# Ben Webster

University of Waterloo

Areas of research:

### Overview

My research is on connections between representation theory, mathematical physics, geometry and topology, including knot homology, the geometry of symplectic singularities, and categorification. I moved to Waterloo and Perimeter in 2017. In the past, I've been an Assistant Professor at the University of Virginia , Northeastern University and the University of Oregon , a C.L.E. Moore Instructor at MIT , and a postdoctoral fellow at the Institute for Advanced Study . I received my Ph.D. in 2007 from UC Berkeley, under the supervision of Nicolai Reshetikhin, and spent the fall of 2006 at the Center for the Topology and Quantization of Moduli Spaces in Aarhus, Denmark. In Spring 2014, I was a Junior Chair at Université Denis-Diderot (Paris 7) with support from Fondation Sciences Mathématiques de Paris. Before that, I studied at Simon's Rock College and the Budapest Semesters in Mathematics Program . Before I moved to Canada, my research was supported by the NSF under a CAREER award , and graduate and postdoctoral fellowships, and by a Sloan Research Fellowship.

### Affiliations

If you are interested in working with me as a PhD student, please send me an email at [email protected] Perimeter Institute is committed to diversity within its community and I welcome applications from underrepresented groups. Information about working with me as a graduate student can be found on my webpage here: https://uwaterloo.ca/scholar/b2webste/info-students

### Research Interests

My research is on the interface between representation theory, algebraic geometry and mathematical physics. Hyperkahler cones appear naturally in physics as branches of moduli spaces of vacua for N=4 d=3 field theories; on the other hand, they are of mathematical interest because of their symplectic resolutions of singularities, and the fact that they can be quantized to produce interesting non-commutative algebras such as universal enveloping algebras and Cherednik algebras. I'm interested in the representation theory of algebras that appear this way, and how it relates to fields like topology and combinatorics; since coming to Perimeter, one of my focuses has been how understanding these varieties in the context of field theory gives us insight into the mathematics.

### Positions Held

- 2017-present Associate Professor, University of Waterloo
- 2016-2017 Associate Professor, University of Virginia.
- 2013-2016 Assistant Professor, University of Virginia.
- 2011-2013 Assistant Professor, Northeastern University.
- 2010-2011 Assistant Professor, University of Oregon.
- 2008-2010 C.L.E. Moore Instructor and NSF Postdoctoral Fellow, M.I.T.
- 2007-2008 Member and NSF Postdoctoral Fellow, Institute for Advanced Study.

### Awards

- Golden Jubilee Research Excellence Award (UW Faculty of Math)
- Cory Family Teaching Award (at University of Virginia; recognizes excellence in lower division teaching; one awarded for all STEM fields at UVA per year)
- International Researcher Collaboration Award from Sydney University
- Sloan Research Fellowship

