About

I work on the representation theory of complex reductive groups, approached through geometric and algebraic methods — in particular the geometric Satake correspondence, symplectic duality, and categorification.

I am the co-managing editor of Mathematische Zeitschrift. I am also the director of the Institut des sciences mathématiques.

Research

Symplectic duality and Coulomb branches

Symplectic duality relates conical symplectic singularities, motivated by Higgs/Coulomb branches in 3d supersymmetric gauge theories. With McBreen and Proudfoot, I formulated a quantum cohomology version of the Hikita conjecture for symplectic dual pairs; with Hilburn and Weekes, we developed a Springer theory built on the Braverman–Finkelberg–Nakajima construction.

2 papers
  1. The quantum Hikita Conjecturewith M. McBreen and N. Proudfoot · Advances in Mathematics
  2. BFN Springer Theorywith J. Hilburn and A. Weekes

Slices in the affine Grassmannian & truncated shifted Yangians

With Webster, Weekes, and Yacobi, I studied slices to spherical Schubert varieties in the affine Grassmannian, their natural Poisson structures, and conjectural quantizations via shifted Yangian quotients — conjectured to be symplectic dual to Nakajima quiver varieties. Follow-up work (with Muthiah, Finkelberg, Pham, Rybnikov, and Tingley) established the highest-weight theory, Coulomb branch description, comultiplication, and the symplectic duality itself as an equivalence of categories. Finally, together with Labelle, Leroux-Lapierre, Pinet, and Weekes, we used this theory to prove conjectures of Hernandez, Leclerc and others concerning the Grothendieck ring of category O for shifted Yangians.

10 papers
  1. Yangians and quantizations of slices in the affine Grassmannianwith B. Webster, A. Weekes, and O. Yacobi · Algebra & Number Theory, 2014
  2. Highest weights for truncated shifted Yangians and product monomial crystalswith P. Tingley, B. Webster, A. Weekes, and O. Yacobi · J. of Combinatorial Algebra, 2019
  3. On a reducedness conjecture for spherical Schubert varieties and slices in the affine Grassmannianwith D. Muthiah and A. Weekes · Transformation Groups, 2018
  4. Appendix to 3d N=4 quiver gauge theories and slices in the affine Grassmannianwith A. Braverman, M. Finkelberg, R. Kodera, H. Nakajima, B. Webster, and A. Weekes · Adv. Theoretical & Math. Physics, 2019
  5. Comultiplication for shifted Yangians and quantum open Toda latticewith M. Finkelberg, K. Pham, L. Rybnikov, and A. Weekes · Advances in Mathematics, 2018
  6. Reducedness of affine Grassmannian slices in type Awith D. Muthiah, A. Weekes, and O. Yacobi · Proceedings of the AMS, 2018
  7. On Category O for affine Grassmannian slices and categorified tensor productswith P. Tingley, B. Webster, A. Weekes, and O. Yacobi · Proc. London Math. Soc., 2018
  8. Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangianswith K. Pham and A. Weekes · Advances in Mathematics, 2022
  9. Lie algebra actions on module categories for truncated shifted Yangianswith B. Webster, A. Weekes, and O. Yacobi · Forum of Mathematics, Sigma, 2024
  10. Category O for truncated shifted Yangians and the bi-infinite Bott-Samelson varietywith A. Labelle, A. Leroux-Lapierre, T. Pinet, and A. Weekes · 2026

Webs, buildings, components of Satake fibres, and skew Howe duality

Webs are planar graphs that give invariant vectors inside tensor products of representations, forming a combinatorial "spider" model of representation categories. With Fontaine and Kuperberg, I linked webs to components of Satake fibres; with Cautis and Morrison, I used skew Howe duality to give an explicit SL(n) spider presentation. Later work with Fontaine and, separately, solo, extended this to a full combinatorial geometric Satake equivalence and its quantum K-theoretic and symmetric categorical versions.

6 papers
  1. Buildings, spiders, and geometric Satakewith B. Fontaine and G. Kuperberg · Compositio Mathematica, 2013
  2. Webs and quantum skew Howe dualitywith S. Cautis and S. Morrison · Mathematische Annalen, 2014
  3. Cyclic sieving, rotation, and geometric representation theorywith B. Fontaine · Selecta Mathematica, 2014
  4. A combinatorial geometric Satake equivalenceAdvances in Mathematics, 2016
  5. Quantum K-theoretic geometric Satakewith S. Cautis · Compositio Mathematica, 2018
  6. Categorical geometric symmetric Howe dualitywith S. Cautis · Selecta Mathematica, 2018

Components of quiver varieties and (affine) MV polytopes

With Baumann, I applied MV polytope theory to components of Lusztig's nilpotent varieties, introducing reflection functors for preprojective algebra modules. With Baumann and Tingley, this was generalized to affine Lie algebras, giving affine MV polytopes determined by rank-2 conditions; further joint work with Dunlap, Gaussent, and Knutson made the rank-2 case explicit and related the MV basis to the dual semicanonical basis.

