About me
Hi, I’m Dario. I’m a student of mathematical physics, working on my PhD at Stellenbosch University under the guidance of Bruce Bartlett. My research focusses on topological quantum field theory (TQFT).
My current research
In Chern-Simons theory, as in any TQFT, one must associate to each surface a corresponding vector space1. Historically, there have been two rather distinct approaches to formulating this vector space in Chern-Simons theory.
On the one hand, this space can be constructed by the usual machinery of geometric quantization - as a space of holomorphic sections of a certain line bundle living over the moduli space of flat $\mathrm{SU}(2)$ connections on the surface2. On the other hand, the space can be considered a space of skeins - linear combinations of diagrams drawn inside the 3-dimensional handlebody bounded by our surface, modulo certain local ‘skein relations’ that hold inside each embedded ball3.
My current PhD work is focussed on writing down a new basis-free isomorphism between the holomorphic section and skein spaces, essentially providing another (simpler) answer to this MathOverflow question. The idea for this isomorphism comes from my supervisor, but several steps are still imprecise or require further proof or verification.
My background
I have a mixed background in pure mathematics and physics. I completed a bachelors in physics, Honours in mathematics4, Master’s in physics, and am working on my PhD in mathematics.
My Master’s thesis investigated a natural quantum-to-classical transition arising (by the mechanism of decoherence) in the 3D fuzzy-sphere model of quantum mechanics on a spacetime with non-commutative geometry. I published this paper about my results. During my Master’s years, I also took courses on general relativity, quantum computing, and solid-state physics.
My Honours mini-thesis was an expository account following a paper by Chelkak, Cimasoni, & Kassel (2017) that derives a certain formula for the Ising model embedded on a surface. During my Honours year I also took courses on QFT, Lie theory, functional analysis, measure theory, statistical physics, and C*-algebras.
My research interests
Subject codes: 57K16, 57R56, 81T45, 81R60, 81P05, 81S22, 53D50, 57K31, 14D21, 18M30
I am generally interested in most topics in mathematical physics, low-dimensional topology, and differential geometry, especially non-commutative geometry, topological quantum field theory, and applications of graphical calculi. I want to learn more about higher categories, homological algebra, and Morse-Cerf theory.
I am open (and eager) to collaborate on these and other topics. Please feel free to reach out!
By TQFT, I mean in the sense of Atiyah (1990) - that is, a certain functor from the cobordism category to the category of vector spaces. ↩
Jørgen Andersen and William Petersen. Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I. Transactions of the American Mathematical Society, 372(8):5713–5745, 2019. ↩
Christian Blanchet, Nathan Habegger, Gregor Masbaum, and Pierre Vogel. Topological quantum field theories derived from the Kauffman bracket. Topology, 34(4):883–927, 1995. ↩
Some South African university faculties separate the fourth year of undergraduate (the ‘senior’ year, in US terminology) into an optional ‘Honours’ degree. ↩