### Recent Publications

- Quantum Frobenius Heisenberg categorification (with Jonathan Brundan and Alistair Savage). J. Pure Appl. Algebra 226 (2022), no. 1, Paper No. 106792, 50 pp. arXiv: 2009.06690
- Foundations of Frobenius Heisenberg categories (with Jonathan Brundan and Alistair Savage). J. Algebra 578 (2021), 115-185. arXiv: 2007.01642
- Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra. to appear in New York Journal of Math. arXiv: 1609.05494
- Heisenberg and Kac-Moody categorification (with Jonathan Brundan and Alistair Savage). Selecta Math. (N.S.) 26 (2020), no. 5, 74. arXiv: 1907.11988
- On graded presentations of Hecke algebras and their generalizations. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 1-38.
- A quantum Mirkovic-Vybornov isomorphism. (with A. Weekes and O. Yacobi) Representation Theory 24 (2020), 38-84.
- On the definition of quantum Heisenberg category (with Jonathan Brundan and Alistair Savage). Algebra and Number Theory 14 (2020), no. 2, 275-321.
- Highest weights for truncated shifted Yangians and product monomial crystals (with J. Kamnitzer, P. Tingley, A. Weekes and O. Yacobi). J. Comb. Algebra 3 (2019), no. 3, 237-303. arXiv: 1511.09131
- Weighted Khovanov-Lauda-Rouquier algebras. Doc. Math. 24 (2019), 209-250. arXiv: 1209.2463
- A categorical action on quantized quiver varieties. Mathematische Zeitschrift 292 (2019), no. 1-2, 611-639.
- On category O for affine Grassmannian slices and categorified tensor products. (with J. Kamnitzer, P. Tingley, A. Weekes and O. Yacobi). Proc. Lond. Math. Soc. (3) 119 (2019), no. 5, 1179-1233. arXiv: 1806.07519
- Categorification of quantum symmetric pairs I. (with H. Bao, P. Shan and W. Wang). Quantum Topology, 9, Issue 4 (2018) pp. 643-714. arXiv: 1605.03780
- Current algebras and categorified quantum groups (with A. Beliakova, K. Habiro and A. Lauda). Journal of the London Mathematical Society, 95, 248-276 arXiv: 1412.1417
- Quantizations of conical symplectic resolutions I: local and global structure (with T. Braden and N. Proudfoot). Astérisque No. 384, 1-73. arXiv: 1208.3863
- Categorified skew Howe duality and comparison of knot homologies. Advances in Mathematics 330 (2018), 876-945. arXiv: 1502.06011
- Quantizations of conical symplectic resolutions II: category O and symplectic duality. (with T. Braden, A. Licata and N. Proudfoot). Astérisque No. 384, 75-179. arXiv: 1407.0964
- Comparison of canonical bases for Schur and universal enveloping algebras. Transformation Groups, 22(3), 869-883. arXiv: 1503.08734
- Geometry and categorification. "Categorification in Geometry, Topology and Physics," 1-22, Contemp. Math., 680, AMS, 2017. arXiv: 1602.05992
- A geometric construction of colored HOMFLYPT homology (with G. Williamson). Geometry & Topology 21-5 (2017), 2557--2600. arXiv: 0905.0486
- On generalized category O for a quiver variety. Mathematische Annalen 368 (2017), no. 1-2, 483-536. arXiv: 1409.4461
- Rouquier's conjecture and diagrammatic algebra. Forum Math. Sigma 5 (2017), e27, 71 pp. arXiv: 1306.0074
- Knot invariants and higher representation theory. Memoirs of the American Mathematical Society 250, no. 1191, pp. 133. arXiv: 1309.3796
- Lie algebra actions on module categories for truncated shifted Yangians. (with J. Kamnitzer, A. Weekes and O. Yacobi). arXiv: 2203.12429
- Gelfand-Tsetlin modules: canonicity and calculations. arXiv: 2011.06029
- Three perspectives on categorical symmetric Howe duality (with an appendix by J. Guan and B. Webster). arXiv: 2001.07584
- Rational Cherednik algebras of $G(\ell,p,n)$ from the Coulomb perspective (with E. Lepage). arXiv: 1912.00046
- Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems. arXiv: 1905.04623
- Gelfand-Tsetlin modules in the Coulomb context. arXiv: 1904.05415
- Homological mirror symmetry for hypertoric varieties II (with Benjamin Gammage and Michael McBreen). arXiv: 1903.07928
- The degenerate Heisenberg category and its Grothendieck ring (with Jonathan Brundan and Alistair Savage). arXiv: 1812.03255
- On category O for affine Grassmannian slices and categorified tensor products. (with J. Kamnitzer, P. Tingley, A. Weekes and O. Yacobi). arXiv: 1806.07519
- Homological Mirror Symmetry for Hypertoric Varieties I. (with Michael McBreen). arXiv: 1806.07519
- Koszul duality between Higgs and Coulomb categories $\mathcal{O}$. arXiv: 1611.06541
- As editor: Representation theory, mathematical physics, and integrable systems. In honor of Nicolai Reshetikhin. Edited by Anton Alekseev, Edward Frenkel, Marc Rosso, Ben Webster and Milen Yakimov. Progress in Mathematics, 340. Birkhäuser/Springer, Cham, [2021], ©2021. xviii+643 pp.
- Geometry and categorification. "Categorification in Geometry, Topology and Physics," 1-22, Contemp. Math., 680, AMS, 2017. arXiv: 1602.05992
- Tensor product algebras, Grassmannians and Khovanov homology. "Physics and mathematics of link homology," 23-58, Contemp. Math., 680, AMS, 2016. arXiv: 1312.7357

