University of Oxford

For continuous maps $f: S^{2n-1} \to S^n$ one can define an integer-valued invariant, the so-called Hopf invariant. The problem of determining for which $n$ there are maps having Hopf invariant one can be related to many problems in topology and geometry, such as which spheres are parallelisable, which spheres are H-spaces (that is, have a product), and what are the division algebras over $\mathbb{R}$.

The best way to solve this problem is using complex K-theory and Adams operations. I will show how all the above problems are related, give an introduction to complex K-theory and it's operations, and show how to use it to solve this problem.

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric

variety X. This combinatorial description can be used to construct moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles on X by Payne and Perling). We study how the moduli spaces of stable equivariant sheaves on X can be used to explicitly compute the fixed point locus of the moduli space of all stable sheaves on X, i.e. the subscheme of invariant stable sheaves on X.

The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.

Will discuss several constructions of the Grothendieck group in different contexts together with Wall's solution of the problem of determining homotopy types of finite CW complexes as a motivating application.

Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$,

$$ Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right) $$

is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$.

I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.