Geometry and Analysis Seminar

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.

Since Donaldson-Thomas proposed a programme for studying gauge theory in higher dimensions, there has
been significant interest in understanding special Yang-Mills connections in Ricci-flat 7-manifolds with holonomy
G_2 called G_2-instantons.  However, still relatively little is known about these connections, so we begin the
systematic study of G_2-instantons in the SU(2)^2-invariant setting.  We provide existence, non-existence and
classification results, and exhibit explicit sequences of G_2-instantons where “bubbling" and "removable
singularity" phenomena occur in the limit.  This is joint work with Goncalo Oliveira (Duke).

We will discuss two recent short-time existence results for (1) mean curvature of surface clusters, where n-dimensional surfaces in R^{n+k}, are allowed to meet at equal angles along smooth edges, and (2) for planar networks, where curves are initially allowed to meet in multiple junctions that resolve immediately into triple junctions with equal angles. The first result, which is joint work with B. White, follows from an elliptic regularisation scheme, together with a local regularity result for flows with triple junctions, which are close to a static flow of the half-planes. The second result, which is joint work with T. Ilmanen and A.Neves, relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow. 
 

The classical Yamabe problem asks to find in a given conformal class a metric of constant scalar curvature. In fully nonlinear analogues, the scalar curvature is replaced by certain functions of the eigenvalue of the Schouten curvature tensor. I will report on quantitative Liouville theorems and fine blow-up analysis for these problems. Joint work with Yanyan Li.
 

We give a noncommutative analogue of Castelnuovo's classic theorem that (-1) lines on a smooth surface can be contracted, and show how this may be used to construct an explicit birational map between a noncommutative P^2 and a noncommutative quadric surface. This has applications to the classification of noncommutative projective surfaces, one of the major open problems in noncommutative algebraic geometry. We will not assume a background in noncommutative ring theory.  The talk is based on joint work with Rogalski and Staffor

 An "open de Rham space" refers to a moduli space of meromorphic connections on the projective line with underlying trivial bundle.  In the case where the connections have simple poles, it is well-known that these spaces exhibit hyperkähler metrics and can be realized as quiver varieties.  This story can in fact be extended to the case of higher order poles, at least in the "untwisted" case.  The "twisted" spaces, introduced by Bremer and Sage, refer to those which have normal forms diagonalizable only after passing to a ramified cover.  These spaces often arise as quotients by unipotent groups and in some low-dimensional examples one finds some well-known hyperkähler manifolds, such as the moduli of magnetic monopoles.  This is a report on ongoing work with Tamás Hausel and Dimitri Wyss.