I will give a proof of the following result known as Connes' trace theorem: Any pseudodifferential operator of order $-n$ acting on the smooth sections of a vector bundle over a compact $n$-dimensional manifold is in the domain of the Dixmier trace and its Dixmier trace coincides with its Wodzicki residue.

The separability problem, which is one of the central problems in the theory of quantum entanglement, asks for simple methods to determine whether a given quantum state is entangled (i.e., contains useful "quantumness") or separable (i.e., not entangled). We discuss how norms and eigenvector perturbation results can be used to approach problems like the separability problem, and we present some recent progress on these problems.

Let $f:M^n\rightarrow \mathbb{R}$ be a function with only nondegenerate critical points. Denote by $Crit_kf$ those critical points of f that have index k, let $c_k$ denote their total number. Consider the free abelian groups $C_k=\mathbb{Z}^{c_k}$, $C_k$ has one generator for each critical point of index k that f has. It is well known that that the strong Morse inequalities $c_k-c_{k-1}+\dots\pm c_0\geq b_k-b_{k-1}+\dots\pm b_0$ for $k=0,\dots,n-1$ and $c_n-c_{n-1}+\dots\pm c_0=b_n-b_{n-1}+\dots\pm b_0$, are equivalent to the existence of boundary homomorphisms $\partial_k:C_k\rightarrow C_{k-1}$ whose homology groups have rank, $b_k=Rank(H_k(M;\mathbb{Z}))$.There are several ways of getting to a boundary operator that will work. In this talk we will discuss one approach to constructing such a chain complex for a manifold M, given a metric g on M and a Morse function f on M. All approaches of which I am aware are based on the following observation. Associated to every Morse function f on M is a dynamical system given by the negative gradient flow of f. To define $\partial_k:C_k\rightarrow C_{k-1}$ we will investigate this dynamical system.

Abstract: The $\Gamma$ index theorem is one of the first index theorems on non-compact spaces. It plays an important role in extensions of index theory in noncommutative geometry and in representation theory of non compact Lie groups. I shall give a proof of this theorem and indicate some applications.