a Sylow p-subgroup of G. The McKay conjecture, which has been open for more than 30 years, states that G and N have the same number of irreducible characters of degree not divisible by p (i.e. of p'-degree). The conjecture has been strengthened in a number of ways, in particular, by Alperin and Isaacs-Navarro. The latter refinement suggests a precise correspondence between irreducible character degrees of G and of N modulo p and up to sign, if one considers only characters of p'-degree. The talk will review some of these generalisations and will consider a possible new refinement, which implies the Isaacs–Navarro conjecture.

Tim Browning: The L₁-norm of certain exponential sums
I will discuss the approach of Balog and Ruzsa for bounding below the L₁-norm of linear exponential sums whose coefficients are supported on the square-free integers. I will discuss how their lower bound can be improved by linking a problem about spacing of fractions a/p² to a problem about counting points of bounded height on elliptic curves. This is joint work with Antal Balog. The main result has been obtained independently by Sergei Konyagin.

Daniel Appel: On the abelianization of standard congruence subgroups of the automorphism group of the rank two free group
For an epimorphism π of the rank two free group F₂ onto a finite group G write Γ(G,π) for the group of all automorphisms f of F₂ for which π f = π. This is called the standard congruence subgroup of Aut(F) associated to G and π.

Congruence subgroups associated to abelian groups are closely connected to certain congruence subgroups of SL(2,Z). I will explain this connection and show how to use it to determine the abelianization of Γ(G,π) for abelian G. If time allows, I will also point out some open problems about the abelianization of Γ(G,π) for arbitrary finite groups G.

Boris Bukh: Sum-product theorems for polynomials
Suppose A is a set of numbers and f(x,y) is a polynomial, how small can f(A,A) be? If f(x,y)=x+y or f(x,y)=xy, then f(A,A) can be very small indeed if A is aprogression. However, Erdős and Szemerédi proved that A+A and AA cannot be simultaneously small when A is a set of real numbers. In this talk, I will survey this and related results, and will discuss several new results for other polynomial functions f. Joint work with Jacob Tsimerman.

Alexander Stasinski: Deligne-Lusztig theory and generalisations
We give a gentle introduction to classical Deligne–Lusztig theory of representations of certain linear groups over finite fields by way of examples. We then go on to sketch some recent generalisations of this to linear groups over finite local rings.

Vicky Neale: Bracket quadratics as bases for the integers
One of the classical problems of additive number theory, known as Waring's problem, is to show that the kth powers form a basis for the integers. That is, for any k there is some s = s(k) such that every positive integer is a sum of s kth powers. Lagrange's theorem, which says that every positive integer is a sum of four squares, is a special case of this. Waring's problem was first solved by Hilbert, and then a few years later Hardy and Littlewood supplied a new proof, using what is now known as their circle method.

I shall describe how to use a new variation of the circle method to show a Waring-type result: that the bracket quadratics n_⌊n√2_⌋ form an asymptotic basis for the integers. That is, there is some s so that every sufficiently large positive integer is a sum of s numbers of the form n_⌊n√2_⌋. The proof uses recent work of Green and Tao on the quantitative distribution of polynomial orbits on nilmanifolds. This is joint work with Ben Green.

Amaia Zugadi-Reizabal: Automorphisms of p-adic trees and Hausdorff dimension
In this talk, we will introduce the group of automorphisms of the p-adic tree and show that it is an important source of groups satisfying rare properties. We will focus on the study of the Hausdorff dimension on the group of p-adic automorphisms and we will present some recent results.

Rainer Dietmann: Probabilistic Galois Theory
On probabilistic grounds, one should expect that 'almost all' integer polynomials of degree n have the full symmetric group S_n as Galois group of their splitting field over the rationals. This conjecture has been confirmed by van der Waerden, and the strongest quantitative form known to date is by Gallagher, using the large sieve from analytic number theory. In this direction, we can prove the following result: Let G be a subgroup of S_n of index m. Then the number of monic integer polynomials of degree n and height at most H having Galois group G can be bounded by O(H^n-1+1/m+ε). Apart from the ε, this recovers Gallagher's result, and for m exceeding 2 (i.e. G different from S_n, A_n) gives stronger bounds than those resulting from Gallagher's sieve approach.

Stephan Baier: A subconvexity bound for GL(3) automorphic L-functions
This is joint work with L. Zhao. We establish a subconvexity bound for Godement–Jacquet L-functions associated with Maass forms for SL(3,Z). Our approach is based on an approach by M. Jutila.

Mohan Shrikhande: A survey of embedding problems of Quasi-Residual Designs
The notion of residual and derived design of a symmetric design goes back to a classic paper of R.C. Bose (1939). A residual design of a symmetric design D is a 2-design obtained from D by removing a block B and replacing every other block A by A \ B. A quasi-residual design is a 2-design which has the parameters of a residual design. A quasi-residual design which is a residual design is called embeddable.

In this survey talk, we begin with some classical results, then discuss some techniques for constructing quasi-residual designs and some different types of non-embeddability conditions. We include some recent results for families of non-embeddable quasi-residual designs. Proofs are provided for some new results and we give some tables of possible parameter sets of non-embeddable quasi-residual designs. This is joint work with T.A. Alraqad.

James McKee: Small-span characteristic polynomials of integer symmetric matrices
Let f(x) be the characteristic polynomial of an integer symmetric matrix. Then all the roots of f(x) are real, and its span is defined to be the difference between the largest root and the smallest root. I shall describe a recent classification of all cases where the span is less than 4. Much of the talk will be devoted to the history of the problem, and to why '4' is such an improtant number in this area.

Kenny Paterson: Breaking and Provably repairing a proven secure protocol
SSH is one of the most widely used secure network protocols. Originally designed as a replacement for insecure remote login procedures, it has since become a general purpose tool for securing Internet traffic. As such, it is used by millions of people on a daily basis. This talk will give a gentle introduction to SSH and its security. We will focus on some recent attacks against SSH due to Albrecht et al. and some new (positive) security results about SSH due to the presenter and Watson. The talk is intended to be accessible to all.