Solutions of the Einstein equations evolve on the constraint manifold which is an infinite-dimensional subset of tensors on a 3-manifold M which satisfy an under-determined geometric system of equations. There are many approaches to solving these equations and attempts at understanding the structure of the manifold. In this talk we will survey this topic including recent work on constructing localized solutions and special types of black hole solutions.
In 1986, a system of equations for compactifications of the heterotic string which preserve supersymmetry was proposed independently by C. Hull and A. Strominger. They are more complicated than the Calabi-Yau compactifications proposed earlier by P. Candelas, G. Horowitz, A. Strominger, and E. Witten, because they allow non-vanishing torsion and they incorporate terms which are quadratic in the curvature tensor. As such they are also particularly interesting from the point of view of both non-Kaehler geometry and the theory of non-linear partial differential equations. While the complete solution of such PDEs seems out of reach at the present time, we describe progress in developing a new general approach based on geometric flows which shares some features with the Ricci flow. In particular, this approach can recover the well-known non-perturbative solutions found in 2006 by J.X. Fu and S.T. Yau.
Fano varieties constitute a fascinating class of algebraic varieties that are important in birational
geometry and beyond. On the other hand, studying mild singularities is an indispensable feature of
modern birational geometry. In this talk I will try to explain how one interwines the two subjects to
prove various local and global boundedness statements regarding linear systems on varieties and
families of Fano varieties.
I will describe applications of a 6-dimensional string theory to the Geometric Langlands Programme and
to the Knot Categorification Programme.
Mumford's geometric invariant theory (GIT) provides a method for constructing quotient varieties for linear actions of reductive groups on projective varieties. The GIT quotient depends on the choice of linearization for the group action, and this dependence was described using 'variation of GIT' (VGIT) by Thaddeus and Dolgachev & Hu in the 1990s. GIT has been extended to non-reductive actions; many of the nice features of classical GIT fail in general, but are satisfied given the extra data of a graded linearization for an action of a linear algebraic group with graded unipotent radical. The aim of this talk is to describe this picture and a version of VGIT which applies to it.
A hyperbolic 3-manifold M carries a flat PSL2(ℂ)-connection whose Chern-Simons invariant has been much studied since the early 1980s. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the classical Chern-Simons invariant, expressed in the language of topological field theory, plays an important role.
The Remodelling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti provides a precise correspondence between open-closed Gromov-Witten invariants of a symplectic toric Calabi-Yau threefold and the invariants of the mirror curve defined by Eynard-Orantin topological recursion. It can be viewed as a version of all genus open-closed mirror symmetry. I will present a proof of the conjecture and describe its implications on the structure of higher genus Gromov-Witten invariants, based on joint work with Bohan Fang and Zhengyu Zong.
In this talk I review the recent progress made in defining homological invariants for 3-manifold using string theory constructions. This generalizes the constructions of homological invariants for knots using M5-branes, to the case of 3-manifolds.
The Vafa-Witten equations on an oriented Riemannian 4-manifold are first order, non-linear equations for a pair of connection on a principal SO3 bundle over a 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. This talk will describe a theorem about the behaviour of
sequences of solutions to the Vafa-Witten equations which have no convergent subsequence. This theorem says in part that a renormalization of a subsequence of the self-dual 2-form components of any given solution sequence converges on the complement of a closed set with Hausdorff dimension at most 2; and the limit defines a harmonic 2-form with values in a real line bundle. This behaviour generalizes Karen Uhlenbeck's compactness theorem for the self-dual Yang-Mills equations; it is similar to what happens in other first order generalizations of the Seiberg-Witten/self-duality equations.
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