Oxford

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.

For a fixed background manifold $M$ and parameter-space $X$, the associated configuration space is the space of $n$-point subsets of $M$ with parameters drawn from $X$ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space.

It is a classical result that the sequence of unordered configuration spaces, as $n$ increases, is homologically stable: for each $k$ the degree-$k$ homology is eventually independent of $n$. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability.

The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations.

If time permits, I will also say something about homological stability with twisted coefficients.

A simple formula is given for the $n$-field tree-level MHV gravitational

amplitude, based on soft limit factors. It expresses the full $S_n$ symmetry

naturally, as a determinant of elements of a symmetric ($n \times n$) matrix.

By recent work of Voevodsky and others, type theories are now considered as a candidate

for a homotopical foundations of mathematics. I will explain what are type theories using the language

of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic

concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.