LECTURE COURSES

I will discuss string topology and its connections with Hochschild homology and cohomology. Some of the key ingredients are:

* the geometry and topology of loop spaces,

* Hochschild homology and cohomology as algebraic approximations to the algebraic/differential topology of loop spaces,

* how to do calculations,

* the implications of these calculations.

This course describes joint work with Professor Jun-Muk Hwang. We have so far obtained a complete classification of general singular fibers of proper holomorphic Lagrangian fibrations which turns out to be fairly parallel to the famous classification of singular fibers of elliptic fibrations by Kodaira. Our argument is very elementary (the prerequisites are just some basic knowledge of complex geometry and algebraic geometry). In the course, I would like to start with Kodaira's result, relevant definitions of holomorphic Lagrangian fibrations, several concrete examples, which illustrate the main result, then precisely formulate the main result, and finally explain an outline of proof - how one can apply techniques of vector fields and elementary birational geometry to obtain the classification.

GUEST LECTURES

First I will construct Ricci-flat conical Kähler metrics on crepant resolutions of Sasaki-Einstein cones in every Kähler class. Applying the deformation method by Poisson structures in generalized geometry, I will deform ordinary Calabi-Yau structures to obtain generalized Calabi-Yau metrical structures on the crepant resolution. If time permits, I will discuss ``pathological" deformations, e.g. obstructed deformations of holomorphic Poisson (symplectic) manifolds which are non-Kählerian.

We give an introduction to recent developments in the geometry of compact manifolds with holonomy G

The solution space to quantum cohomology differential equation has an integral structure given by the K-group and the Gamma class. In this talk I will discuss the compatibility between this Gamma-integral structure and the quantum Lefschetz principle. We will see that part of the Gamma-integral solutions to the quantum differential equation of toric complete intersections can be expressed explicitly as periods of the mirror.

I will talk about joint work with Masaki Tsukamoto to develop infinite energy Yang-Mills gauge theory. We study moduli spaces of ASD connections over the infinite cylinder S

Polyhedral Kähler manifolds are complex manifolds that are built from a collection of Euclidean simplexes, they admit a flat Kähler metric that acquires singularities along a collection of divisors. We will discuss different situations where polyhedral Kähler metrics can be used, including rigidity results on positively curved polyhedral manifolds, construction of complex surfaces of CAT(0) type, and properties of extremal line arrangements.

STUDENT LECTURES

Calibrated submanifolds of annuli with small energy are expressible as the graphs of small normal vector fields on subannuli of the annuli.

Kodaira's classification of complex surfaces lacks completeness for surfaces with first Betti number 1. However by works of Kodaira, Inoue, and Teleman, all surfaces with first Betti number 1 and second Betti number have been found. I am currently interested in classifying those surfaces by diffeomorphisms. In this talk, I would like to briefly introduce my current progress.

After introducing the notion of Fueter sections of hyperkähler bundles over 3-manifolds, I'll explain their relevance to a conjectural counting invariant arising from the study of G

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Last update: 21 December 2010