L3

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.

We describe how one can recover the Mori--Mukai

classification of smooth 3-dimensional Fano manifolds using mirror

symmetry, and indicate how the same ideas might apply to the

classification of smooth 4-dimensional Fano manifolds. This is joint

work in progress with Corti, Galkin, Golyshev, and Kasprzyk.