K-theory 1000-1M09KT

K-theory is a mathematical discipline whose name comes from a research method, rather than the studied objects (e.g. groups, vector spaces, topological spaces). Roughly speaking the method consists of introducing an algebraic structure (an abelian group, ring etc.) in a class of objects under consideration (K comes from the German "Klasse"). Properties of the given objects are formulated in algebraic terms (e.g. Invertible, zero-divisor etc.)
This approach was introduced by the great French mathematician Alexandre Grothendieck in the middle 1950's in his seminal work in algebraic geometry. A few years later Michael Atiyah (GB), Friedrich Hirzebruch (D) and Isadore Singer (USA) carried over the method into topology and the theory of elliptic differential operators. The Periodicity Theorem proved earlier by Raul Bott (USA) provided an extension of the topological functor K to a generalized cohomology theory. J.Frank Adams used it to prove that a reasonable multiplication in a real vector space can be introduced only in the classical cases: real numbers, complex numbers and quaternions.
In the years 1960/70 John Milnor (USA) and Daniel Quillen (USA) carried over K-theory into a purely algebraic setting, defining it for arbitrary rings. The methods found application also in functional analysis; more precisely in the theory of C*-algebras, developed by Allain Connes (F). Due to analogies with classical differential geometry, which is related to the commutative algebra of smooth functions, Connes' theory is called a noncommutative geometry.
The importance of K-theoretical ideas in XX century mathematics can be illustrated by the Fields medals awarded to mathematicians who contributed to them: J.Milnor (1962), A.Grothendieck, M.F.Atiyah (1966), D.Quillen (1978), A.Connes (1982), V.Voevodsky (2002).
The course will concentrate on the topological K-theory, which is the starting point for further generalizations, which also will be noticed. The final list of topics discussed in the lecture will depend on prerequsities and interests of participants.

Topics:
1. Vector bundles and their homotopical classification
2. Vector bundles as projective modules over the ring of continous functions
3. Functor K. Examples in algebra and topology.
4. Half exactness of the topological functor K. Generalized cohomology theories.
5. Bott periodicity.
6. Multiplicative structure in K-theory.
7. Thom isomorphism in topological K-theory.
8. Adams operations in topological K-theory.
9. Atiyah-Singer index theorem for elliptic operators (info)
10. Adams theorem on lineary independent vector fields on spheres.

Prerequisities: basic courses in Algebra and Topology, Algebraic Topology I.