Research

I study representation theory, categorification, and algebraic combinatorics.

Representation theory is a field which focuses on studying algebraic structures like groups and algebras by focusing on ways to represent them as matrices acting on a vector space. This approach turns out to be extremely powerful and broadly applicable, partly because it allows us to use our thorough knowledge of linear algebra to investigate more nonlinear phenomena.

Algebraic combinatorics studies algebraic structures by attempting to understand their combinatorial properties. Techniques and objects from combinatorics, such as symmetric functions and diagrammatic algebra, have been found to have important applications to representation theory. This connection appears in several places in my research, particularly in my study of graphical calculi and symmetric functions.

Some specific topics of recent interest for me:

  • Heisenberg categorification
  • Trace decategorification
  • Symmetric functions

Papers

Published/Accepted

  • The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph (with H. Kvinge and C. Oguz), 2017. In Seminaire Lotharingien de Combinatoire. arXiv 1712.09626.
  • The type B extended nilHecke algebra and symmetric functions , 2017. To appear in Journal of Pure and Applied Algebra. arXiv 1710.10698.
  • Trace of the twisted Heisenberg category (with C. Oguz), 2017. In Communications in Mathematical Physics. arXiv 1702.08108.
  • Cocenters of the Hecke Clifford and spin Hecke algebras, 2016. In Journal of Algebra. arXiv 1605.05744
  • Gelfand Models for Diagram Algebras (with T. Halverson), 2015. In Journal of Algebraic Combinatorics. arXiv 1302.6150

Preprints

  • Traces of tensor product categories (with C. Leonard), in preparation.
  • The spin Heisenberg category (with W. Wang), in preparation.

Conferences and Seminars

Invited talks

  • Ottawa-Lyon-Sao Paulo Workshop on Representation Theory. July 2018, University of Ottawa and Carleton University. Traces of tensor product categories.
  • Virginia Tech Algebra Seminar, October 2017. Trace and center of the twisted Heisenberg category. Link to slides.
  • AMS Special Session on Combinatorics and Representation Theory of Reflection Groups: Real and Complex, September 2017, Denton, TX.
  • AMS Special Session on Representation Theory and Integrable Systems, April 2017, Bloomington, IN.
  • University of Virginia Algebra Seminar, February 2017. Trace of the twisted Heisenberg category. Link to slides.
  • University of Virginia Algebra Seminar, December 2014. Gelfand Models for diagram algebras. Link to slides.

Contributed talks

  • Interactions of quantum affine algebras with cluster algebras, current algebras and categorification. June 2018, Catholic University of America. Trace and center of the twisted Heisenberg category. Link to slides.
  • 2018 Joint Mathematics Meetings - AMS Contributed Paper Session on Lie Theory and Related Topics. January 2018, San Diego, CA.
  • University of Virginia Graduate Seminar, September 2016. An introduction to categorification.
  • Southeastern Lie Theory Conference: Algebraic groups, Quantum groups, and Geometry, May 2016, University of Virginia. Traces of Hecke-Clifford and spin Hecke algebras.
  • University of Virginia Graduate Seminar, February 2016. Spin representation theory.

Conferences attended

  • US-Mexico Conference on Representation Theory and Noncommutative Algebra, May 2016, University of Southern California.
  • Southeastern Lie Theory Conference: Algebraic and Combinatorial Representation Theory, October 2015, North Carolina State University.
  • Summer School and Workshop on Lie Theory, July 2015, East China Normal University.
  • AMS Southeast Spring Sectional, March 2015, Georgetown University.