7 papers
  1. Preprojective algebras and MV polytopeswith P. Baumann · Representation Theory, 2012
  2. Modules with 1-dimensional socle and components of Lusztig quiver varieties in type Awith C. Sadanand · Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, 2011
  3. Affine Mirković–Vilonen polytopeswith P. Baumann and P. Tingley · Publications IHÉS, 2014
  4. Rank 2 affine MV polytopeswith P. Baumann, T. Dunlap, and P. Tingley · Representation Theory, 2013
  5. Réflexions dans un cristalwith P. Baumann and S. Gaussent · C. R. Acad. Sci. Paris, 2012
  6. The Mirković–Vilonen basis and Duistermaat–Heckman measureswith P. Baumann and A. Knutson · Acta Mathematica, 2021
  7. Heaps, crystals, and preprojective algebra moduleswith A. Dranowski, B. Elek, and C. Morton-Ferguson · Selecta Mathematica, 2024

Geometric categorical g actions & derived equivalences

With Cautis and Licata, I developed "geometric categorical sl(2) actions," showing they yield strong categorical sl(2) actions and, via an extension of Chuang–Rouquier, equivalences of derived categories of coherent sheaves — applied to cotangent bundles of Grassmannians and affine Schubert varieties. This was extended to general Kac–Moody g (with Cautis) and to quiver-variety and D-module settings (with Cautis, Licata, and Dodd).

6 papers
  1. Categorical geometric skew Howe dualitywith S. Cautis and A. Licata · Inventiones Mathematicae, 2010
  2. Coherent sheaves and categorical sl(2) actionswith S. Cautis and A. Licata · Duke Math. Journal, 2010
  3. Derived equivalences for cotangent bundles to Grassmannians via categorical sl(2) actionswith S. Cautis and A. Licata · J. Reine Angew. Math., 2013
  4. Braid groups and geometric categorical g actionswith S. Cautis · Compositio Mathematica, 2012
  5. Coherent sheaves on quiver varieties and categorificationwith S. Cautis and A. Licata · Mathematische Annalen, 2013
  6. Associated graded of Hodge modules and categorical sl(2) actionswith S. Cautis and C. Dodd · Selecta Mathematica, 2021

Knot homology via derived categories of coherent sheaves

Extending Khovanov's categorification of quantum knot invariants, Cautis and I built a program using derived categories of coherent sheaves on varieties from the geometric Langlands program — recovering Khovanov homology in the sl(2) case and conjecturally Khovanov–Rozansky homology for sl(m), with later work on the symplectic and coloured-link versions.

4 papers
  1. Knot homology via derived categories of coherent sheaves I, sl(2) casewith S. Cautis · Duke Math. Journal, 2008
  2. Knot homology via derived categories of coherent sheaves II, sl(m) casewith S. Cautis · Inventiones Mathematicae, 2008
  3. The Beilinson–Drinfeld Grassmannian and symplectic knot homologyClay Mathematics Proceedings, 2011
  4. Knot homology via derived categories of coherent sheaves IV, coloured linksQuantum Topology, 2017

Mirković–Vilonen cycles and polytopes

By the geometric Satake isomorphism, MV cycles in the affine Grassmannian give a basis for representations of complex reductive groups. My thesis gave an explicit description of MV cycles and polytopes, linking them combinatorially to Berenstein–Zelevinsky data and the canonical basis, and constructed a bijection between convolution-morphism fibre components and hives.

5 papers
  1. Mirković–Vilonen cycles and polytopesAnnals of Mathematics, 2010
  2. The crystal structure on the set of Mirković–Vilonen polytopesAdvances in Mathematics, 2007
  3. Hives and the fibres of the convolution morphismSelecta Mathematics, 2007
  4. Computing fusion products of MV cycles using the Mirković–Vybornov isomorphismwith R. Bai and A. Dranowski · J. of Combinatorial Algebra, 2023
  5. Fixed points of reverse Hessenberg convolution varieties2024

Crystals, coboundary categories & the moduli spaces of real curves

With Henriques, I related the octahedron recurrence to gl(n) crystals and defined a commutor making the crystal category a coboundary category, on which the cactus group acts — connecting to the topology of the moduli space of genus-0 real curves (with Etingof and Rains). With Tingley, I gave an alternate definition of the commutor via Kashiwara's involution and its identification with Drinfeld's unitarized R-matrix; with Halacheva, Rybnikov, and Weekes, the cactus group action was identified with monodromy of Bethe eigenvectors. More recently, with Ilin, Li, Przytycki, and Rybnikov, we studied of the moduli space of cactus flower curves whose fundamental group is the virtual cactus group. Then we used this cactus flower space to study inhomogeneous and trigonometric Gaudin models.

