It is well known that a normal trace on $B(H)_{+}$ is proportional to the usual trace. On the other hand, it has been an open question whether or not a trace is proportional to the usual one on the set where that trace is finite. In 1966 Dixmier gave negative examples to this problem. Traces $f$ constructed by him have the following properties: $f(a) = 0$ for every operator a of finite rank, but $f(b) = 1$ for some compact operator $b$. Such traces are called Dixmier trace. In this talk I am going to talk about constructing Dixmier traces and deal with some examples.

Study of the topological and geometric properties of a (Riemannian) manifold by investigating the spectral properties of the geometric elliptic operators, or in general elliptic complexes, is the approach of the spectral geometry. Witten, in his famous paper "Supersymmetry and Morse theory", used the spectral properties of the perturbed de Rham complex, so called Witten complex, to prove the Morse inequalities. In this talk we shall cover his proof. The idea of the proof is to use the approximations of the eigenvalues of the corresponding laplacian. In the next step, we will have an overview on Bismut's proof. Bismut puts Witten's idea in another format. He proves the inequalities by studying the long term behavior of the heat kernel.

I will review the paper "Superconnections, Thom classes, and equivariant differential forms" by Mathai and Quillen, and explain how their results can be used to simplify the cohomological formula for the index of an elliptic operator on a compact manifold.

We introduce a new cohomology theory constructed using homotopy theoretic methods that bears some formal resemblance to K-theory -- it is equipped with Adams operations, a form of periodicity, and an analogue of the J-homomorphism. However, the information that it encodes is of a higher "chromatic level" than K-theory, and so may be suitable for studying higher chromatic phenomena in stable homotopy theory. Unfortunately, its geometric meaning is far from apparent, although there are some hints of a relationship with n-gerbes.