Abstract
The aim of these lectures is to study Connes' cyclic cohomology.
It consists of two main parts, definitions and basic properties of
cyclic cohomology and cyclic cohomology in the context of
noncommutative differential geometry.
Let $A$ be an algebra. We begin the first part with Hochschild
cohomology $HH^*(A,M)$ of A with coefficient in an $A$bimodule
$M$. Then, Connes' cyclic cochain complex $C^*_{\lambda}(A)$ of
$A$ appears as a subcomplex of Hochschild cochain complex
$C^*(A,A^{*})$. In order to observe the relation between cyclic
cohomology and Hochschild cohomology, which is illustrated in
Connes' exact sequence, we define cyclic cochain bicomplex
$CC^{**}(A)$ of $A$. We reduce $CC^{**}(A)$ to a shorter cochain
bicomplex known as $(b,B)$bicomplex. Periodic and negative cyclic
cohomology of $A$ is discussed next.
The second part is devoted to Connes' noncommutative differential
calculus. We begin this part by introducing universal differential
forms. Then, we define cycles over an algebra $A$ and their
characters. The main theorem of this part states that cyclic
cocycles in cyclic cocomplex of $A$ are exactly the characters of
cycles over $A$. Afterwards, we explain briefly how a Fredholm
module over an involutive algebra $A$ defines a cycle over $A$.
This allows us to define the ConnesChern character of a Fredholm
module over $A$ as a cyclic cocycle in the periodic cyclic
cohomology of $A$. The pairing of these cyclic cocycles with
elements of Ktheory of $A$ is the noncommutative version of index
formula.
Information:
Date:  Every Wednesday, Nov. 14, 21, 28, 2007, 14:0016:00  Place:  Niavaran Bldg., Niavaran Square, Tehran, Iran 
