Todd's PhD: summary and links
My PhD in maths at the University of Oxford took 3.5 years from October 2017 to May 2021. I studied a new invariant of a class of symplectic manifolds, and wrote two papers with my findings.
Key facts | |
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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 |
Title: | Equivariant Seidel maps and a flat connection on equivariant symplectic cohomology |
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.