This is an unedited transscript of the handwritten notes of my talk at the Master Class.

The talk will describe Ornstein-Uhlenbeck semigroups and explore an example in infinite dimensions related to the renormalization group of quantum field theory. An Ornstein-Uhlenbeck semigroup describes a diffusion process with constant diffusion and linear drift. The resulting effect is Gaussian convolution followed by rescaling. The first goal of the talk is to contrast the situation of detai...

In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.

An algorithm is presented for computing eigenvalues of regular selfadjoint Sturm-Liouville (SL) problems with matrix coefficients and separated boundary conditions.

A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.