Honors Program
Document Type
Thesis
Date of this Version
Spring 3-24-2024
Citation
White, Nicholas. "A Cohomological Perspective to Nonlocal Operators." Undergraduate Honors Thesis. University of Nebraska - Lincoln. 2024.
Abstract
Nonlocal models have experienced a large period of growth in recent years. In particular, nonlocal models centered around a finite horizon have been the subject of many novel results. In this work we consider three nonlocal operators defined via a finite horizon: a weighted averaging operator in one dimension, an averaging differential operator, and the truncated Riesz fractional gradient. We primarily explore the kernel of each of these operators when we restrict to open sets. We discuss how the topological structure of the domain can give insight into the behavior of these operators, and more specifically the structure of their kernels. In particular, we define a nonlocal analogue to the 0th de Rham cohomology group and compare its structure to the classical case.
Comments
Copyright Nicholas White 2024.