Research overview:
Research overview:
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I am a geometer studying problems in mathematics which have their origins in quantum field theory. In particular, I am interested in holomorphic Poisson manifolds and related spaces. A primary goal of my research is to understand the quantizations of these spaces in a non-perturbative manner.
You can find my publications on arXiv.
Projects:
Projects:
PhD research:
PhD research:
In my thesis work, I am studying Picard groups of holomorphic Poisson manifolds. The Picard 2-group of an integrable Poisson manifold is the 2-group of Morita autoequivalences of the source simply connected Lie groupoid integrating it. Lately I've been studying this 2-group in the holomorphic category, where we have some finiteness results if the manifold is compact.
More about this:
Here is a poster I presented at Poisson 2024 about this project: Picard Poster.
Hodge theoretic quantization:
Hodge theoretic quantization:
With Brent Pym I've been studying Kontsevich's formality morphism using mixed Hodge theory. To any log Poisson variety, we associate a canonical mixed Hodge structure which, under formality, is identified with another mixed Hodge structure associated to the quantum algebra produced by Kontsevich's formula. In many cases of interest, this allows us to determine the deformation quantization exactly, without using Feynman graphs.
More about this:
Semiclassical Hodge theory for log Poisson manifolds, joint with Brent Pym: arxiv.org/abs/2408.16685. Submitted.
My MSc thesis, which works this out for non-compact tori: Thesis.
A talk I gave in at Gone Fishing 2023 about this: Gone Fishing Talk.
Undergraduate summer research projects:
Undergraduate summer research projects:
The following papers are from summer research problems I worked on in my undergraduate years:
Combinatorial proofs of the type A quiver component formulas, joint with Jenna Rajchgot. Accepted to Selecta Mathematica. Arxiv 2503.09888.
Explicit Poincaré duality in the cohomology ring of the SU(2) character variety of a surface, joint with Lisa Jeffrey and Steven Rayan. Published in Expositiones Mathematicae. Arxiv 2007.00196.
Geodesics in a Kerr-Newman-(anti-)de Sitter Instanton, joint with Steven Rayan. Published in Journal of Geometry and Physics. Arxiv 1805.08887.
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