[based on joint work with Li Guo and  Bin Zhang]

 We apply to  the study of exponential sums on lattice points in
convex rational polyhedral cones, the generalised algebraic approach of
Connes and Kreimer to  perturbative quantum field theory.  For this purpose
we equip the space of    cones   with a connected coalgebra structure.
The  algebraic Birkhoff factorisation of Connes and Kreimer   adapted  and
generalised to this context then gives rise to a convolution factorisation
of exponential sums on lattice points in cones. We show that this
factorisation coincides with the classical Euler-Maclaurin formula
generalised to convex rational polyhedral cones by Berline and Vergne by
means of  an interpolating holomorphic function.
We define  renormalised conical zeta values at non-positive integers as the
Taylor coefficients at zero of the interpolating holomorphic function.  When
restricted to Chen cones, this  yields yet another way to renormalise
multiple zeta values  at non-positive integers.

When high-frequency acoustic or electromagnetic waves are incident upon an obstacle, the resulting scattered field is composed of rapidly oscillating waves. Conventional numerical methods for such problems use piecewise-polynomial approximation spaces which are not well-suited to capture the oscillatory solution. Hence these methods are prohibitively expensive in the high-frequency regime. Much work has been done in developing “hybrid numerical-asymptotic” (HNA) boundary element methods which utilise approximation spaces containing oscillatory functions carefully chosen to capture the high-frequency asymptotic behaviour of the solution. The computational cost of this approach is significantly smaller than that of conventional methods, and for many problems it is independent of the frequency. In this talk, I will outline the HNA method and discuss its extension to scattering by penetrable obstacles.​

Everything measurable about a quantum system, as modelled by a noncommutative operator algebra, is captured by its commutative subalgebras. We briefly survey this programme, and zoom in one specific incarnation: any bilinear associative function on the set of n-by-n matrices over a field of characteristic not two, that makes the same vectors orthogonal as ordinary matrix multiplication and gives the same trace as ordinary matrix multiplication, must in fact be ordinary matrix multiplication (or its opposite). Model-theoretic questions about the hypotheses and scope of this theorem are raised.

In this talk we will review some of the main ideas around Hamilton's program for the Ricci flow and see how they fit together to provide a proof of the Poincaré conjecture in dimension 3. We will then analyse this tools in the context of 4-manifolds.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a
free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.