Riemann surfaces were ?rst de?ned in Riemann�s 1851 PhD dissertation where he laid down the geometric foundations of the theory of functions of one complex variable. The theory was advanced much further in Riemann�s 1857 epoch-making paper on the theory of abelian functions. It took mathematicians many years to fully grasp, develop, and extend Riemann�s ideas to its modern status... (go to the course page)

9140 -- Representation Theory of Finite Groups (Summer 2011)

While representation theory of finite abelian groups, and the corresponding character theory , goes back to the classical works of Lagrange, Gauss, and Dirichlet in early 19th century, the non-abelian representation theory and character theory started in 1890's in a series of ground breaking papers by Frobenius... (go to the course page)

9506B -- Index Theory (Winter 2011)

9160 -- Smooth Manifolds (Fall 2009)

The notion of a smooth manifold plays an important role in many areas of mathematics and physics, including: di?erential geometry, algebraic and dif- ferential topology, global analysis, classical mechanics, general relativity and high energy physics.... (go to the course page)

9054/4154 -- Functional Analysis (Fall 2009)

Linear Functional Analysis is mostly concerned with the study of continuous linear maps between topological vector spaces and as such it comprises an in?nite dimensional analogue of linear algebra. It is only through a ?ne and delicate mixture of linearity and continuity (analysis) that such a miraculous foray into the in?nite dimensional realm is possible. Some of the most impor- tant classes of topological vector spaces, with increasing order of generality, include: Hilbert spaces, Banach spaces, Frechet spaces and locally convex spaces... (go to the course page)

9170 -- Quantum Computing (Summer 2009), Reading Course

9610A -- Topics in Complex Geometry (Holomorphic Structures in Noncommutative Geometry) (Fall 2008)

While noncommutative geometry is a well developed subject, we have so far only glimpses of a noncommutative complex geometry. This year's topics in complex geometry will be centered around ``complex and holomorphic noncommutative geometry". There is no general theory as yet... (go to the course page)

557b -- Mathematics of Quantum Computing (2001-2002)

Existing computers and the corresponding mathematical theory of computing are based on laws of classical physics, as opposed to quantum physics. In 1981, Richard Feynman made a simple yet deep observation: while classical computers seem to be inefficient in tackling computations in quantum mechanics (e.g. solutions of Schr�dinger equation), (hypothetical) computers working based on principles of quantum mechanics should be extremely efficient at least for classical problems!... (go to the course page)

5XXb -- Homological Algebra (1999-2000)

The index theorem of Atiyah and Singer is a milestone of modern mathematics. This result computes a global invariant , namely the index ( = dimention kernel-dimension cokernel ) of an elliptic differential operator on a compact manifold, in terms of the local data defined by the operator and the manifold . Special cases of this theorem include results such as the Gauss-Bonnet-Chern theorem , Hirzebruch's signature theorem, ... (go to the course page)

537b -- Index Theory (1998-99)

Homological algebra, as we understand it today, is the study of chain complexes in an abelian category .The subject has its roots in topology and algebra specifically in the Riemann-Poincar� definition of the homology of a space and in Hilbert`s theorem of Syzygies. Before 1950, mathematicians had already difined homology and cohomology theories for various kinds of geometric and algebraic structures, without any clear link among them. The discovery of derived functors by Cartan and Eilenberg around 1951 ... (go to the course page)

536a -- Cyclic Cohomology and Noncommutative Geometry (1998-99)

The ideas of non commutative space , noncommutative geometry and accompanying invariants like K-theory, K-homology and cyclic cohomology grew out of attempts to solve several outstanding problems in topology ( Novikov conjecture ) and analysis ( extensions of Weyl-Von Neumann theorem , extensions of index theorem to foliated spaces ) . Cyclic cohomology replaces de Rham homology in noncommutative setting ... (go to the course page)

713b -- Noncommutative Geometry and Cyclic Homology (1997-98)

This course is an introduction to current ideas and methods of noncommutative geometry and cyclic homology as has been developed in Alain Connes' program since 1980. ... (go to the course page)

732b -- Differential Forms in Topology (1996-97)

Differential forms play a predominant role in many areas of mathematics like topology , geometry , analysis , and in modern theoretical physics . They have the unique feature that they stand at a crossroad of analytic and algebraic methods to probe topological invariants of a space.... (go to the course page)