Ben Webster

Ben Webster's picture
Associate Faculty
University of Waterloo - Department of Pure Mathematics

Area of Research:
Mathematical Physics
Email: bwebster@perimeterinstitute.ca

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

Department of Pure Mathematics, University of Waterloo

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; now that I'm at Perimeter, I also hope to investigate further 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)
  • NSERC Discovery Grant "Diagrammatic and geometric techniques in representation theory." (2018-2023)
  • 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
  • NSF conference grant "Algebra, Combinatorics, and Representation Theory."
  • NSF CAREER grant: "Representation theory of symplectic singularities." (2012-2017)
  • NSA Young Investigator Grant

Recent Publications

  • Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra. to appear in New York Journal of Math. arXiv: 1609.05494
  • A categorical action on quantized quiver varieties. Mathematische Zeitschrift 292 (2019), no. 1-2, 611-639.
  • On graded presentations of Hecke algebras and their generalizations. to appear in Algebraic Combinatorics. arXiv: 1305.0599
  • Highest weights for truncated shifted Yangians and product monomial crystals (with J. Kamnitzer, P. Tingley, A. Weekes and O. Yacobi). to appear in Journal of Combinatorial Algebra. arXiv: 1511.09131
  • Weighted Khovanov-Lauda-Rouquier algebras. Doc. Math. 24 (2019), 209-250. arXiv: 1209.2463
  • 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
  • Categorified skew Howe duality and comparison of knot homologies. Advances in Mathematics 330 (2018), 876-945. arXiv: 1502.06011
  • 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
  • Geometry and categorification. "Categorification in Geometry, Topology and Physics," 1-22, Contemp. Math., 680, AMS, 2017. arXiv: 1602.05992
  • Comparison of canonical bases for Schur and universal enveloping algebras. Transformation Groups, 22(3), 869-883. arXiv: 1503.08734
  • 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
  • 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 II: category O and symplectic duality. (with T. Braden, A. Licata and N. Proudfoot). Astérisque No. 384, 75-179. arXiv: 1407.0964
  • Quantizations of conical symplectic resolutions I: local and global structure (with T. Braden and N. Proudfoot). Astérisque No. 384, 1-73. arXiv: 1208.3863
  • Heisenberg and Kac-Moody categorification (with Jonathan Brundan and Alistair Savage). arXiv: 1907.11988
  • 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 the definition of quantum Heisenberg category (with Jonathan Brundan and Alistair Savage). arXiv: 1812.04779
  • 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
  • A quantum Mirkovic-Vybornov isomorphism. (with Alex Weekes and Oded Yacobi) arXiv: 1706.03841
  • Koszul duality between Higgs and Coulomb categories $\mathcal{O}$. arXiv: 1611.06541
  • 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

  • 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
  • Berkeley (String-Math seminar): Mathematical hints of 3-d mirror symmetry
  • Luminy (Symplectic representation theory): Gelfand-Tsetlin theory and Coulomb branches
  • 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)
  • Notre Dame (Thematic Program in Geometric Representation Theory): Coulomb branches and applications (4 talk series)
  • Oberwolfach (Interactions between Algebraic Geometry and Noncommutative Algebra): Coulomb branches and KLR algebras
  • Macquarie (AustMS Representation Theory Keynote): Representation theory of symplectic singularities
  • Sydney (Future Directions in Representation Theory): Symplectic duality and KLR algebras
  • Northeastern (GASC Seminar): Coherent sheaves and Coulomb branches
  • U. Toronto (Geometric Representation Theory Seminar): The extended BFN category and shifted Yangians
  • 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: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