Research Interests
Representation theory, with an emphasis on geometric, categorical, and diagrammatic techniques.
Some objects/topics that appear in my research:
- perverse sheaves
- Hecke category in its various incarnations (parity sheaves on Kac-Moody flag varieties, Soergel bimodules, moment graph sheaves, Elias-Williamson diagrammatic category)
- Koszul duality phenomenon for the Hecke category and related categories
- modular representation theory of reductive groups
- diagrammatic presentations of various categories in representation theory
- curved dg/A∞ (co)algebras and their (bi)(co)modules; bar-cobar Koszul duality
- link homology (Khovanov, Khovanov-Rozansky, HOMFLYPT)
Papers
Most of my papers are available on the arXiv e-print server. You can also find me on Google Scholar.
In preparation
- [9] Curved Hecke categories, II (with M. Hogancamp)
- [8] Curved Hecke categories, I (with M. Hogancamp)
- [7] A diagrammatic categorification of the 0-Hecke algebra (with M. Khovanov) (on hold)
Preprints
- [3] Free-monodromic mixed tilting sheaves on flag varieties (with P. N. Achar, S. Riche, and G. Williamson), submitted. [arXiv]
- [1] Mixed perverse sheaves on moment graphs, submitted. [arXiv]
Published / to appear
-
[6] Ext-enhanced monoidal Koszul duality for GL_2 (with M. Hogancamp)
To appear in RIMS Kôkyûroku Bessatsu, proceedings for RIMS symposium on 10th CFT Seminar: A Conference on Vertex Algebras and Related Topics. [arXiv] -
[5] On monoidal Koszul duality for the Hecke category
Proceedings of the XXII Latin American Algebra Colloquium, Rev. Colombiana Mat. 53 (2019), suppl., 195–222. [arXiv] [journal] [MR] -
[4] Koszul duality for Kac–Moody groups and characters of tilting modules (with P. N. Achar, S. Riche, and G. Williamson)
J. Amer. Math. Soc. 32 (2019), 261–310. [arXiv] [journal] [MR] -
[2] Modular Koszul duality for Soergel bimodules
Advances in Mathematics 428 (2023) 109158. [arXiv] [journal] [MR]
Not meant for publication
-
[a] Diagrammatics for Ext-enhanced Soergel bimodules in type A_1
Unfinished notes, available here.
Other mathematical writing
-
[0] Modular Koszul duality for Soergel bimodules, Ph.D. Thesis, Stanford University, 2017.
Available as papers [1] and [2]. [Stanford Libraries]
Book
- Introduction to Soergel bimodules (with B. Elias, U. Thiel, G. Williamson), RSME Springer Series vol. 5, 2020. (xxv+590 pages) [Springer Link] [Springer] [doi] [Amazon] [MR]
Some supplementary material can be found here.
Collaborators
Pramod N. Achar, Ben Elias, Matthew Hogancamp, Lars Thorge Jensen, Mikhail Khovanov, Simon Riche, Ulrich Thiel, Geordie Williamson
Some talk videos/notes
Aug 2020, Curved Hecke categories, I and II, in the QUACKS conference: [video] [slides]Apr 2020, Towards a triply-graded Koszul duality for the Hecke category, in the (online) GRT learning seminar winter 2020: Symplectic Duality at University of Toronto: [notes]
Nov 2017, Modular Koszul duality for Kac–Moody groups, at Perimeter Institute: [video]
Research done as an undergrad
Until early grad school, I thought I might want to do number theory. I wrote my undergraduate thesis at Princeton [-1] under Peter Sarnak. Even earlier, in summer 2009, I worked on [-2] for the fractal REU at UConn.
- [-1] A note on Riemann surfaces of large systole, J. Ramanujan Math. Soc. 28 (3): 359–377 (2013). [arXiv] [Princeton Library] [MR]
- [-2] Modified Hanoi Towers Groups and Limit Spaces (with G. Stadnyk and B. Steinhurst), International Journal of Algebra and Computation 21 (6): 867–887 (2011). [arXiv] [journal] [MR]