The Farrell-Jones Conjecture gives a structural picture of the K- and L-theory of group rings. It has many applications to the classification of manifolds and questions about group rings. Proofs of instances of this conjecture often combine methods from algebraic topology and K-theory with methods from geometric group theory. In this course we will explain the conjecture, its applications and methods of proof.
In this course we discuss the combinatorial aspects of K-theory. We begin with a discussion of the Barratt-Priddy-Quillen Theorem and its connection between the topology of the sphere spectrum and the combinatorics of the symmetric groups. From this, we proceed to the construction of Waldhausen A-theory, and discuss several of its applications. Throughout the discussion we emphazise the connections between the combinatorial construction of K-theory and its topological implications.
This course aims to cover some very basic aspects of K-theory with emphasis on the applications to condensed matter physics and/or string theory. Some basic aspects of the theory of Clifford algebras, their representations and periodicity, and the relation to the classification of super-division algebras will be discussed. This course will also cover a physical incarnation of Milnor's proof of Bott periodicity using Clifford modules. Thus classifying spaces appear as large N limits of classical Cartan symmetric spaces.
I want to review some recent results on the comparison of algebraic and topological K-theory for topological rings, such as operator ideals or rings of continuous functions on a topological space. A main theme here is automatic homoty invariance of a certain algebraically defined functors. I will mention Karoubi's Conjecture (as proved by Mariusz Wodzicki) and a recent result on homotopy invariance of negative algebraic K-theory of the ring of continuous functions on a topological space.
The theory of motives began in the early sixties when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the eighties when the Moscow school (Bellinson, Bondal, Kapranov, Manin and others) started the study of algebraic varieties via their derived categories of coherent sheaves. In the last decade several important developments have been established by Kontsevich. The purpouse of this course is to give a rigorous overview of the theory of noncommutative motives. Several applications to neighboring areas of mathematics will also be described in detail.
A C*-algebra is approximately finite-dimensional (AF) if it is the inductive limit of a sequence of finite-dimensional C*-algebras. We will construct the dimension group of a unital AF C*-algebra by providing its K_0-group with a structure of scaled ordered group. We will then discuss Elliott's theorem, which shows that two unital AF C*-algebras are isomorphic if and only if their dimension groups are isomorphic as ordered groups.