### Seminars

- Northeastern (Canada-Mexico-USA Conference in Representation Theory, Noncommutative Algebra, and Categorification): Coulomb branches and KLRW algebras
- McMaster (Colloquium): Representation theory and a little bit of quantum field theory
- Northeastern (GASC Seminar): Representation theory and a little bit of quantum field theory
- Boston University (geometry seminar): Mirror symmetry for hypertoric varieties
- Queens (Colloquium): Representation theory and a little bit of quantum field theory
- Zoom (Geometry and Physics Seminar, Zhejiang University): 3d mirror symmetry and symplectic singularities
- Zoom (Berkeley String-Math Seminar): Noncommutative resolutions of Coulomb branches
- Zoom (NSF-FRG workshop on Categorical braid group actions and categorical representation theory): Knot homology from coherent sheaves on Coulomb branches
- Zoom (BIRS Workshop on Perspectives on Knot Homology): Knot homology from coherent sheaves on Coulomb branches
- Zoom (Yale Geometry, Symmetry and Physics Seminar): Knot homology and coherent sheaves on Coulomb branches
- Zoom (Northwestern Geometry and Physics Seminar): Hypertoric mirror symmetry
- Zoom (Learning Seminar on Categorification): Coulomb branches and cylindrical KLRW algebras
- Zoom (Quantum groups, Representation theory, Superalgebras, and Tensor categories): Tensor products and categorification
- Zoom (QUACKS): Howe to translate Gelfand-Tsetlin
- Zoom (Western Hemisphere Colloquium on Geometry and Physics): 3d mirror symmetry and its discontents
- Zoom (Gone Fishin'): Coulomb branches in math and physics
- Zoom (UC Davis Coulomb seminar): Tilting bundles from positive characteristic
- Fields Insitute (GRT Seminar): S(ymplectic) duality: where do we stand?
- Fields Institute (Workshop on Higher Structures in Geometry and Physics): Line defects and tilting bundles
- KSU [remote] (Kansas State M-Seminar): Tilting bundles and 3-dimensional field theory
- CRM Montreal (Workshop on quiver varieties and representation theory): Quiver varieties as Coulomb branches
- Warsaw (Symposium on Integrable Systems): Representation theory, topology and quantum field theory
- Luminy (Symplectic representation theory): Gelfand-Tsetlin theory and Coulomb branches
- Berkeley (String-Math seminar): Mathematical hints of 3-d mirror symmetry
- VUB (Aspects of Higher Representation Theory): Heisenberg v. Kac-Moody
- Oberwolfach (Enveloping Algebras and Geometric Representation Theory): The classification of Gelfand-Tsetlin modules and the Braverman-Finkelberg-Nakajima construction
- U. Virginia (Representation theory, Combinatorics, and Geometry): Coherent sheaves on Hilbert schemes through the Coulomb lens
- SUNY Buffalo (Algebra Seminar): Representation theory of symplectic singularities
- ECNU, Shanghai (Workshop on Lie Theory and Representation Theory): Coulomb branches and Cherednik algebras
- ECNU, Shanghai (Summer School on Lie Theory and Representation Theory): Categorification: Heisenberg and Kac-Moody (8 talk series)
- Oberwolfach (Interactions between Algebraic Geometry and Noncommutative Algebra): Coulomb branches and KLR algebras
- Notre Dame (Thematic Program in Geometric Representation Theory): Coulomb branches and applications (4 talk series)
- Sydney (Future Directions in Representation Theory): Symplectic duality and KLR algebras
- Macquarie (AustMS Representation Theory Keynote): Representation theory of symplectic singularities
- U. Toronto (Geometric Representation Theory Seminar): The extended BFN category and shifted Yangians
- Northeastern (GASC Seminar): Coherent sheaves and Coulomb branches
- U. Buffalo (AMS Sectional): Representation theory and the Coulomb branch
- U. of Oregon (WARTHOG): Symplectic duality (the abelian case) (five day workshop)
- Newton Institute (Quantum topology and categorified representation theory): Representation theory and the Coulomb branch
- GWU (Knots in Washington): Knot invariants via quantizations of Hecke modifications
- UCLA (Gauge Theory and Categorification): Knot invariants via quantizations of Hecke modifications
- Rice (Colloquium): Representation theory of symplectic singularities
- Univ. of Texas (Colloquium): Representation theory of symplectic singularities
- Waterloo (Colloquium): Representation theory of symplectic singularities
- UNC (Symplectic Varieties and Geometric Representation Theory): Representation theory and the Coulomb branch
- PIRSA:22060082, Discussion on knot homology, 2022-06-27, QFT for Mathematicians 2022
- PIRSA:22060078, Welcome and Opening Remarks, 2022-06-27, QFT for Mathematicians 2022
- PIRSA:19090115, Tilting bundles and 3-dimensional field theory, 2019-09-26, Mathematical Physics
- PIRSA:19060004, Welcome and Opening Remarks, 2019-06-17, QFT for Mathematicians
- PIRSA:19040120, Mathematical hints of 3-d mirror symmetry, 2019-04-15, Mathematical Physics
- PIRSA:17090065, An extended category for the BFN construction, 2017-09-25, Mathematical Physics
- PIRSA:17030051, Three-dimensional mirror symmetry: a mathematical perspective, 2017-03-02, Colloquium