UK-Japan Winter School
New Methods in Geometry
King's College London, 10-13 January 2011


ABSTRACTS OF LECTURE COURSES AND GUEST LECTURES


LECTURE COURSES

Simon Donaldson (Imperial College London): Exceptional holonomy, gauge theory and calibrated geometry
Lecture 1. The algebraic and differential geometric foundations of exceptional holonomy and calibrated geometry. Cylindrical manifolds and the inclusions SU(3) ⊂ G2 ⊂ Spin(7). Submanifolds and Yang-Mills theory. Very brief review of other enumerative theories based on solutions of elliptic equations, such as Gromov-Witten invariants.
Lecture 2. A discussion of results (due to Tian, Price, Nakajima and others) on convergence of Yang-Mills connections away from a co-dimension 4 subset, and related background in PDE theory. Estimates over balls with small normalised energy, monotonicity formulae.
Lecture 3. More detail on enumerative theories: transversality, index theory and compactness. An account of work of Haydys giving a (conjectural) model for sequences of connections developing a codimension-4 singularity, relations with Seiberg-Witten equations. Significance for the possibilities of defining enumerative theories based on Yang-Mills theory over manifolds of exceptional holonomy.

John Jones (University of Warwick): String Topology
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.

Keiji Oguiso (Osaka University): Classification of general singular fibers of proper holomorphic Lagrangian fibrations via characteristic curves
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

Ryushi Goto (Osaka University): Holomorphic Poisson and generalized Calabi-Yau metrical structures
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.

Mark Haskins (Imperial College London): Compact G2 manifolds, associative submanifolds and complex 3-folds
We give an introduction to recent developments in the geometry of compact manifolds with holonomy G2, focusing on recent work with Corti, Nordstrom and Pacini; we prove the existence of many compact 7-manifolds with holonomy G2 that contain rigid associative submanifolds. The main ingredients in the proof are: an appropriate noncompact version of the Calabi conjecture for asymptotically cylindrical complex manifolds, gluing methods and a certain class of complex projective 3-folds (weak Fano 3-folds). In this construction the associative submanifolds are closely related to special holomorphic curves in these weak Fano 3-folds.

Hiroshi Iritani (Kyoto University): Quantum cohomology and periods
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.

Shinichiroh Matsuo (Tokyo University): Instanton approximation, periodic ASD connections, and mean dimension
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 S3 x R. Our moduli spaces contain not only finite energy ASD connections but also infinite energy ones, and thus they are of infinite dimension. For such infinite dimensional spaces, Gromov introduced a new invariant called "mean dimension". We estimate the mean dimension of the infinite dimensional moduli spaces of ASD connections.

Dmitri Panov (King's College London): Polyhedral Kähler manifolds
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

Yohsuke Imagi (Kyoto University): Energy of calibrated submanifolds of annuli
Calibrated submanifolds of annuli with small energy are expressible as the graphs of small normal vector fields on subannuli of the annuli.

Shota Murakami (Keio University): On diffeomorphism classes of complex surfaces with first Betti number 1 and second Betti number 0
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.

Thomas Walpuski (Imperial College London): Fueter sections and G2-instantons
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 G2-instanton moduli spaces. If time permits, I'll discuss what is known about Fueter sections in general and how they might be of independent interest.



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Last update: 21 December 2010