This summer research project will explore the profound relationship between the geometry and topology of manifolds, as encoded by vector bundles. Known as Chern-Weil theory, this approach to studying the interplay of geometry and topology has been foundational to the modern study of geometry, to the development of gauge theories of particle physics (such as the standard model of particle physics), and to the ongoing development of non-commutative geometry.
The project will involve learning about the language of vector bundles, connections and curvature, as well as de Rham cohomology. All these topics can be phrased using differential geometry, in particular exterior calculus. These tools are central to modern geometry and related fields, and so well worth learning in any case.
The process of learning, and the transition to the research topic, will be facilitated by the detailed study of concrete examples. This will serve both to make the abstract theory comprehensible and to develop the ability to synthesise the theory into computations.
The aim of the research project would be to completely work out the ramifications of Chern-Weil theory for some low-dimensional examples, as well as to develop an intuition for what happens in more complex settings. As an application, we will work out the ramifications for the index theory of elliptic operators in low dimensions.
University of Wollongong
Jamie Bell is a third-year undergraduate student at the University of Wollongong studying a Bachelor of Mathematics Advanced. Majoring in pure mathematics, Jamie has a keen interest in a diverse range of pure topics from PDE and geometry, to topology and analysis. At undergraduate level, he has been exposed to differential geometry, Galois theory, functional analysis and noncommutative geometry through the completion of independent research projects. Jamie intends to complete his honours in pure mathematics in 2022. Outside of his studies, he is an avid mountain biker and hiker, and enjoys teaching others about mathematics through his tutoring business.