My research interests include canonical bases in geometric representation theory, cluster algebras, and categorification.

Ongoing projects

1. Spectral representation theory. We adapt the Khovanov stable homotopy type construction of Lawson-Lipshitz-Sarkar to the category of Blanchet foams, towards realizing a spectral action of the 2-quantum group $U_q(sl_n)$ in the category of symmetric spectra by spectral bimodules. Joint with Meng Guo, Aaron Lauda and Andy Manion.

1. Approximating the Kazhdan-Lusztig basis by eigenbases. We generalize the classical upper-triangularity result of Garsia-McLarnan relating the standard Gelfand–Tsetlin (GT) eigenbasis and the Kazhan–Lusztig (KL) basis in Specht modules, to a distinguished family of eigenbases. We study the space of all bases (including KL) which are upper-triangular w.r.t. this family. Joint with Oded Yacobi.

1. Perfect bases in cluster algebras. We continue to study cluster structures hiding within the MV basis. In particular, we compare the MV basis and the $\theta$-basis of HMS, towards settling Kamnitzer’s conjecture that the two are the same.

1. Derived geometric Satake equivalence. We extend the notion of Duistermaat–Heckman measure in the context of geometric Satake to the derived setting, identifying equivariant multiplicities of MV cycles w.r.t. a bigger torus and certain invariants of quantized universal
   centralizers.

1. Asymptotic characters of modules for truncated shifted Yangians. We study characters on category O of truncated shifted Yangians, which quantize slices in the affine Grassmannian. We compare with characters on an equivalent category of modules for KLRW algebra. Joint with Alexis Leroux-Lapierre.

Past projects

1. Extremal tensor products of Demazure crystals with Sami Assaf and Nicolle Gonzalez arXiv:2210.10236
2. Heaps, crystals, and preprojective algebra modules with Balazs Elek, Joel Kamnitzer and Calder Morton-Ferguson arXiv:2202.02490
3. Computing fusion products of MV cycles using the Mirkovic-Vybornov isomorphism with Roger Bai and Joel Kamnitzer arXiv:2106.07101

Thesis work

1. Comparing two perfect bases arXiv:2105.14420 and Appendix to The Mirkovic-Vilonen basis and Duistermaat-Heckman measures with Joel Kamnitzer and Calder Morton-Ferguson arXiv:1905.08460
2. Generalized orbital varieties for Mirkovic-Vybornov slices as affinizations of Mirkovic-Vilonen cycles arXiv:1905.08174