I will describe the results of two projects on the construction of Calabi-Yau threefolds and certain real Lagrangian submanifolds. The first concerns the construction of a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which exploits crepant resolutions of such hypersurfaces. A central ingredient here is the construction of a certain 'integral affine structure with singularities' on the boundary of a class of polytopes from which one can form a topological model, due to Gross, of the corresponding Calabi-Yau threefold X. In general, such topological models carry a canonical (anti-symplectic) involution i and in the second project, which is joint work with H. Argüz, we describe the fixed point locus of this involution. In particular, we prove that the map i*-1 on graded pieces of a Leray filtration of H^3(X,Z2) can be identified with the map D -> D^2, where D is an element of H^2(X',Z2) and X' is mirror-dual to X. We use this to compute the Z2 cohomology group of the fixed locus, answering a question of Castaño-Bernard--Matessi.

Recent years have seen a great deal of progress in understanding the behavior of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite complete understanding of their evolution starting from a random initial condition, with a universality picture for their critical behavior. Here we will consider their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own : when $p$ shrinks to 0 they display some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics.

I'll discuss our recent proof of a conjecture of Talagrand, a fractional version of the "expectation-threshold" conjecture of Kahn and Kalai. As a consequence of this result, we resolve various (heretofore) difficult problems in probabilistic combinatorics and statistical physics.

I will discuss some ongoing work on three-dimensional supersymmetric gauge theories and their relationship to (equivariant) quantum K-theory. I will emphasise the interplay between the physical and mathematical motivations and approaches, and attempt to build a dictionary between the two.  As an interesting example, I will discuss the quantum K-theory of flag manifolds. The QK ring will be related to the vacuum structure of a gauge theory with Chern-Simons interactions, and the (genus-0) K-theoretic invariants will be computed in terms of explicit residue formulas that can be derived from the relevant supersymmetric path integrals.

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, by means of a virtual class $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ of the moduli spaces ${\mathcal M}_\alpha^{\rm st}(\tau)\subseteq{\mathcal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects in some homology theory. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds.

We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces ${\mathcal M},{\mathcal M}^{\rm pl}$, where my big vertex algebras project http://people.maths.ox.ac.uk/~joyce/hall.pdf gives $H_*({\mathcal M})$ the structure of a graded vertex algebra, and $H_*({\mathcal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\mathcal M})$. The virtual classes $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\mathcal M}^{\rm pl})$. In most such theories, defining $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\mathcal M}_\alpha^{\rm st}(\tau)\ne{\mathcal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\mathcal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles.

This is joint work with Jacob Gross and Yuuji Tanaka.

There are many known ways to compute the homology of the moduli space of algebraic vector bundles on a curve. For higher-dimensional varieties however, this problem is very difficult. It turns out that the moduli stack of objects in the derived category of a variety X, however, is topologically simpler than the moduli stack of vector bundles on X. We compute the rational homology of the moduli stack of complexes in the derived category of a smooth complex projective variety. For a certain class of varieties X including curves, surfaces, flag varieties, and certain 3- and 4-folds we get that the rational cohomology is freely generated by Künneth components of Chern characters of the universal complex––this allows us to identify Joyce's vertex algebra construction with a super-lattice vertex algebra on the rational cohomology of X in these cases. 

Certain exotic Lorentzian manifolds, including some of importance to string theory, possess an unusual geometric feature called an "evanescent ergosurface". In this talk I will introduce this feature and motivate the study of the wave equation on the associated geometries. It turns out that the presence of an evanescent ergosurface prevents the energy of waves from being uniformly bounded in terms of their initial energy; I will outline the proof of this statement. An immediate corollary is that there do not exist manifolds with both an evanescent ergosurface and a globally timelike Killing vector field.