Research
Most of my work is in the general areas of quantum algebra and mathematical physics. The mathematical topics I think the most about are tensor categories, operator algebras, and higher algebra. Recently I have been very interested in using all of these tools to rigorously study quantum spin systems (which I think of as discrete quantum field theories) and topological order. I sometimes work on purely mathematical problems inspired by physics, but I also enjoy collaborating with physicists to develop and apply abstract mathematical tools to concrete physical problems.
-My research is supported by NSF Grant DMS-2100531 (previously 1901082).
-Here is a link to my Google Scholar page.
Publications and Preprints:
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Composing topological domain walls and anyon mobility (with Peter Huston, Fiona Burnell, and David Penneys). arXiv:2208.14018
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K-theoretic classification of inductive limit actions of fusion categories on AF-algebras. (with Quan Chen and Roberto Hernández Palomares). arXiv:2207.11854
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Equivariant fusion subcategories. (with César Galindo). arXiv:2111.09116
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A categorical Connes' 𝛘(M). (with Quan Chen and David Penneys). arXiv:2111.06378
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Q-system completion for C* 2-categories (with Quan Chen, Roberto Hernández Palomares and David Penneys). J. Funct. Anal. arxiv:2105.12010
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Remarks on anomalous symmetries of C*-algebras. Comm. Math. Phys. arxiv:2011.13898
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A 3-categorical perspective on G-crossed braided categories (with David Penneys and David Reutter). To appear in J. London Math. Soc. arxiv:2009.00405
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Triangle presentations and tilting modules for SL2k+1. J. Comb. Algebra. (2021). arxiv:2005.07172
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Extension theory for braided-enriched fusion categories (with Scott Morrison, David Penneys, and Julia Plavnik). Int. Math. Res. Not. arxiv:1910.03178
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Computing fusion rules for spherical G-extensions of fusion categories (with Marcel Bischoff). Selecta Math. arxiv:1909.02816
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Rank-finiteness for G-crossed braided fusion categories (with Scott Morrison, Eric C. Rowell, and Dmitri Nikshych). Transform. Groups. (2020). arxiv:1902.06165
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Spontaneous symmetry breaking from anyon condensation (with Marcel Bischoff, Yuan-Ming Lu, and David Penneys). J. High Energ. Phys. (2019). arxiv:1811.00434
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Fusing binary interface defects in topological phases: the Vec(Z/pZ) case (with Daniel Barter and Jacob C. Bridgeman). To appear in J. Math. Phys. (2019). arxiv: 1810.09469
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Domain walls in topological phases and the Brauer-Picard ring for Vec(Z/pZ) (with Daniel Barter and Jacob C. Bridgeman). Comm. Math. Phys. (2019). arxiv: 1806.01279
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Vanishing of categorical obstructions for permutation orbifolds (with Terry Gannon). Comm. Math. Phys. (2019). arXiv:1804.08343
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Free oriented extensions of subfactor planar algebras (with Shamindra Kumar Ghosh and Madhav B. Reddy). Internat. J. of Math. (2018). arXiv:1805.08971
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Annular representations of free product categories (with Shamindra Kumar Ghosh and Madhav B. Reddy). J. Noncommut. Geom. (2020). arxiv:1803.06817
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Fusion rules for Z/2Z permutation gauging (with Cain Edie-Michell and Julia Plavnik). J. Math Phys. (2019). arXiv:1804.01657
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Q-systems and compact W*-algebra objects (with David Penneys), to appear in a volume of Contemporary Mathematics series. arXiv:1707.02155
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Realizations of algebra objects and discrete subfactors (with David Penneys). Adv. Math. (2019). arXiv:1704.02035
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Operator algebras in rigid C*-tensor categories (with David Penneys). Comm. Math. Phys. (2017). arxiv: 1611.04620
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Classification of Thurston-relation subfactor planar algebras (with Zhengwei Liu and Yunxiang Ren). Quantum Topol. (2019). arxiv:1606.00779
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Eigenvalues of rotations and braids in spherical fusion categories (with Daniel Barter and Henry Tucker). J. Algebra (2018). arxiv:1611.00071
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Quantum G2 categories have property (T). Internat. J. Math. (2016). arXiv: 1504.08338
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Annular representation theory for rigid C*-tensor categories (with Shamindra Kumar Ghosh). J. Funct. Anal. (2016). arXiv:1502.06543
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Discrete homotopy theory and critical values of metric space (with Jim Conant, Victoria Cur- nutte, Conrad Plaut, Kristen Pueschel, Maria Lusby, and Jay Wilkins). Fund. Math. (2014). arXiv:1205.2925
Recorded Talks:
Here are links to recordings of talks I've given at some conferences:
1). K-theoretic classification of inductive limit actions of fusion categories on AF-algebras, HIM Bonn, DE (video)
2). A survey of G-crossed braided fusion categories, Simons Center for Geometry and Physics, Stony Brook, NY (video)
2). Spontaneous symmetry breaking from anyon condensation, Banff International Research Station, CA (video)
3). Operator Algebras in Rigid C*-tensor Categories, Isaac Newton Institute, Cambridge, UK (video)