Topology Seminar

Spring 2004

Time/place: Tuesdays 2-3:30 pm, Goldsmith 117

 

This semester the topic of the Topology seminar will be J-holomorphic curves.

Outline (tentative): The goal of this seminar is to learn some basics of J-holomorphic (aka pseudo-holomorphic) curves in symplectic manifolds. We will study the fundamentals in enough detail to justify the definition of quantum cohomology; these include basic results about symplectic manifolds, definition and essential properties of J-holomorphic curves, construction of moduli spaces, and important properties of these moduli spaces such as generic smoothness and compactness. References for the outline below:

1. Aebischer et al., Symplectic Geometry, Birkhauser (1994).
2. Audin et al., Holomorphic Curves in Symplectic Geometry, Birkhauser (1994).
3. McDuff and Salamon, Holomorphic Curves and Quantum Cohomology, AMS (1994).

Here is a possible outline of topics; they vary in size and depth and will probably need some reorganization to ensure a reasonable division of labor.

• Symplectic linear algebra: Sp(n) ∩ O(n) =U(n), etc.
• General symplectic manifolds: Darboux's theorem, existence of compatible almost complex structures. A few examples: Kähler manifolds.
• Basics of J-holomorphic curves. Definitions [Aebischer, 6.1.2] and simplest examples (eg lines/curves in CPⁿ).
• Construction of moduli spaces (several talks)
  • Analytic background on Sobolev spaces, definition of moduli space M. [Aebischer 6.2]
  • Generic smoothness, parameterized moduli space, dimension of M. [Aebischer 6.2.3]
• Regularity (using Sobolev spaces and analysis) [Aebischer 6.1.3, Audin V.1-V.2].
• Local properties of J-holomorphic maps; positivity of intersections

Now we come to the Gromov compactness theorem (several talks). Choice of two approaches: Gromov approach via Schwarz lemma/moduli space of surfaces or PDE/analytic approach via Sobolev spaces.

• Gromov approach
  • Schwarz lemma [Audin VII and Aebischer 7.1]
  • Moduli of surfaces (Teichmuller space) [Aebischer 7.2]
  • Gromov compactness theorem [Aebischer 7.3]
• PDE approach
  • Regularity theory [Aebischer 6.1.3, Audin V.1-V.2]
  • Local properties of holomorphic curves [Aebischer6.3.1, Audin V.3.2, VI.1-2]
  • Compactness (several talks) [Aebischer 6.3.2-5]
• Applications of Gromov compactness (non-squeezing theorem, structures on CP²).
• Definition of quantum cohomology [McDuff/Salamon intro, 7-8.]

Schedule

Date	Speaker	Title
January 27 February 3	Georgi Gospodinov	Symplectic linear algebra
February 13 Time:10:30-12 Place: Room 116	Dave Auckly, Kansas State University	An analytical framework for a geometric functional
Abstract: The Faddeev-Hopf functional has been studied numerically by a small number of mathematical physicists, to determine if it could be the basis of a more efficient emperical model of nucleons. This functional is defined for maps from a three-manifold into the two-sphere. This talk will describe a funky-looking representation of two-sphere valued maps, and present a proof of the fact that there is a minimizer of the Faddeev functional in each homotopy class. (This is joint work with Lev Kapitanski)

Fall 2003 schedule