Clay Adams

Thesis Advisor: Dr. Konrad Aguilar (Pomona College - Department of Mathematics and Statistics)

Second Reader: Dr. Alfonso Castro (Harvey Mudd College - Department of Mathematics)

Contact: ccadams@hmc.edu

Nonstandard Analysis and the Spectral Theorem

At the moment, I'm interested in the intersection of nonstandard analysis and the spectral theorem for operators on a Hilbert space. Nonstandard analysis is an application of model theory analysis, particularly by the construction of the hyperreals, an ordered field extending the real numbers containing both "infinite" and "infinitesimals." In 1976, Moore provided an alternate proof of the spectral theorem for bounded operators on a Hilbert space, passing through the hyperreals as a key element of the proof. In 2021, Goldbring generalized this nonstandard proof to the spectral theorem for unbounded operators on a hilbert space.  Currently, I'm learning more about functional analysis and nonstandard analysis to better understand this work.