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: A flat output of a control system allows to express its state and its inputs as a function of the flat output and its derivatives. It can be used, for example, to solve motion planning problems. We propose a variation of the definition of flatness for linear differential systems to linear differential-delay systems with time-varying coefficients which utilises a prediction operator $\pi$. We characterize $\pi$-flat outputs and provide an algorithm to efficiently compute such outputs.

(Joint work with Jean Levine, Felix Antritter and Franck Cazaurang)