Oxford

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.

In Riemannian geometry there are several notions of rank

defined for non-positively curved manifolds and with natural extensions

for groups acting on non-positively curved spaces.

The talk shall explain how various notions of rank behave for

mapping class groups of surfaces. This is joint work with J. Behrstock.

I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on $R^4$, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group $U(1)$. I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to a structural result about vector spaces.

This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.