Categorification is the art of taking some algebraic structure you love and upgrading it to a categorical structure. I will say more once I finish this website!

- Diagrammatics for Coxeter groups and their braid groups

Joint with Geordie Williamson. Preliminary version.

We provide a presentation by generators and relations for the 2-groupoid attached to any Coxeter group or its braid group. This is equivalent to the fundamental 2-groupoid of its classifying space. Our method uses topology, and an investigation of combinatorial models for the 3-skeleton of this classifying space. The presentation for the braid group relies upon the K(pi,1)-conjecture.

- Quantum Satake in type A: part I

Preliminary version.

The geometric Satake equivalence is an equivalence of monoidal categories between perverse sheaves on the affine Grassmannian and representations of the Langlands dual group. First, I reformulate this equivalence in an algebraic way, as an equivalence of 2-categories between maximally singular Soergel bimodules and representations of the Langlands dual Lie algebra with a choice of central character. In type*A*both 2-categories have nice subcategories which can be described by generators and relations: maximally singular Bott-Samelson bimodules, and tensor products of fundamental representations. I provide an explicit equivalence between these subcategories, which gives a new proof of geometric Satake. Moreover, both subcategories admit a*q*-deformation, leading to an equivalence I call the quantum Satake equivalence.

This is the first of two papers, discussing**sl**_{2}and**sl**_{3}. Part II will give the general case, but as it relies on*Diagrammatics for singular Soergel bimodules in type A*, which has not yet appeared, its own appearance may take some time.

- Kazhdan-Lusztig conjectures and shadows of Hodge theory

Joint with Geordie Williamson. Preliminary version.

This is a survey paper, based on a lecture given by Geordie in memory of Hirzebruch. We give an introduction to our paper*The Hodge theory of Soergel bimodules*, and illustrate the results in the case of the dihedral group H2, the smallest Coxeter group which is not a Weyl group.

- Soergel bimodules for universal Coxeter groups

Joint with Nicolas Libedinsky. Preliminary version.

In the Dihedral Cathedral (below), I described the connection between singular Soergel bimodules for the infinite dihedral group and the two-colored Temperley-Lieb category. Using this, I was able to describe the idempotents projecting to any indecomposable Soergel bimodule using Jones-Wenzl projectors in the Temperley-Lieb algebra. In this paper, we achieve similar results for any universal Coxeter group (a group generated by involutions with no relations). We define the multi-colored Temperley-Lieb category, develop the theory of its top projectors analogous to Jones-Wenzl projectors, connect it to singular Soergel bimodules, and describe the idempotents for all indecomposable Soergel bimodules.

- Soergel calculus

Joint with Geordie Williamson. Preliminary version.

Soergel bimodules form an algebraic categorification of the Hecke algebra, defined for any Coxeter group. Soergel bimodules are summands of Bott-Samelson bimodules, which have a combinatorial description. We give a diagrammatic presentation for the monoidal category of Bott-Samelson bimodules, by generators and relations. We give a diagrammatic description of Libedinsky's Light Leaves, and prove that they form a cellular basis for morphism spaces. We give a new, simpler proof of Soergel's categorification results, which works in additional generality.

This paper is the culmination of two other works on Soergel calculus in special cases. However, this paper relies heavily on*The two-color Soergel calculus*.

- The two-color Soergel calculus

Preliminary version. Also known as the dihedral cathedral.

We give a diagrammatic presentation for the monoidal category of Bott-Samelson bimodules attached to Coxeter groups of rank 2. We do the same for singular Soergel bimodules. We connect singular Soergel bimodules for the infinite dihedral group with the Temperley-Lieb algebra and representations of**sl**_{2}, giving the simplest case of the quantum Satake equivalence. Using Jones-Wenzl projectors, we provide idempotents projecting to all the indecomposable Soergel bimodules.

This paper is based on my PhD thesis.

- On cubes of Frobenius extensions

Joint with Geordie Williamson. Preliminary version.

We describe using diagrammatics the natural transformations between induction and restriction functors associated to systems of Frobenius extensions. This gives a set of generators and relations which applies to any 2-category of singular Soergel bimodules. In the current version one relation is left unproven, though this shall be amended soon.

- An approach to categorification of some small quantum groups II

Joint with You Qi. Submitted.

Khovanov's theory of*p*-complexes and Hopfological algebra implies that the Grothendieck group of a*p*-dg-algebra will naturally form a module over the integers adjoined a*p*-th root of unity. In the prequel to this paper, a*p*-dg-algebra structure was placed on the nilHecke algebra, categorifying the positive half of quantum**sl**_{2}at a root of unity. In this paper, a*p*-dg-algebra structure is placed on Lauda's categorification of the entirety of quantum**sl**_{2}, categorifying it at a root of unity. Along the way, we classify all*p*-differentials on this category, and develop techniques to prove categorification results in the*p*-dg-algebra world.

- The Hodge theory of Soergel bimodules

Joint with Geordie Williamson. Submitted.

