# Overview

I've been inspired by two important geometric constructions in representation theory: *affine Grassmannians* and the geometric Satake equivalence on the one hand, and the theory of *Nakajima quiver varieties* on the other. At first glance these two stories seem to have little to do with one another, but recent developments have shown that this is not the case: they are related through the theory of *symplectic duality *as defined by Braden, Licata, Proudfoot and Webster. Much of my research is focused on aspects of symplectic duality, its origins in physics, and how all of this relates back to representation theory.

In a little more detail, for the past few years my research has been focused on the study of *Coulomb and Higgs branches* of 3d N=4 SUSY gauge theories. These are spaces coming from physics, which have recently been introduced into mathematics through the groundbreaking work of Braverman, Finkelberg, and Nakajima on Coulomb branches. (Higgs branches are simpler to define, and their construction has been understood for some time.) I am especially interested in *quiver gauge theories. *For quiver gauge theories, the Coulomb branch is related to the affine Grassmannian, while the Higgs branch is a Nakajima quiver variety. So quiver gauge theories provide a unifying "link" between of these two important constructions in representation theory.

It is generally expected that the Coulomb branch and Higgs branch of any particular 3d N=4 gauge theory are symplectic dual to one another. Its also worth noting that this duality is related to 3d mirror symmetry in physics.

The Coulomb side of things is very recent, at least mathematically, and still under very active development. I spend a lot of my time thinking about these spaces and their properties, as well as associated algebraic objects.

# Papers

## Published articles

*Lie algebra action on module categories for truncated shifted Yangians*, with Joel Kamnitzer, Ben Webster, and Oded Yacobi,*Forum of Mathematics, Sigma***12**(2024), e18. [Journal version]*BFN Springer Theory**,*with Justin Hilburn and Joel Kamnitzer,*Communications in Mathematical Physics***402**(2023), 765-832. [Journal version]*Coulomb branches for quiver gauge theories with symmetrizers**,*with Hiraku Nakajima,*Journal of the European Mathematical Society***25**(2023), no. 1, 203-230. [Journal version]*Symplectic leaves for generalized affine Grassmannian slices*, with Dinakar Muthiah,*Annales scientifiques de l'école normale supérieure***56**(2023), iss. 1, 287-298. [Journal version]*Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians**,*with Joel Kamnitzer and Khoa Pham,*Advances in Mathematics***399**(2022) 108281 [Journal version]*Quiver gauge theories and symplectic singularities*,*Advances in Mathematics***396**(2022) 108185 [Journal version]*The equations defining affine Grassmannians in type A, and a conjecture of Kreiman, Lakshmibai, Magyar and Weyman*, with Dinakar Muthiah and Oded Yacobi,*International Mathematics Research Notices***2022**(2022), iss. 3, 1922-1972*On a conjecture of Pappas and Rapoport about the standard local model for GL(d)**,*with Dinakar Muthiah and Oded Yacobi,*Journal für die Reine und Angewandte Mathematik***772**(2021), 175-185*Crystals and monodromy of Bethe vectors*, with Iva Halacheva, Joel Kamnitzer, and Leonid Rybnikov,*Duke Mathematical Journal***169**(2020), no. 12, 2337-2419 [Journal version]*A quantum Mirkovic-Vybornov isomorphism*, with Ben Webster and Oded Yacobi,*Representation Theory***24**(2020), 38-84 [Journal version]- Appendices to
*Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian*, with Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima and Ben Webster,*Advances in Theoretical and Mathematial Physics***23**(2019), no. 1, 75-166 [Journal version] - Appendices to
*Shifted quantum affine algebras: integral forms in type A*, with Michael Finkelberg and Alexander Tsymbaliuk,*Arnold Mathematical Journal***5**(2019), no. 2, 197-283 [Journal version] *On category O for affine Grassmannian slices and categorified tensor products*, with Joel Kamnitzer, Peter Tingley, Ben Webster and Oded Yacobi,*Proceedings of the London Mathematical Society***119**(2019), issue 5, 1179-1233 [Journal version]*Highest weights for truncated shifted Yangians and product monomial crystals*, with Joel Kamnitzer, Peter Tingley, Ben Webster and Oded Yacobi,*Journal of Combinatorial Algebra***3**(2019), vol. 3, 237-303 [Journal version]*Reducedness for affine Grassmannian slices of type A**,*with Joel Kamnitzer, Dinakar Muthiah and Oded Yacobi,*Proceedings of the American Mathematical Society***146**(2018), 861-874 [Journal version]*Comultiplication for shifted Yangians and quantum open Toda lattice*, with Michael Finkelberg, Joel Kamnitzer, Khoa Pham and Leonid Rybnikov,*Advances in Mathematics***327**(2018), 349-389 [Journal version]*On a reducedness conjecture for spherical Schubert varieties in the affine Grassmannian*, with Joel Kamnitzer and Dinakar Muthiah,*Transformation Groups***23**(2018), issue 3, 707-722 [Journal version]*Yangians and quantizations of slices in the affine Grassmannian*, with Joel Kamnitzer, Ben Webster and Oded Yacobi,*Algebra and Number Theory***8**(2014), no. 4, 857-893 [Journal version]

## Preprints

*Braid group actions, Baxter polynomials, and affine quantum groups*, with Noah Friesen and Curtis Wendlandt*Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices*, with Dinakar Muthiah*Generators for Coulomb branches of quiver gauge theories*