L3

Triply graded link homology (introduced by Khovanov and Rozansky) is a

categorification of the HOMFLYPT polynomial. In this talk I will discuss

recent joint work with Ben Webster which gives a geometric construction of this invariant in terms of equivariant constructible sheaves. In this

framework the Reidemeister moves have quite natural geometric proofs. A

generalisation of this construction yields a categorification of the

coloured HOMFLYPT polynomial, constructed (conjecturally) by Mackay, Stosic and Vaz. I will also describe how this approach leads to a natural formula for the Jones-Ocneanu trace in terms of the intersection cohomology of Schubert varieties in the special linear group.

Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

In the first part of the talk we briefly describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson.

Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an analytic and an algebraic element in a relative algebraic K-group. As a first application we give some numerical evidence for ETNC in the case of the base change of an elliptic curve defined over the rational numbers. In this special case ETNC is an equivariant version of the Birch and Swinnerton-Dyer conjecture

In this lecture, I will illustrate the cobordism hypothesis by presenting some examples. Exact content to be determined, depending on the interests of the audience.