Autumn 2009

29th September | Tuvi Etzion (Technion) Folding, Tiling, and Applications to Multidimensional Coding |

6th October | Imre Leader (Cambridge) Higher Order Tournaments (abstract) |

13th October | Anton Evseev (Cambridge) A refinement of the McKay conjecture (abstract) |

20th October | Tim Browning (Bristol) The L _{1}-norm of certain exponential sums (abstract) |

3rd November | Daniel Appel (RHUL) On the abelianization of standard congruence subgroups of the automorphism group of the rank two free group (abstract) |

10th November | Boris Bukh (Cambridge) Sum-product theorems for polynomials (abstract) |

17th November | Alexander Stasinski (Southampton) Deligne-Lusztig theory and generalisations (abstract) |

24th November | Vicky Neale (Cambridge) Bracket quadratics as bases for the integers (abstract) |

1st December | Amaia Zugadi-Reizabal () Automorphisms of p-adic trees and Hausdorff dimension (abstract) |

8th December | Christian Elsholtz (RHUL) Multidimensional problems in additive combinatorics |

Spring 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) |

18th February (Thursday, 1pm MCrea 229) | 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) |

Imre Leader: 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: 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: I will discuss the approach of Balog and Ruzsa for bounding below the L_{1}-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 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: For an epimorphism π of the rank two free group *F*_{2} onto a finite group *G* write Γ(*G*,π) for the group of all automorphisms *f* of *F*_{2} 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: 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 a progression. However, Erdös 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: 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: One of the classical problems of additive number theory, known as Waring's problem, is to show that the *k*th 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* *k*th 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* root 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* root 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: 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: 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*of index

_{n}*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*) gives stronger bounds than those resulting from Gallagher's sieve approach.

_{n}Stephan Baier: 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: 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: 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: 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.