MTH 868 Geometry and Topology, Prof. Wang
Description. This course is an introduction to manifolds; their topology and geometry. We begin with the definition, properties and examples of a manifold, its tangent and cotangent bundles and tensor bundles. We will then study vector fields, differential forms, integration on manifolds, Riemannian metrics and general vector bundles. The text is John Lee's 'Introduction to Smooth Manifolds', Springer.

MTH 914 Lie Groups,I, Prof. Pearlstein
Description:

MTH 935 Complex Manifolds,I, Prof. Wolfson
Description: This is an introduction to complex geometry. We will cover the following topics in varying details: holomorphic functions of several variables, complex manifolds, holomorphic vector bundles and Chern classes, Kahler metrics, sheafs and cohomology, Hodge theory, Riemann-Roch theorem and proof for curves.

MTH 960 Algebraic Topology, I, Prof.Ivanov
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 sequences.

MTH 991 Cluster algebras and Poisson geometry, Prof. Shapiro
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).

MTH 993 Special Topics in Geometry - Seiberg-Witten and Gromov-Witten Theory, Prof. Parker
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.

MTH 994 Special Topics in Applied Mathematics - Algebraic Curves, Prof. Li

MTH 996 Special Topics in Topology - Monopole Floer homology, Prof. Fintushel
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.