Senior Honors Thesis Presentation: Schatten Trace Class and Fredholm Determinant: Report on a Determinant Identity

Abstract: Finite dimensional complex Hilbert spaces admit a series of convenient properties. However, those properties don’t all translate to an infinite complex countable Hilbert space. In particular, the Fredholm determinant of ABA^{-1}B^{-1} is not always 1. Nonetheless there exists sufficient conditions for which this property does hold. These conditions are conjectured by A. Kitaev and studied in A. Elgart and M. Fraas’ paper On Kitaev’s Determinant Formula. In my presentation, I will review the core background concepts in operator theory and spectral theory needed to understand their work. I will then report on the results of A. Elgart and M. Frass and propose a new formula for calculating the trace of certain commutators that is compatible with Kitaev’s conjecture and can be used to construct examples for the conjecture.

Advisor: Xiang Tang