Soergel bimodules for a Weyl group are none other than the intersection cohomologies of Schubert varieties, and therefore satisfy a number of Hodge-theoretic properties. In particular, they satisfy the Hodge-Riemann bilinear relations with respect to some Lefschetz operator. de Cataldo and Migliorini used these properties to give new proofs. de Cataldo and Migliorini used these properties to give a new proof of the decomposition theorem of Beilinson-Bernstein-Deligne, for perverse sheaves over the real numbers. Inspired by this, we prove that Soergel bimodules for any Coxeter group have these same Hodge-theoretic properties, despite the lack of underlying geometry! We use this to prove Soergel's conjecture on decompositions of Soergel bimodules over the real numbers, which implies the positivity of Kazhdan-Lusztig polynomials, as well as the Kazhdan-Lusztig conjectures for Weyl groups.

- A diagrammatic category for generalized Bott-Samelson bimodules and a diagrammatic categorification of induced trivial modules for Hecke
algebras

Preliminary version.

Given any element of a Coxeter group one can construct its reduced expression graph, which has a vertex for each reduced expression, and an edge for each braid relation. To any path in the reduced expression graph one can associate a morphism between Bott-Samelson bimodules. In type*A*, Manin and Schechtmann provide a partial orientation on this graph, called the higher Bruhat order, with especially nice properties. We show that any two oriented paths between the same vertices give equal morphisms between Bott-Samelson bimodules. We use this to construct idempotents projecting to the indecomposable Soergel bimodule associated to longest elements in type*A*. We also provide a thick calculus for generalized Bott-Samelson bimodules.

- A diagrammatic Temperley-Lieb categorification

Appeared in Int. J. Math. Math. Sci. (2010) (link)

The Temperley-Lieb algebra is a quotient of the Hecke algebra in type*A*. We categorify it using a quotient of the category of Soergel bimodules. We also categorify all cell modules.

- Rouquier complexes are functorial over braid cobordisms

Joint with Dan Krasner. Appeared in Homology, Homotopy, and Applications (2010) (link)

Rouquier complexes are complexes of Soergel bimodules that give a categorical action of the braid group. Khovanov used these complexes (in type*A*) to produce a triply graded knot homology theory. We construct explain chain maps for each braid cobordism, and construct explicit homotopies to prove that this assignment of complexes to braids lifts to a functor from the braid cobordism category to the homotopy category of Soergel bimodules.

- Diagrammatics for Soergel categories

Joint with Mikhail Khovanov. Appeared in Int. J. Math. Math. Sci. (2010) (link)

We give a diagrammatic presentation of morphisms between Bott-Samelson bimodules in type*A*. This was the paper which began the investigation into Soergel bimodules using generators and relations.

- Finding minimal permutation representations of finite groups

Joint with Lior Silberman and Ramin Takloo-Bighash. Appeared in Experimental Math. (2010) (link)

For any finite group there is a set of minimal size on which it can act faithfully. Such an action is called a minimal permutation representation. We provide a greedy algorithm for determining such a minimal permutation representation and computing its size, which works for any nilpotent group. We study some statistics of the distribution of minimal sizes.

This was based on my undergraduate thesis, under Ramin Takloo-Bighash.

- An approach to categorification of quantum groups at roots of unity III

Joint with You Qi. In preparation, no draft available.

We place a*p*-dg-algebra structure on the thick calculus of Khovanov-Lauda-Mackaay-Stosic, and categorify the Lusztig form of quantum**sl**_{2}at a*p*-th root of unity.

- Braid group actions by spherical functors are strict

Joint with Joseph Grant. In preparation, no draft available.

By using an algebraic reformulation of braid group actions by spherical functors found in Grant's earlier work, we deduce the strictness of these actions from a result of Rouquier.

- Special Koszul complexes

Email to request rough draft.

We define the notion of a quasi-regular sequence in a commutative algebra. We show that the Koszul complex for a quasi-regular sequence has an unusual property: its homology can be split into local terms. This can be used to investigate Hochschild homology of certain Bott-Samelson bimodules.

- Singular Soergel calculus in type
*A*

Joint with Geordie Williamson. In preparation, no draft available.

We provide a presentation by generators and relations of the 2-category of singular Soergel bimodules in type*A*.

- An algebraic approach to folding and unequal parameters

Joint with Geordie Williamson. Email to request rough draft.

We investigate equivariant Soergel bimodules and their weighted Grothendieck groups. We provide a new proof of traditional folding results: that equivariant Soergel bimodules categorify Hecke algebras with unequal parameters in the quasi-split case. We generalize this result to cases without any corresponding geometry.

The actual result is not especially surprising, being mostly a reproof. However, we use diagrammatic technology to perform a number of computations to give new explicit proofs, of a very different nature than the familiar geometric proofs.

- Categorifying the Lusztig-Vogan representation

Joint with Geordie Williamson. Email to request rough draft.

Using the weighted Grothendieck group of a certain category of equivariant Soergel bimodules, we categorify the Lusztig-Vogan representation of any Coxeter group. This proves the signed positivity of Lusztig-Vogan polynomials, conjectured by Lusztig and Vogan.

Similarly to the previous paper on folding, the result is not surprising, and Lusztig and Vogan have also written a paper to the same effect. However, we use different diagrammatic techniques.

Links to my coauthors:

Geordie Williamson

You Qi

Nicolas Libedinsky

Joseph Grant

Mikhail Khovanov

Daniel Krasner

Ramin Takloo-Bighash

Lior Silberman