PhD
Supervisor: | Alex Ritter | |
Institution: | Mathematical Institute, Oxford University | |
PhD examiners: | Paul Seidel, Frances Kirwan | |
Graduation: | May 2021 | |
Field of research: | Mathematics / Symplectic topology / Floer theory |
Subject
My PhD was in symplectic topology, a field of pure mathematics.
I looked at symplectic manifolds with Hamiltonian circle actions. The simplest example is a sphere with the action of rotation. The symmetry of the circle action adds extra structure to many standard symplectic invariants (for example quantum cohomology and symplectic cohomology). I specifically investigated the Seidel maps introduced in Seidel’s 1997 paper and developed equivariant analogues of them.
I defined equivariant Seidel maps, and found that they satisfied many of the same properties as the non-equivariant versions. An important exception is the intertwining relation, which was not satisfied by my new maps. I could quantify the error term precisely, however, with what I called a weighted equivariant Seidel map. The properties of the maps could be written down succinctly using an algebraic structure called a flat connection.
Writing
Papers
- An intertwining relation for equivariant Seidel maps (pdf, 782 kB)
- Shift operators and connections on equivariant symplectic cohomology (pdf, 867 kB)
Thesis
PhD coursework
I wrote these short subject introductions as part of the broadening requirements.
- Introduction to Higher Categories (pdf, 329 kB)
- The Atiyah-Singer Index Theorem (pdf, 176 kB)
Undergraduate thesis
Talks
I presented my PhD work at the Geometry and Analysis Seminar in April 2021. I gave four talks at the department’s Junior Geometry and Topology Seminar.
- Slides for Geometry and Analysis Seminar, 26 April 2021 (pdf, 1.1 MB)
Experience
I attended 9 conferences and workshops in 6 countries.
I taught over 100 hours of mathematics to undergraduate students. This mostly took the form of two-on-one tutorials to first- and second-year undergraduates, but I also taught third- and fourth-year undergraduate students as a class teaching assistant. I was a Stipendiary Lecturer of Mathematics for Corpus Christi College in the 2019–2020 academic year.
I marked the Oxford Mathematics Admissions Test (MAT) in 2018, and participated in admissions interviews in December 2019.
Recommendations
I managed all of the resources for my research (papers, books, etc) with Zotero. This software automatically extracts bibliographic information from library, journal and preprint websites and caches the corresponding pdf document with the click of a button.
I wrote my LaTeX documents using the online service Overleaf, which provides a vastly simpler experience than compiling LaTeX documents on your own computer. I also found the tool tikzcd-editor useful when creating commutative diagrams.