Omar Kidwai
Research Fellow
School of Mathematics
University of Birmingham
Birmingham, United Kingdom
Contact:
Email: lastname.firstname at gmail.com
About Me:
Since June 2022, I have been a Research Fellow at the University of Birmingham, mentored by Marta Mazzocco. Before that I was a JSPS Postdoctoral Fellow with Kohei Iwaki - at Nagoya University followed by the University of Tokyo. Before that, I spent a year as a postdoc in the group of Marco Gualtieri at the University of Toronto. I completed my DPhil in 2018 at the University of Oxford, where I was a student of Lotte Hollands and Andrew Dancer in the Mathematical Institute.
My research lies at the interface of algebraic geometry, differential equations, and quantum field theory. The physical motivations come from four-dimensional N=2 supersymmetric QFT, which turns out to be closely related to the geometry of ODEs in the complex domain. As a result, I often spend time thinking about quadratic (and higher) differentials on Riemann surfaces. These induce certain graphical objects drawn on the surface called spectral networks, which play a fundamental role in the exact WKB analysis of the differential equations involved, among other things.
Recently, I've spent most of my time trying to understand how to count these networks (formalized as so-called BPS structures describing Donaldson-Thomas invariants), using the formalism of topological recursion (TR), which a priori appears in a totally different context. In simple cases, we found an easy and algebraic way of calculating or predicting these. More recently, I have been working on generalizing this to the refined case, the higher rank case, and understanding more carefully their DT theory -- stay tuned.
Preprints:
1. Donaldson-Thomas invariants for the Bridgeland-Smith correspondence, with Nick Williams, arXiv:2401.10093
2. Refined BPS structures and topological recursion - the Weber and
Whittaker curves, with Kento Osuga, arXiv:2311.17046
Publications:
3. Quantum curves from refined topological recursion: The genus 0 case, with Kento Osuga. Adv. Math. 432 (2023) 109243, arXiv:2204.12431
4. Topological recursion and uncoupled BPS structures II: Voros symbols and the τ-function, with Kohei Iwaki. Commun. Math. Phys. 399, 519-572 (2023), arXiv:2108.06995
5. Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies, with Kohei Iwaki. Adv. Math. 398 (2022) 108191, arXiv:2010.05596
6. Higher length-twist coordinates, generalized Heun s opers, and
twisted superpotentials, with Lotte Hollands. Adv. Theor. Math. Phys. 22 (2019) 1713 1822, arXiv:1710.04438
Teaching:
Summer 2019: MAT223H, Linear Algebra I
Spring 2019: MAT244H, Introduction to Ordinary Differential Equations
Fall 2018: MAT244H, Introduction to Ordinary Differential Equations
