Description: This is a year long course
in algebraic topology. Topics to be covered may include: Cohomology,
universal coefficient theorem, Kunneth formula, Poincare duality.
deRham theory, homotopy theory, cellular and simplicial approximations,
Whitehead and Hurewicz theorems, fiber bundles, Postnikov
towers, obstruction theory, characteristic classes and spectral
Description: The goal is to give an introduction into newly emerging theory of cluster algebras (mostly from the point of view of Poisson structures and Integrable systems).
Description:Physicists have many examples of gauge theories that are very different yet, in some limit, yield equivalent quantum field theories. In mathematics, one example is now well-understood: Taubes' proved that a certain limit of the Seiberg-Witten equations concentrate (as solutions of a non-linear PDE) along sets that are completely characterized by Gromov-Witten invariants (solutions of a completely different PDE). The course will introduce both sides of this correspondence, including the needed geometry and analysis, and examine the concentration phenomenon.
1. Complex curves and Deligne-Mumford space
2. J-holomorphic maps and Gromov-Witten invariants
3. Seiberg-Witten equations and their moduli space
4. Taubes' concentration phenomenon and the proof that SW=Gr.
If time permits, we will finish with an introduction to the "Embedded Contact Homology" of Huchings and Taubes.
Description:The object of study in this course will be the Seiberg-Witten version of Floer homology. No previous knowledge of 4-manifolds, Seiberg-Witten invariants, or any type of Floer homology will be required, but of course, those of you just beginning with this will need to put in a bit of work. The text will be 'Monopoles and Three manifolds', by Kronheimer and Mrowka (Chapter 0 and parts of 7,9,10) My current idea for a course outline is:
1. Review of homology via Morse theory, connections, spin^c-structures, basic Seiberg-Witten theory
2. Properties of Seiberg-Witten invariants of 4-manifolds
3. Monopole Floer homology of 3-manifolds
4. Gluing theorems and calculations of Seiberg-Witten invariants using monopole Floer homology and some applications
If there is enough time we will also study Kronheimer and Mrowka's paper on excision.