9 papers
  1. Crystals and coboundary categorieswith A. Henriques · Duke Math. Journal
  2. The octahedron recurrence and gl(n) crystalswith A. Henriques · Advances in Mathematics, 2006
  3. The cohomology ring of the moduli space of stable curves of genus 0 with marked pointswith P. Etingof, A. Henriques, and E. Rains · Annals of Mathematics, 2010
  4. A definition of the crystal commutor using Kashiwara's involutionwith P. Tingley · J. of Algebraic Combinatorics, 2009
  5. The crystal commutor and Drinfeld's unitarized R-matrixwith P. Tingley · J. of Algebraic Combinatorics, 2009
  6. Crystals and monodromy of Bethe vectorswith I. Halacheva, L. Rybnikov, and A. Weekes · Duke Math. Journal, 2020
  7. The moduli space of cactus flower curves and the virtual cactus groupwith A. Ilin, Y. Li, P. Przytycki, and L. Rybnikov · 2023
  8. Gaudin models and moduli space of flower curveswith A. Ilin and L. Rybnikov · Advances in Mathematics, 2025
  9. Cactus flower spaces and monodromy of Bethe vectorswith L. Rybnikov · 2025

Survey articles

Editorial

Managing editor, Mathematische Zeitschrift. Submit papers by email.

Talks

Recorded talks and lecture notes.

Teaching

Programs and workshops organized

Upcoming — SLMath (Simons Laufer Mathematical Sciences Institute) program:
Representation Theory Under the Influence of Quantum Field Theory, Berkeley, CA Aug 17 – Dec 18, 2026
Co-organized with David Ben-Zvi, Tudor Dimofte, Iva Halacheva, Pavel Safronov, and Peng Shan.

PhD students & postdocs

PhD students

  • Alexis Leroux-Lapierre expected PhD 2026
  • Jeremy Peters PhD 2026
  • Kathlyn Dykes PhD 2022now Researcher, Tutte Institute
  • Yuguang (Roger) Bai PhD 2021now Security Engineer, CertiK
  • Khoa Pham PhD 2020now Senior Data Scientist, RBC
  • Anne Dranowski PhD 2020now Researcher, Charles Schwab
  • Bulent (Ozgür) Esentepe PhD 2019now Postdoctoral Fellow, Graz University
  • Chia-Cheng Liu PhD 2019now Applied Scientist, Amazon Canada
  • Iva Halacheva PhD 2016now Assistant Professor, Northeastern University
  • Alex Weekes PhD 2016now Associate Professor, University of Sherbrooke
  • Daniel Rowe PhD 2015now Associate Professor, Northern Michigan University
  • Bradley Hannigan-Daley PhD 2014now Principal Data Scientist, NextRoll
  • Stephen Morgan PhD 2014now Senior Data Analyst, NSW Department of Education
  • Bruce Fontaine PhD 2012now Staff Software Engineer, Databricks

Postdocs

  • Jonathan Boretsky 2025–present
  • Théo Pinet 2024–2026now Zelevinsky Postdoctoral Fellow, Northeastern University
  • Artem Kalmykov 2024–2026now Postdoctoral Fellow, IST Austria
  • Yu Li 2022–2025now Postdoctoral Research Associate, University of Notre Dame
  • Aleksei Ilin 2021–2023now Postdoctoral Fellow, IST Austria
  • Oscar Kivinen 2020–2021now Professor, Aalto University
  • Changjian Su 2018–2021now Assistant Professor, Tsinghua University
  • Kostiantyn Tolmachov 2018–2021now Postdoctoral Fellow, University of Hamburg
  • Balazs Elek 2018–2020
  • Michael McBreen 2018–2019now Assistant Professor, Chinese University of Hong Kong
  • Alexander Shapiro 2016–2018now Reader, University of Edinburgh
  • Dinakar Muthiah 2013–2015now Senior Lecturer, University of Glasgow
  • Peter Samuelson 2012–2015now Associate Professor, UC Riverside
  • Karene Chu 2012–2013now Assistant Director of Education, MIT
  • Christopher Dodd 2011–2014now Assistant Professor, University of Illinois
  • Jaimal Thind 2011–2013now Assistant Professor, University of Toronto
  • Oded Yacobi 2010–2013now Associate Professor, Sydney University

Archive