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 3 pm, unless stated otherwise. Tea will be served after the seminar at 4 pm in Room 237 of the McCrea Building.

**Seminars for the spring term 2010.**

- 12th January:
**Graham Brightwell**(LSE): Forward Processes: Searching for a Lost Child - 19th January:
**Olof Sisask**(QMUL): Almost-periodicity results in additive combinatorics - 26th January:
**Rainer Dietmann**(RHUL): Probabilistic Galois Theory (abstract) - 2nd February:
**John Talbot**(UCL): Triangles in tripartite graphs - 9th February:
**Stephan Baier**(Bristol): A subconvexity bound for GL(3) automorphic L-functions (abstract) **Thursday**18th February, 1pm MCrea229:**Mohan Shrikhande**(Central Michigan University): A survey of embedding problems of Quasi-Residual Designs (abstract)- 23rd February:
**James McKee**(RHUL): Small-span characteristic polynomials of integer symmetric matrices (abstract) - 2nd March:
**Robert Johnson**(QMUL): Minimizing the average resistance in a graph - 9th March:
**Nikolay Nikolov**(Imperial College London): Boundedly generated groups and wreath products - 16th March:
**Kenny Paterson**(RHUL): Breaking and Provably repairing a proven secure protocol (abstract)

**Abstracts:**

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.

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.

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.

Kenny Paterson: SSH: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.

**Useful links**:

Campus map (The McCrea Building is number 17)