Research Summary


    Categorified Coulomb branches

  • Canonical bases for Coulomb branches of 4d N=2 gauge theories (arXiv) with H. Williams.
  • Ind-geometric stacks (arXiv) with H. Williams.
  • Tamely presented morphisms and coherent pullback (arXiv) with H. Williams.
  • Cluster theory of the coherent Satake category (arXiv) with H. Williams, J. Amer. Math. Soc. 32 (2019), 709--778.




    Categorical Heisenberg actions

  • Heisenberg categorification and Hilbert schemes (arXiv) with A. Licata, Duke Math Journal 161 (2012), no. 13, 2469--2547.
  • Vertex operators and 2-representations of quantum affine algebras (arXiv) with A. Licata.
  • Braid group actions via categorified Heisenberg complexes (arXiv) with A. Licata and J. Sussan, Compositio Math. 150 (2014), 105--142.
  • On a categorical Boson-Fermion correspondence (arXiv) with J. Sussan, C.M.P. 336 (2015), 649--669.
  • W-algebras from Heisenberg categories (arXiv) with A. Lauda, A. Licata and J. Sussan, J. Inst. Math. Jussieu (2016), 1--37.
  • The Elliptic Hall algebra and the deformed Khovanov Heisenberg category (arXiv) with A. Lauda, A. Licata, P. Samuelson and J. Sussan, Selecta Math. 24 (2018), 4041--4103.




    Equivalences and categorical sl(2) actions

  • Coherent sheaves and categorical sl(2) actions (arXiv) with J. Kamnitzer and A. Licata, Duke Math Journal 154 (2010) no. 1, 135--179.
  • Categorical geometric skew Howe duality (arXiv) with J. Kamnitzer and A. Licata, Inventiones Math. 180 (2010) no. 1, 111--159.
  • Derived equivalences for cotangent bundles of Grassmannians via categorical sl(2) actions (arXiv) with J. Kamnitzer and A. Licata, J. Reine Angew. Math. 675 (2013), 53--99.
  • Equivalences and stratified flops (arXiv) Compositio Math. 148 (2012) no. 1, 185--209.
  • Flops and about: a guide (arXiv) EMS Congress Reports (2011), no. 8, 61--101.
  • Associated graded of Hodge modules and categorical sl(2) actions (arXiv) with C. Dodd and J. Kamnitzer, Selecta Math. 27 (2021), no. 2, Paper No. 22, 55 pp.
  • Categorical geometric symmetric Howe duality (arXiv) with J. Kamnitzer, Selecta Math. 24 (2018), no. 2, 1593--1631.




    Quantum groups and higher representation theory

  • Braiding via geometric Lie algebra actions (arXiv) with J. Kamnitzer, Compositio Math. 148 (2012) no. 2, 464--506.
  • Coherent sheaves on quiver varieties and categorification (arXiv) with J. Kamnitzer and A. Licata, Math. Annalen 357 (2013) no. 3, 805--854.
  • Implicit structure in 2-representations of quantum groups (arXiv) with A. Lauda, Selecta Math 21 (2015), 201--244.
  • Loop realizations of quantum affine algebras (arXiv) with A. Licata, J. Math. Phys. 53 (2012), no. 12, 18pp.
  • Webs and quantum skew Howe duality (arXiv) with J. Kamnitzer and S. Morrison, Math. Annalen 360 (2014), 351--390.
  • Rigidity in higher representation theory (arXiv)
  • Quantum K-theoretic geometric Satake: the SL(n) case (arXiv) with J. Kamnitzer, Compositio Math. 154 (2018), no. 2, 275--327.
  • Exotic t-structures and actions of quantum affine algebras (arXiv) with C. Koppensteiner, J. Eur. Math. Soc. 22 (2020), no. 10, 3263--3304.
  • Curved Rickard complexes and link homologies (arXiv) with A. Lauda and J. Sussan, J. Reine Angew. Math. 769 (2020), 87--119.




    Knot homologies via derived categories of coherent sheaves

  • Knot Homology Via Derived Categories of Coherent Sheaves I, sl(2) case (arXiv) with J. Kamnitzer, Duke Math Journal 142 (2008) no.3, 511--588.
  • Knot Homology Via Derived Categories of Coherent Sheaves II, sl(m) case (arXiv) with J. Kamnitzer, Inventiones Math. 174 (2008) no. 1, 165--232.
  • Clasp technology to knot homology via the affine Grassmannian (arXiv) Math. Annalen 363 (2015), 1053--1115.
  • Knot homology via derived categories of coherent sheaves IV, coloured links (arXiv) with J. Kamnitzer, Quantum Topology 8 (2017) no. 2, 381--411.
  • Remarks on coloured triply graded link invariants (arXiv) Algebr. Geom. Topol. 17 (2017), no. 6, 3811-3836.




    The abelian monodromy extension property

  • The Abelian Monodromy Extension property for families of curves (arXiv) Math. Annalen 344 (2009) no. 3, 717--747.
  • The solvable monodromy extension property and varieties of log general type (arXiv) Clay Math. Proc.: A Celebration of Algebraic Geometry Clay Math. Proc. 18, (2013), 119--129.




    The geometric McKay correspondence in dimension three

  • A derived approach to geometric McKay correspondence in dimension three (arXiv) with T. Logvinenko, J. Reine Angew. Math. 636 (2009), 193--236.
  • Derived Reid's recipe for abelian subgroups of SL(3) (arXiv) with A. Craw and T. Logvinenko, J. Reine Angew. Math. 727 (2017), 1--48.