Abstract: The index of a bounded operator $T\in B(H)$ of a Hilbert space $H$ is defined as the difference between the dimensions of kernel and cokernel. That is, $${\rm Ind}(T):=\dim(\ker T)-\dim({\rm coker}T)$$ This index, if defined, is called the Fredholm index. The Fredholm index of an operator on a finite dimensional Hilbert space $H$ by the dimension theorem in linear algebra. However, the case of infinite dimensional Hilbert spaces requires more delicate analysis and an operator with nonzero index exists. The celebrated local index formula in noncommutative geometry (Connes and Moscovici 1995) relates the index of Dirac type operators and the residue cocycle in the cyclic cohomology. In the classical case, this formula equates topology and geometry. In my talk, I will prove two special cases of local index formula following closely the chapter 5 in Noncommutative geometry and particle physics by Walter Van Suijlekom. If the time is allotted, I will demonstrate the strength of the formula using simple classical spectral triples such as the circle $S^1$.

Abstract: We present a chain lemma for tensor products of any number of quaternion algebras over fields of cohomological dimension 2. We discuss the connection to other objects, such as quadratic forms and the symplectic group.