This page lists this term's Pure Mathematics Seminars. All are welcome. All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 4.00pm, unless stated otherwise. Tea is served before the seminar at 3.30 in Room 237 of the McCrea Building.

**Seminars for the summer term 2008.**

- 29th April:
**Tuvi Etzion**(Technion, Haifa), Designs and Coding theory in projective space (abstract) - 6th May:
**Louigi Addario-Berry**(Oxford), Killed random branching walks (abstract) - 13th May:
**James McKee**(RHUL), Integer symmetric matrices of small spectral radius and small Mahler measure (abstract) - 3rd June:
**David Mireles-Morales**(RHUL), A geometric interpretation of the infrastructure of a real function field (abstract) - 10th June:
**Peter Symonds**(Manchester), Group actions on polynomial rings (abstract)

**Abstracts:**

**Tuvi Etzion **(Technion, Haifa), Designs and Coding Theory in Projective Space

Designs in projective spaces were defined in 1987 by Thomas. Recent interest in the area with applications in coding (by Ahlswede, Aydinian, and Khachatrian [2001]) and in cryptography (Wang, Xing, and Safavi-Naimi [2003]) drew some attention But, the area became very active in the last year due to a recent application for error-correction in network coding given last year by Koetter and Kschischang. First, we will show that there is a strong necessary condition for the existence of designs with λ= 1, which make the existence of nontrivial or simple ones very unlikely. We continue by showing several lower and upper bounds on the sizes of error-correcting codes, which are akin to known bounds in the Hamming scheme. Interesting codes which are better than the Reed-Solomon like codes of Koetter and Kschischang will be presented. Finally, we will examine the existence of algebraic and combinatorial structure of such codes. In particular we will examine the existence of codes which form large linear spaces and codes which have complements or quasi-complements.

**Louigi Addario-Berry** (Oxford), Killed random branching walks

Joint work with Nicolas Broutin. The problem is related to searching in trees. Suppose we are given a complete binary tree (a rooted tree in which the root has degree 3 and every other vertex has degree 2) with independent, identically distributed random edge weights (say copies of some random variable X). The depth d(v) of a vertex v is the number of edges on the path from v to the root. We give each vertex v the label S_{v} which is the sum of the edge weights on the path from v to the root. For positive integers n, we let M_{n} be the maximum label of any vertex at depth n, and let M*= max {M_{n}: n =0,1,...}. It is of course possible that M* is infinity.

Under suitable moment assumptions on X, it is known that there is a constant A

such that M_{n}/n --> A almost surely and in expectation. We call the cases A>0, A=0, and A< 0 supercritical, critical, and subcritical, respectively. When A <= 0 it makes sense to try to find the vertex of maximum weight M* in the whole tree.

One possible strategy is to only explore the subtree T_{0 }containing the root consisting only of vertices of non-negative weight.

With probability bounded away from zero this strategy finds the vertex of maximum weight. We derive precise information about the expected running time strategy. Equivalently, we derive precise information about the random variable |T_{0}|. In the process, we also derive rather precise information about M*. This answers a question of David Aldous.

**James McKee** (RHUL), Integer symmetric matrices of small spectral radius and small Mahler measure

This is joint work with Chris Smyth (Edinburgh). We determine all integer symmetric matrices of sprectral radius at most 2.019, and all whose Mahler measure is at most 1.3. In particular, we solve the strong version of Lehmer's problem for integer symmetric matirces: the Mahler measure is either 1 or at least "Lehmer's number" 1.17628...

**David Mireles-Morales** (RHUL), A geometric interpretation of the infrastructure of a real function field

Let $C$ be a hyperelliptic curve given by a plane real model $C:y^{2}=F(x)$. Let $R$ denote the set of principal reduced ideals in the ring $C[x,y]$. Every ideal in $R$ can be assigned an integer called the distance of the ideal. The distance gives $R$ an `almost group' structure. Why $R$ fails to be a group (or why it displays an `almost group' structure) has so far not been completely understood.

In this talk we will construct a map from the infrastructure into the class group of the curve that preserves the `group-like' structure. This will allow us to give an explicit interpretation of the distance and give a precise description of why $R$ fails to be a group. We also prove that the problem of computing the distance of a random ideal is equivalent to the discrete logarithm problem in the class group of the curve.

**Peter Symonds** (Manchester), Group actions on polynomial rings

We consider a group acting on a polynomial ring over a finite field by linear substitutions and we want to understand the ring as a module for the group.

We consider some examples that lead to a Structure Theorem that has various interesting consequences. For example only finitely many different isomorphism classes of indecomposable summands can occur. We also obtain an a priori bound on the degrees of the generators.

**Useful links**:

Campus map (The McCrea Building is number 17)