Geometry and Analysis Seminar

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.

I will describe an ongoing research project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its space of stability conditions. More specifically we expect to find a complex hyperkahler structure on the total space of the tangent bundle. These ideas are closely related to the work of Gaiotto, Moore and Neitzke from a decade ago. The main analytic input is a class of Riemann-Hilbert problems involving maps from the complex plane to an algebraic torus with prescribed discontinuities along a collection of rays.

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

An 8-dimensional Riemannian manifold with holonomy group contained in Spin(7) is Ricci-flat, but not Kahler. The condition that the holonomy reduces to Spin(7) is equivalent to a complicated system of non-linear PDEs. In the non-compact setting, symmetries can be used to reduce this complexity. In the case of cohomogeneity one manifolds, i.e. where a generic orbit has codimension one, the non-linear PDE system
reduces to a nonlinear ODE system. I will discuss recent progress in the construction of 1-parameter families of complete cohomogeneity one Spin(7) holonomy metrics. All examples are asymptotically conical (AC) or asymptotically locally conical (ALC).

Exceptional holonomy manifolds come with certain geometric data that include a Ricci flat metric. Finding examples is therefore very difficult. The task can be made more tractable by imposing symmetry.  The focus of this talk is the case of torus symmetry. For a particular rank of the torus, one gets a natural parameterisation of the orbit space in terms of so-called multi-moment maps. I will discuss aspects of the local and global geometry of these 'toric' exceptional holonomy manifolds. The talk is based on joint work with Andrew Swann.

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.
 

Morse theory has a long history with many spectacular applications in different areas of mathematics. In this talk I will explain an extension of the main theorem of Morse theory that works for a large class of functions on singular spaces. The main example to keep in mind is that of moment maps on varieties, and I will present some applications to the topology of symplectic quotients of singular spaces.
 

It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller components are generalizations of this that exist for the moduli space of representations of the fundamental group into certain real simple Lie groups of higher rank. As for the usual Teichmüller space, these components consist entirely  of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of (still somewhat conjecturally) all possible Teichmüller components, and a parametrization of them using Higgs bundles.

We  show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

Conical Symplectic Resolutions form a broad family of holomorphic symplectic manifolds that are of interest to mathematical physicists, algebraic geometers, and representation theorists; Nakajima Quiver Varieties and Hypertoric Varieties are known as their special cases. In this talk, I will be focused on the Symplectic Geometry of Conical Symplectic Resolutions, and its non-symplectic applications. More precisely, I will talk about my work on finding Exact Lagrangian Submanifolds inside CSRs, and work in progress (joint with Alexander Ritter) about the construction of Symplectic Cohomology on CSRs.