(This is Part II of a two-part talk.)

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

A field extension $F/K$ in characteristic $p$ is purely inseparable if for each $x$ in $F$, some power $x^{p^n}$ belongs to $K$. Using methods from homotopy theory, we construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p$ belongs to $K$ for all $x$ in $F$). This is joint work with Waldron.

It is expected that complete noncompact Calabi-Yau manifolds are in some sense governed by their asymptotics at infinity. In the maximal volume growth case, the asymptotics at infinity are given by Calabi-Yau cones. We are interested in deformations of such metrics that fix the asymptotic cones at infinity. In the asymptotically conical case, Conlon-Hein proved uniqueness under such deformations. Their method is based on the corresponding linearized problem, namely the study of subquadratic harmonic functions. We generalize their work to the maximal volume growth case, allowing the tangent cones at infinity to have non-isolated singularities. Part of the talk is based on work in progress joint with Gabor Szekelyhidi.

In the mean curvature flow, translating solutions are an important model for singularity formation. In this talk, I will describe the asymptotic structure of 2D mean curvature flow translators embedded in R^3 which have finite total curvature, which turns out to be highly rigid. I will outline the proof of this asymptotic description, in particular focusing on some novel and unexpected features of this proof.