COURSE DESCRIPTION
This course aims to build a knowledge base on regularization techniques for moduli spaces arising from geometric PDE's (in particular pseudholomorphic curves). Regularization techniques of the type discussed here were first developed in the 1980s as part of the construction of Gromov-Witten invariants (from pseudoholomorphic curves) and Donaldson invariants (from Yang-Mills instantons). They have since seen rapid extensions in the construction of a wealth of invariants for symplectic or contact manifolds as well as low dimensional manifolds, such as various Floer theories, Fukaya's A-infty category, symplectic field theory, and gauge theoretic TQFTs. Due to the speed of the development of these applications, the subtleties of the underlying regularization techniques have not been studied systematically yet.
The goals of this course are to
- promote a conceptual understanding of the issues involved in regularization,
- give overviews of most existing approaches in symplectic topology,
- provide guiding questions and test cases for judging the rigour of any given approach,
- demonstrate good practices of rigour in manageable simplified settings,
- develop a basic common language for the mathematical discourse,
- provide a forum for the clarification of mathematical questions.
Topics and their weights will depend on the development of the course discussion, but certainly include
- an overview of moduli spaces of pseudoholomorphic curves and their analytic description,
- gluing of pseudoholomorphic curves in transverse situations,
- geometric regularization techniques such as ``choosing a generic almost complex structure'',
- Kuranishi type regularization techniques, based on finite dimensional reductions,
- Euler class type regularization techniques, based on obstruction bundles or stabilizations,
- polyfold regularization techniques, based on a generalized notion of Fredholm sections.
Prerequisites:
The perfect preparation (and motivation) is to have seen the broad outlines of ``counting'' or ``integrating over'' a moduli space of pseudoholomorphic curves, and have some basic familiarity with the underlying Gromov compactness and Fredholm theory. Lecture notes for a course (MIT 18.156 / Berkeley MATH278 in fall 2013) covering these basics can be found at the Resources tab of Piazza. However, I will aim to make the course as self-contained as possible. Minimal prerequisites are familiarity with the language of manifolds, vector bundles, Banach spaces, metric spaces, singular homology, and the willingness to take black boxes for granted or read up on them.
Logistics
Lectures can be joined in person on Monday/Wednesday at UC Berkeley (11-12:30 PST in Evans 730) or viewed live via browser/tablet/smartphone/room system using https://bluejeans.com/153111379 during 2-3:30 EST. (Note: You do need to download a plugin, so please test this well in advance.) With some delay, recordings will be made available on this site and become part of a Program at the Simons Center for Geometry and Physics, which is partially supporting this course.
To join the forum use access code jhol101.
Credit
At UC Berkeley, course credit can be obtained by enrolling in MATH276-002, and participating actively and regularly. Graduate students at other institutions, who are interested in gaining course credit, should find a local faculty or staff member who can grant course credit, and have them contact Katrin Wehrheim directly.