Parametrized spectra are topological objects that represent
twisted forms of cohomology theories.  In this talk I will describe a theory
of parametrized spectra as highly structured bundle-like objects.  In
particular, we can make sense of the structure "group" of a bundle of
spectra.  This point of view leads to new examples and a good framework for
twisted equivariant cohomology theories.  

Given a 3-dimensional TQFT, the "conformal blocks" are the
values of that TQFT on closed Riemann surfaces.
The construction that we'll present (joint work with Douglas &
Bartels) takes as only input the value of the TQFT on discs. Towards
the end, I will explain to what extent the conformal blocks that we
construct agree with the conformal blocks constructed e.g. from the
theory of vertex operator algebras.

 We describe a framework for defining and classifying TQFTs via
surgery. Given a functor 
from the category of smooth manifolds and diffeomorphisms to
finite-dimensional vector spaces, 
and maps induced by surgery along framed spheres, we give a set of axioms
that allows one to assemble functorial coboridsm maps. 
Using this, we can reprove the correspondence between (1+1)-dimensional
TQFTs and commutative Frobenius algebras, 
and classify (2+1)-dimensional TQFTs in terms of a new structure, namely
split graded involutive nearly Frobenius algebras 
endowed with a certain mapping class group representation. The latter has
not appeared in the literature even in conjectural form. 
This framework is also well-suited to defining natural cobordism maps in
Heegaard Floer homology.

Vector bundles over a compact manifold can be defined via transition 
functions to a linear group. Often one imposes 
conditions on this structure group. For example for real vector bundles on 
may  ask that all 
transition functions lie in the special orthogonal group to encode 
orientability. Commutative K-theory arises when we impose the condition 
that the transition functions commute with each other whenever they are 
simultaneously defined.

We will introduce commutative K-theory and some natural variants of it, 
and will show that they give rise to  new generalised 
cohomology theories.

This is joint work with Adem, Gomez and Lind building on previous work by 
Adem, F. Cohen, and Gomez.

Consider a nonsingular projective variety $X$ defined by a system of $R$ forms of the same degree $d$. The circle method proves the Hasse principle and Manin's conjecture for $X$ when $\text{dim}X > C(d,R)$. I will describe how to improve the value of $C$ when $R$ is large. I use a technique for estimating mean values of exponential sums which I call a ``moat lemma". This leads to a novel and intriguing system of auxiliary inequalities.