Arun Debray

a.debray@gmail.com
 

Hello! I am a mathematician, working in algebraic topology with an eye towards quantum field theory and condensed-matter physics. Since Fall 2024, I have been an Assistant Professor at the University of Kentucky. From 2021–2024, I was a postdoc at Purdue University. Before that, I was a graduate student at UT Austin, advised by Dan Freed. Before grad school, I was an undergrad at Stanford University, graduating in 2015.

I find mathematics so fascinating partly because different subfields and concepts are very inter­connected, reinforcing and influencing each other.
“Good mathematicians see analogies between theorems or theories; the very best ones see analogies between analogies.” – Stefan Banach

Quick Links

Papers and Preprints

  1. Weak topological phases in the presence of interactions, joint with Omar Antolín Camarena, Cameron Krulewski, Natalia Pacheco-Tallaj, Daniel Sheinbaum, and Luuk Stehouwer. 2024. arXiv:2410.10031.

  2. Smith homomorphisms and Spinh structures, joint with Cameron Krulewski. Accepted for publication, Proc. Amer. Math Soc, 2024. arXiv:2406.08237.

  3. The Smith fiber sequence and invertible field theories, joint with Sanath K. Devalapurkar, Cameron Krulewski, Yu Leon Liu, Natalia Pacheco-Tallaj, and Ryan Thorngren. 2024. arXiv:2405.04649.

  4. Differential cohomology (encyclopedia article). In Encyclopedia of Mathematical Physics, 2nd (ed. Richard Szabo and Martin Bojowald), Academic Press, 2025. arXiv:2312.14338.

  5. Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders, joint with Weicheng Ye and Matthew Yu. 2023. arXiv:2312.13341.

  6. Global anomalies & bordism of non-supersymmetric strings, joint with Ivano Basile, Matilda Delgado, and Miguel Montero. JHEP, volume 2024, number 92. 2024. arXiv:2310.06895.

  7. A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases, joint with Sanath K. Devalapurkar, Cameron Krulewski, Yu Leon Liu, Natalia Pacheco-Tallaj, and Ryan Thorngren. 2023. arXiv:2309.16749.

  8. Adams spectral sequences for non-vector-bundle Thom spectra, joint with Matthew Yu. 2023. arXiv:2305.01678.

  9. Bordism for the 2-group symmetries of the heterotic and CHL strings. In Higher Structures in Geometry, Topology and Physics, Contemp. Math. volume 802, pp. 227–297, 2024. arXiv:2304.14764.

  10. The Chronicles of IIBordia: Dualities, Bordisms, and the Swampland, joint with Markus Dierigl, Jonathan J. Heckman, and Miguel Montero. Accepted for publication, Advances in Theoretical and Mathematical Physics. 2023. arXiv:2302.00007.

  11. What bordism-theoretic anomaly cancellation can do for U, joint with Matthew Yu. Comm. Math. Phys. volume 405, number 154, 2024. arXiv:2210.04911.

  12. Constructing the Virasoro groups using differential cohomology, joint with Yu Leon Liu and Christoph Weis. Int. Math. Res. Not., volume 2023, number 21, pp. 18537–18574. 2023. arXiv:2112.10837.

  13. The anomaly that was not meant IIB, joint with Markus Dierigl, Jonathan J. Heckman, and Miguel Montero. Fortschr. Phys. volume 70, issue 1. 2022. arXiv:2107.14227.

  14. Stable diffeomorphism classification of some unorientable 4-manifolds. Bull. London Math. Soc., volume 54, issue 6, pp. 2219–2231. 2022. arXiv:2102.03965.

  15. Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle, 2021. arXiv:2102.02941.

  16. The low-energy TQFT of the generalized double semion model. Comm. Math. Phys. volume 375, issue 2, pp. 1079–1115. 2020. arXiv:1811.03583.

  17. The Arf-Brown TQFT of Pin− Surfaces, joint with Sam Gunningham. In Topology and Quantum Theory in Interaction, Contemp. Math. volume 718, pp. 49–87. 2018. arXiv:1803.11183.

Books

  1. Differential Cohomology: Categories, Characteristic Classes, and Connections, edited jointly with Araminta Amabel and Peter J. Haine. 2021. arXiv:2109.12250.

Older Stuff

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These projects are a mix of math and computer science, and include a few side projects. A few of these will migrate to my GitHub soon.

  • Dropbox API Explorer, Summer 2015: my summer internship project at Dropbox. The API Explorer is a tool to learn the Dropbox v2 API and quickly test examples, and is generated from the API spec. The project uses React and was written in TypeScript. See also the associated blog post.

  • Modular Representation Theory and the CDE Triangle, 2014-15: my senior thesis, written under the direction of Akshay Venkatesh as part of the honors major in mathematics. I discussed modular representation theory and a collection of results on modular representations encapsulated in a diagram called the CDE triangle, presenting the background theory of modular representations needed to state and prove these results and using these results in several explicit examples.

  • AT&T Foundry, Summer 2014: In this internship, I applied techniques from abstract algebra and statistics to problems within network theory, including understanding conditions on a network for a linear code to exist and implementing probabilistic power conservation techniques within sensor networks.

  • Sleep Data Tracker, Summer 2014. In order to better understand my sleep schedule (to the degree that “sleep schedule” isn't an oxymoron), I wrote a script to track my sleep habits and collate statistics on them. The statistics side was written in Haskell, and the data plotting in Python (using Matplotlib for the plotting). The results are automatically written up; here is my data from over the summer. At some point, I intend to clean this program up to the point where other people can use it.

  • Astronomical Implications of Machine Learning, Fall 2013: Working with Raymond Wu, we used supervised learning to develop a classifier for stellar lightcurves to detect whether they indicated the presence of exosolar planets. We achieved 82% classification accuracy. This project was for Stanford's CS 229 (machine learning) class.

  • Quick Error Detection in LLVM, Summer 2013: I worked in a team, under direction of David Lin and Sundaram Anthanarayanan, to implement and improve Quick Error Detection, a technique to insert error-checking routines for hardware errors into software with the LLVM compiler infrastructure. This project was for Stanford's Computer Science Undergraduate Research Internship (CURIS) research program.

  • Class Numbers of Binary Quadratic Forms, Summer 2012: As part of this program, I did research on class numbers of binary quadratic forms, investigating their maxima and the structure of the class groups. The program included weekly presentations and writing a concluding paper. This project was part of the Stanford Research Institute in Mathematics (SURIM).

Other