We describe the representation theory of loop groups in
terms of K-theory and noncommutative geometry. For any simply
connected compact Lie group G and any positive integer level l we
consider a natural noncommutative universal algebra whose 0th K-group
can be identified with abelian group generated by the level l
positive-energy representations of the loop group LG.
Moreover, for any of these representations, we define a spectral
triple in the sense of A. Connes and compute the corresponding index
pairing with K-theory. As a result, these spectral triples give rise
to a complete noncommutative geometric invariant for the
representation theory of LG at fixed level l. The construction is
based on the supersymmetric conformal field theory models associated
with LG and it can be generalized, in the setting of conformal nets,
to many other rational chiral conformal field theory models including
loop groups model associated to non-simply connected compact Lie
groups, coset models and the moonshine conformal field theory. (Based
on a joint work with Robin Hillier)

We will introduce the Thurston norm in the setting of 3-manifold groups, and show how the techniques coming from L2-homology allow us to extend its definition to the setting of free-by-cyclic groups.
We will also look at the relationship between this Thurston norm and the Alexander norm, and the BNS invariants, in particular focusing on the case of ascending HNN extensions of the 2-generated free group.

In 1878, Jordan showed that there is a function f on the set of natural numbers such that, if $G$ is a finite subgroup of $GL(n,C)$, then $G$ has an abelian normal subgroup of index at most $f(n)$. Early bounds were given by Frobenius and Schur, and close to optimal bounds were given by Weisfeiler in unpublished work in 1984 using the classification of finite simple groups; about ten years ago I obtained the optimal bounds. Crucially, these are "absolute" bounds; they do not address the wider question of divisibility of orders.

In 1887, Minkowski established a bound for the order of a Sylow p-subgroup of a finite subgroup of GL(n,Z). Recently, Serre asked me whether I could obtain Minkowski-like results for complex linear groups, and posed a very specific question. The answer turns out to be no, but his suggestion is actually quite close to the truth, and I shall address this question in my seminar. The answer addresses the divisibility issue in general, and it turns out that a central technical theorem on the structure of linear groups from my earlier work which there was framed as a replacement theorem can be reinterpreted as an embedding theorem and so can be used to preserve divisibility.

We hope to bring together all Oxford researchers interested in Cryptography, in Quantum Computing and in the interactions between the two.

Please register at:  http://oxford-cryptography-day.eventbrite.co.uk