Oxford

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.

In recent years, surprising connections between type theory and homotopy theory have been discovered. In this talk I will recall the notions of intensional type theories and identity types. I will describe "infinity groupoids", formal algebraic models of topological spaces, and explain how identity types carry the structure of an infinity groupoid. I will finish by discussing categorical semantics of intensional type theories.

The talk will take place in Lecture Theatre B, at the Department of Computer Science.

I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of  Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.

Algebraic theories, locally presentable categories and their application to type theories. The seminar will take place in Lecture Theatre A of the Department of Computer Science.

For a fixed n we will investigate homomorphisms Out(F_n) to

Out(F_m) (i.e. free representations) and Out(F_n) to

GL_m(K) (i.e. K-linear representations). We will

completely classify both kinds of representations (at least for suitable

fields K) for a range of values $m$.

Abstract:

Motivated by the correlation functions-Wilson loop correspondence in maximally supersymmetric Yang-Mills theory, we will investigate a conjecture of Alday, Buchbinder, and Tseytlin regarding correlators of null polygonal Wilson loops with local operators in general position.  By translating the problem to twistor space, we can show that such correlators arise by taking null limits of correlation functions in the gauge theory, thereby providing a proof for the conjecture.  Additionally, twistor methods allow us to derive a recursive formula for computing these correlators, akin to the BCFW recursion for scattering amplitudes.