Efficient factorization or efficient computation of class 
numbers would both suffice to break RSA.  However the talk lies more in 
computational number theory rather than in cryptography proper. We will 
address two questions: (1) How quickly can one construct a factor table 
for the numbers up to x?, and (2) How quickly can one do the same for the 
class numbers (of imaginary quadratic fields)? Somewhat surprisingly, the 
approach we describe for the second problem is motivated by the classical 
Hardy-Littlewood method.

 In the last few years, it has become clear that there are striking connections between supersymmetry and geometric representation theory.  In this talk, I will discuss boundary conditions in three dimensional gauge theories with N = 4 supersymmetry.  I will then outline a physical understanding of a remarkable conjecture in representation theory known as `symplectic duality.

Additive combinatorics enable one to characterise subsets S of elements in a group such that S+S has

small cardinality. In particular a theorem of Vosper says that subsets of integers modulo a prime p

with minimal sumsets can only be arithmetic progressions, apart from some degenerate cases. We are

interested in q-analogues of these results, namely characterising subspaces S in some algebras such

that the linear span of its square S^2 has small dimension.

Analogues of Vosper's theorem will imply that such spaces will have bases consisting of elements in

geometric progression.

We derive such analogues in extensions of finite fields, where bounds on codes in the space of

quadratic forms play a crucial role. We also obtain that under appropriately formulated conditions,

linear codes with small squares for the component-wise product can only be generalized Reed-Solomon

codes.



Based on joint works with Christine Bachoc and Oriol Serra, and with Diego Mirandola.

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of N scalars with a random potential of fourth order. We show that the scaling of the typical minimal bounce action with N is readily understood. We discuss the extension to more realistic landscape models as well as the effects of gravity. 

The requirement to compute Jordan blocks for multiple eigenvalues arises in a number of physical problems, for example panel flutter problems in aerodynamical stability, the stability of electrical power systems, and in quantum mechanics. We introduce a general method for computing a 2-dimensional Jordan block in a parameter-dependent matrix eigenvalue problem based on the so called Implicit Determinant Method. This is joint work with Alastair Spence (Bath).

For many years, sum-factored algorithms for finite elements in rectangular reference geometry have combined low complexity with the mathematical power of high-order approximation.  However, such algorithms rely heavily on the tensor product structure inherent in the geometry and basis functions, and similar algorithms for simplicial geometry have proven elusive.

Bernstein polynomials are totally nonnegative, rotationally symmetric, and geometrically decomposed bases with many other remarkable properties that lead to optimal-complexity algorithms for element wise finite element computations.  The also form natural building blocks for the finite element exterior calculus bases for the de Rham complex so that H(div) and H(curl) bases have efficient representations as well.  We will also their relevance for explicit discontinuous Galerkin methods, where the element mass matrix requires special attention.