Geometry and Analysis Seminar

We discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics and the mathematics of gauge theories. The
correspondence between the gauge theory, the underlying algebraic geometry of its space of vacua, the combinatorics of dimers and toric varieties, as
well as the number theory of dessin d'enfants becomes particularly intricate under this light.

The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT).  Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. Methods from non-reductive GIT allow us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves of fixed singularity type.

What is the correct combinatorial object to encode a linear representation?  Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies.  We suggest that already in very simple cases these miss a crucial feature.  The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration.  Some striking similarities to, and key differences from, the theory of toric varieties ensue.  Finite and non-finite generation phenomena emerge naturally.  We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.

 The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric.  Thus studying Sen’s conjectures requires a good understanding of the asymptotic structure of M_k and its metric.  This is a challenging analytical problem, because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k.  For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases.  In this talk, I shall introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.

I will define the notion of "sheaf of categories with a local action of Hochschild cochains" over a stack. (This notion is analogous to D-modules, in the same way as the notion of "sheaf of categories" is analogous to quasi-coherent sheaves.) I will prove that both categories appearing in geometric Langlands carry this structure over the stack of de Rham {\check{G}}-local systems. Using this, I will explain how to glue D-mod(Bun_G) out of *tempered* D-modules associated to smaller Levi subgroups of G.

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).

The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.

We review the concept of solitons in the Ricci flow, and describe various methods for generating examples, including some where the equations

may be solved in closed form

Generalized holomorphic bundles are the analogues of holomorphic vector bundles in the generalized geometry setting. In this talk, I will discuss the deformation theory of generalized holomorphic bundles on generalized Kaehler manifolds. I will also give explicit examples of moduli spaces of generalized holomorphic bundles on Hopf surfaces and on Inoue surfaces. This is joint work with Shengda Hu and Mohamed El Alami

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small  sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal
discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact
5 dimensional sphere has a unique Milnor filling up to normalization. This generalizes a Theorem by Mumford.

Title: Integrals and symplectic forms on infinitesimal quotients

Abstract: Lie algebroids are models for "infinitesimal actions" on manifolds: examples include Lie algebra actions, singular foliations, and Poisson brackets.  Typically, the orbit space of such an action is highly singular and non-Hausdorff (a stack), but good algebraic techniques have been developed for studying its geometry.  In particular, the orbit space has a formal tangent complex, so that it makes sense to talk about differential forms.  I will explain how this perspective sheds light on the differential geometry of shifted symplectic structures, and unifies a number of classical cohomological localization theorems.  The talk is
based mostly on joint work with Pavel Safronov.