L3

Joint work with: Sina Greenwood, Brian Raines and Casey Sherman

Abstract: We say a space $X$ with property $\C P$ is \emph{universal} for orbit spectra of homeomorphisms with property $\C P$ provided that if $Y$ is any space with property $\C P$ and the same cardinality as $X$ and $h:Y\to Y$ is any (auto)homeomorphism then there is a homeomorphism$g:X\to X$ such that the orbit equivalence classes for $h$ and $g$ are isomorphic. We construct a compact metric space $X$ that is universal for homeomorphisms of compact metric spaces of cardinality the continuum. There is no universal space for countable compact metric spaces. In the presence of some set theoretic assumptions we also give a separable metric space of size continuum that is universal for homeomorphisms on separable metric spaces.

See http://www.essex.ac.uk/maths/people/fremlin/chap53.pdf (538S)

The section conjecture of Grothendieck's anabelian geometry speculates about a description of the set of rational

points of a hyperbolic curve over a number field entirely in terms of profinite groups and Galois theory.

In the talk we will discuss local to global aspects of the conjecture, and what can be achieved when sections with

additional group theoretic properties are considered.

Derived moduli stacks extend moduli stacks to give families over simplicial or dg rings. Lurie's representability theorem gives criteria for a functor to be representable by a derived geometric stack, and I will introduce a variant of it. This establishes representability for problems such as moduli of sheaves and moduli of polarised schemes.

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.

A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring

satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative

differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology

is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes

without encountering gluing issues.