A Categorification of the Vandermonde Determinant: In the spirit of Bar-Natan's description of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers, we construct a complex of colored smoothings of the 2-strand torus link in the shape of the Bruhat order on the symmetric group, and apply a TQFT to obtain a chain complex whose Euler characteristic is equal to the Vandermonde determinant evaluated at the given sequence. Applying this process to n-strand torus links yields categorifications of certain generalized Vandermonde determinants.
To appear in Journal of Knot Theory and its Ramifications Dallas Volume.
Preprint available at arXiv:1811.08090.
Categorical Diagonalization: This is an expository chapter in the MSRI proceedings on Soergel bimodules based on the work of Elias and Hogancamp. This chapter is written jointly with Dmitry Vagner and Nachiket Karnick. Beginning with the scenario in which a linear operator is diagonalized, we develop refined notions of the involved properties---the vanishing of a polynomial in the endofunctor---and structures---the acquiring of an idempotent decomposition of the space the operator acts on---and discuss how to go between the two. Most importantly, we recall Lagrange interpolation, which, from a collection of distinct eigenvalues, produces an idempotent decomposition. We then categorify this picture by finding categorified decompositions corresponding to functors on monoidal homotopy categories. We then proceed to look at several examples, culminating in the diagonalizability of the full twist functor on Rouqier Complexes in the representation theory of Hecke algebras.
Chapter 23 in the Springer text "Introduction to Soergel Bimodules"
Preprint available here: Categorical Diagonalization
Thin Posets, CW Posets, and Categorification: Many well-known categorifications are constructed in a way similar to Khovanov's categorification of the Jones polynomial. In this paper we explore a category structure on the set of all such categorifications, and the cohomology functor on this category. General structure is this category are explored and computational techniques developed.
Preprint available at arXiv:1911.05600.
Submitted December 2019.
Torsion in Thin Regions of Khovanov Homology: Joint with Lowrance, Sazdanovic, Summers. Using the Turner, Lee, and Bockstein spectral sequences on Khovanov homology, we give conditions guaranteeing that locally thin regions of Khovanov homology contain only torsion of order 2. Based on Turners rational 3-torus link computations for Khovanov homology, and Benheddi's computations modulo 2, we show that the integral Khovanov homology of 3-strand torus links has only torsion of order 2, and give an explicit computation of integral Khovanov homology.
Preprint available at arXiv:1903.05760.
Submitted January 2019.
A Broken Circuit Model for Chromatic Homology Theories: We give a categorification of a famous result of Whitney: the broken circuit theorem for the chromatic polynomial. We show analogues of Whitney's theorem in chromatic homology and chromatic symmetrc homology. As applications, we arrive at homological bounds for these theories.
Preprint available at arXiv:1911.13226.
Submitted December 2019.
On the Strength of Chromatic Symmetric Homology for Graphs: Joint with Sazdanovic, Yip, and Stella. In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius characteristic is a q,t generalization of the chromatic symmetric function. We exhibit three pairs of graphs where each pair has the same chromatic symmetric function but distinct homology over the complex numbers as symmetric group modules. We also show that integral chromatic symmetric homology contains torsion, and based on computations, conjecture that integral-torsion in bigrading (1,0) detects nonplanarity in the graph.
Preprint available at arXiv:1911.13297.
Submitted to Journal of Algebraic Combinatorics December 2019.
A Categorification of the Laguerre Polynomials: We continue the work of Khovanov and Sazdanovic in categorifying orthogonal polynomials. Similar to their categorification of Hermite and Chebyshev polynomials, in this paper Laguerre polynomials are realized as elements of the Grothendieck group of a category of projective modules over a diagrammatic algebra.
Preprint available here: A Categorification of the Laguerre Polynomials
Notes on Derived Categories and Mirror Symmetry in String Theory: This is an expository paper written for an NCSU class in homological algebra taught by Tye Lidman. We begin by introducing derived categories and describing why they are useful in a general setting. The rest of the paper will be dedicated to understanding an equivalence of the A-model and B-model of Dirichlet branes in string theory known as homological mirror symmetry. The B-model is realized mathematically via coherent sheaves on a Kahler manifold, and the A-model is realized via Lagrangian submanifolds of a Kahler manifold. Roughly, the statement of mirror symmetry in string theory is that the bounded derived category of coherent sheaves on M is equivalent to the bounded derived category of Lagrangian submanifolds of the mirror manifold (the Fukaya category).
Available here: Derived Categories and Mirror Symmetry in String Theory