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 spring term 2008.**

- 8th January: No Seminar
- 15th January:
**Andrew Booker**(Bristol),*"Vinogrodov's three primes revisited"* - 22th January:
**Sebastian von Wuthenau**(Cambridge),*"Geodesics in Surface Tessellations with Bounded Combinatorial Curvature"* - 29th January:
**Daniel Appel**(RHUL),*"Congruence Subgroups of the Automorphism Group of a Free Group"*Abstract:

We briefly recall the notion of a free group F and its automorphism group Aut(F). We then introduce congruence subgroups of Aut(F). These finite index subgroups of Aut(F) arise as stabilisers in a natural action of Aut(F) on the set of epimorphisms from F onto finite groups.

The Congruence Subgroup Problem asks whether every finite index subgroup of Aut(F) contains a congruence subgroup. We present related work on the structure of certain congruence subgroups of Aut(F) and we explain some connections to classical problems concerning generating pairs of finite groups. - 5th February:
**Leslie Goldberg**(University of Liverpool),*"Inapproximability of the Tutte polynomial"* - 12th February:
**Martin Bridson**(Oxford),*"Dimension rigidity for group actions"*Abstract:

Following a brief discussion of rigidity and super-rigidity for

group actions, I shall focus on actions of SL(n,Z) and explain

how to prove theorems of the following type:

THM 1: SL(n,Z) cannot act non-trivially by homeomorphisms on a

sphere of dimension less than (n-1) or a contractible manifold

of dimension less than n (these being the dimensions of the standard

linear actions).

THM 2: For every compact manifold M there exists an integer N(M) such

that SL(n,Z) cannot act by homeomorphisms on M if n>N(M).

The heart of the proof involves Smith theory (which I shall explain)

and an understanding of the torsion in SL(n,Z). - 19th February:
**Kenny Paterson**(RHUL) inaugural lecture in the Main Lecture Theatre at 5:30 - 26th February:
**Chris Brookes**(Cambridge),*"Iwasawa algebras"* - 4th March:
**Rachel Camina**(Cambridge),*"Recognising Z_p[[t]]-analytic pro-p groups"*Abstract:

We define a group theoretic criterion on a pro-p group that guarantees the existence of an analytic structure over Z_p[[t]]. - 11th March:
**Wouter Castryck**(Leuven),*"Effectively determining the number of points on curves over finite fields"*Abstract:

This talk is about the problem of developing an algorithm that, given a curve over a finite field, outputs the number of points on it. Partly due to its cryptographical relevance, the problem has seen detailed study in the past 20 years. A complete solution still seems far off, but over finite fields of fixed (small) characteristic, already fairly large classes of curves can be handled using p-adic cohomology, by following combined ideas of Takakazu Satoh, Kiran Kedlaya and Alan Lauder. I will sketch these ideas, and summarize the state of the art in point counting, anno 2008. - 18th March:
**Roger Heath-Brown**(Oxford),*"Sums and differences of three k-th powers"*Abstract:

It is easy to show that any integer can be written as x^2+y^2-z^2 in infinitely many ways. We can write 1 as a sum of three cubes (allowing negative values) in infinitely many ways. This talk will look at the number of respresentations of integers as sums and differences of three k-th powers in genereral.

Abstract:

In a 1742 letter to Euler, Goldbach conjectured that every integer at least 6 is the sum of three prime numbers. I will give some background to the problem and touch on some work in progress, joint with H. Helfgott, toward the first complete proof of the "easy" half of the conjecture.

Abstract:

Baues and Peyerimhoff introduced a notion of combinatorial curvature of a corner in a tessellation and showed that a tessellation in the Euclidean plane has empty cut locus if all corners have negative combinatorial curvature. The cylinder has the same curvature as the Euclidean plane, but cylinder tessellations with negative combinatorial curvature can have non-empty cut locus. The main difference with the Euclidean plane is that the cylinder has a larger variety of paths between two faces. This allows examples with non-empty cut locus that have more sides in every face. If we want a face f in the cut locus of a face g, then the neighbours of f can be approached by geodesics starting at g following paths belonging to the different ``classes of paths'', creating as few planar cycles as possible. This suggest than a generalization of this result to other surfaces or other dimensions would need to take in consideration the different varieties of paths of the surface.

Our main result for cylinder tessellations is that every cylinder tessellation satisfying at least one of the following conditions:

a) Each face has at least 5 sides and each vertex has valency at least 4,

b) Each face has at least 9 sides.

has the property that any finite geodesic can be expanded as a geodesic. We show that this result is sharp by providing examples of geodesics that can not be extended as geodesics in tessellations with any of the following properties:

a) Each face has at least 4 sides and each vertex has valency at least p, for any positive integer p;

b) Each face has at least 8 sides.

We also conjecture a generalization of this result to other surfaces.

Abstract:

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y - take as input a graph G, and output a value which is a good approximation to T(G;x,y). The precise formulation of the problem uses notions from computational complexity, which will be explained in the talk. Our main interest is in determining for which points (x,y) the computational problem is feasible. Jaeger, Vertigan and Welsh have completely mapped the complexity of *exactly* computing the Tutte polynomial. They have shown that this is feasible along the hyperbola (x-1)(y-1)=1 and at four "special" points. At any other fixed point (x,y), the computation is infeasible, in the sense that the problem is #P-hard. We aim to map the complexity of *approximately* computing the polynomial. We give several results, including a large region of intractability which includes the specialisation of the Tutte polynomial to the "flow polynomial" F(G, lambda) for lambda>2. This implies, for instance, that subject to complexity-theoretic assumptions, there is no fully-polynomial randomised approximation scheme for counting nowhere-zero 6 flows, even though the corresponding decision problem can be solved in polynomial time. The talk will discuss these and other results, and will also point out open questions, such as whether there is an approximation scheme for T(G;2,0), which is the number of acyclic orientations of G.

(joint work with Mark Jerrum)

*"From Fish to Phishing".*Abstract:

Cryptography, a thriving academic discipline at the intersection of mathematics and computer science, plays an important role in securing many facets of everyday life. This talk will discuss problems that crop up when cryptography is used in the real world, beginning with Fish, an important cipher from World War II, and ending with Phishing, a modern phenomenon in which fraudsters trick victims into revealing personal information such as credit card details. It will show what cryptography can (and cannot) do for us.

**Useful links**:

Campus map (The McCrea Building is number 17)