C2

In 1937 Vinogradov showed that every sufficiently large odd number is the sum of three primes, using bounds on the sums of additive characters taken over the primes. He was improving, rather dramatically, on an earlier result of Schnirelmann, which showed that every sufficiently large integer is the sum of at most 37 000 primes. We discuss a natural analogue of this question in the multiplicative group (Z/pZ)* and find that, although the current unconditional character sum technology is too weak to use Vinogradov's approach, an idea from Schnirelmann's work still proves fruitful. We will use a result of Selberg-Delange, an application of a small sieve, and a few easy ideas from additive combinatorics. 

In this talk I will discuss the notion of o-minimality, which can be approached from either a model-theoretic standpoint, or an algebraic one.  I will exhibit some o-minimal structures, focussing on those most relevant to number theorists, and attempt to explain how o-minimality can be used to attain an assortment of results.

In this talk I will describe Arthur's classification of automorphic representations of symplectic and orthogonal groups using automorphic representations of $\mathrm{GL}_N$.

A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras.  The talk is partly based on joint work with Xuhua He.

On calm clear nights a minimum in air temperature can occur just above the ground at heights of order 0.5m or less. This is contrary to the conventional belief that ground is the point of minimum. This feature is paradoxical as an apparent unstable layer (the height below the point of minimum) sustains itself for several hours. This was first reported from India by Ramdas and his coworkers in 1932 and was disbelieved initially and attributed to flawed thermometers. We trace its history, acceptance and present a mathematical model in the form of a PDE that simulates this phenomenon.