MTH 868 Geometry and Topology, Prof. Wolfson
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. Wang
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. Kalfagianni
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 Homological Algebra, Prof. Ivanov
Description: Homological Algebra is one of the main algebraic tools (for algebra itself and for other branches of mathematics) developed during the 20th century, and is currently an active area of research by itself. I am plannig to devote the Fall semester to the classical part of the theory (homology of rings with coefficients in modules, some applications to groups and commutative algebra, connections with topology), and the Spring semester to more modern topics (such as derived categories, triangulated categories, non-abelian homological algebra). If time permits, an introduction to algebraic K-theory will be given. The prerequisites for the course are a good command of a basic algebra course. The textbooks are
J. Rotman, An Introduction to Homological Algebra, 2nd edition, 2009 (Fall),
S.I. Gelfand, Yu. I. Manin, Methods of homological algebra (Spring, tentatively).

MTH 996 Special Topics in Topology - Low dimensional Topology, Prof. Hedden
Description: his class will explore low-dimensional topology (dimensions less than 5) through the use of modern invariants from gauge theory, symplectic geometry, and quantum algebra. Special emphasis will be on the Heegaard Floer invariants of Ozsvath and Szabo. These are extremely powerful invariants of three- and four- manifolds which were developed in an attempt to compute the smooth four-manifold invariants of gauge theory via cut and paste techniques. The aim will be to introduce and develop the machinery of these and other invariants and then focus on applications of the invariants to a host of problems. For instance, many remarkable applications have been found in knot theory, Dehn surgery, and three-dimensional contact geometry. It seems likely that further applications abound. A little knowledge of low-dimensional topology would be helpful for the course but not necessary (e.g. basic algebraic topology, a smattering of concepts from three- and four-manifold theory). In general, the course will adapt to suit the interests of the participants, but topics I would like to cover include the recent invariants for three-manifolds with boundary developed by Lipshitz, Ozsvath, and Thurston, and the sutured Floer homology invariants of Juhasz. Both of these invariants are parts of a newly emerging theory which allows for easier computation of the Heegaard Floer invariants. Understanding these new invariants will undoubtedly lead to new applications in low-dimensional topology. To get to this may be a bit ambitious and, should interest be present, we could have an offshoot seminar dedicated to these and other advanced topics in the subject and allow the normal class hours to go at a reasonable pace.