Research

We show that every all-Z tiling of the m=2 amplituhedron A(n, k, 2) (equivalently, every positroid tiling of the hypersimplex) consists of M(n, k, 2) tiles, where M(n, k, 2) is a particular specialization of MacMahon's number. This is evidence towards a conjecture of Karp-Williams-Zhang that every tiling of A(n, k, m) consists of M(n, k, m) of tiles. Along the way, we give formulas for volumes of tree positroid polytopes in terms of the number of circular extensions of a partial cyclic order. Our main tool is Parke-Taylor functions, which appear in physics. As a corollary of our results, we obtain a number of identities among Parke-Taylor functions, again phrased in terms of circular extensions of partial cyclic orders.

A companion paper to "Cluster algebras and tilings for the m=4 amplituhedron." We describe facets of BCFW tiles, exhibit a tiling containing  a non-BCFW tile, and show that BCFW tiles are positive parts of cluster varieties.

We prove various results involving the BCFW recursion for scattering amplitudes and the m=4 amplituhedron. First, we show that images of BCFW cells are tiles for the amplituhedron. Next, we prove the cluster adjacency conjecture for BCFW tiles, showing that the facets of each BCFW tile lie on hypersurfaces cut out by compatible cluster variables of Gr(4,n). We also give a semi-algebraic description of BCFW tiles, again using compatible cluster variables. Along the way, we give explicit formulas for many high-degree cluster variables for Gr(4,n) using a cluster quasi-homomorphism, which may be of independent interest. Finally, we prove that any way of running the BCFW recursion gives a tiling of the m=4 amplituhedron. This proves one of the original conjectures of Arkani-Hamed and Trnka on the amplituhedron and generalizes a result of Even-Zohar, Lakrec, and Tessler for a single way of running the recursion.

Demazure weaves have recently been used to give a cluster structure on braid varieties (a different approach than in the papers below). In this paper, we consider open positroid varieties, which have a cluster structure given by reduced plabic graphs. For each reduced plabic graph, we construct a Demazure weave. T-duality on plabic graphs (defined in Parisi-SB-Williams) makes a surprise appearance. We show that the target seed for the plabic graph is the same as the seed given by the Demazure weave. We also show that the Muller-Speyer twist, an automorphism of the open positroid variety, is a quasi-cluster automorphism (the Donaldson-Thomas transformation) of the target cluster structure. This shows that the source and target cluster structures quasi-coincide, that is, differ only by rescaling by frozens.

We show that braid varieties for any simple reductive algebraic group have a cluster structure. This paper is independent of "Braid variety cluster structures, I" but uses similar ideas and proof strategies. Our main tools are the Deodhar geometry of braid varieties and a "deletion-contraction" inductive argument. 

Leclerc gave a conjectural cluster structure on type ADE Richardson varieties using cluster category techniques. We show that in type A, his conjectural cluster structure really is a cluster structure. In particular, we show Leclerc's cluster algebra is related to the cluster structure in "Braid variety cluster structures, I" by a twist automorphism, recently introduced by Galashin and Lam.

We introduce 3-dimensional generalizations of Postnikov's plabic graphs and use them to establish cluster structures for type A braid varieties. Our results include known cluster structures on open positroid varieties and double Bruhat cells, and establish new cluster structures for type A open Richardson varieties.

This is a follow-up paper to "1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity". We show that certain dual canonical basis elements of C[SL(m)], written in terms of Kazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations, are positive when evaluated on k-positive matrices (matrices whose minors of size k×k and smaller are positive). We also make some conjectures on the form of cluster variables for the big open double Bruhat cell in SL(m).

We prove various conjectures about the m=2 amplituhedron, including the sign-flip characterization of Arkani-Hamed--Thomas--Trnka, the cluster adjacency conjecture of Lukowski--Parisi--Spradlin--Volovich, and the conjecture of Lukowski--Parisi--Williams relating triangulations of the amplituhedron to triangulations of the hypersimplex via T-duality. We also discuss new cluster structures in the amplituhedron. Finally, we discuss twistor coordinate sign-stratification of the amplituhedron, and show that when m=2, the number of sign chambers is the Eulerian number.

We consider Laurent polynomial expressions for cluster variables in terms of an arbitrary initial cluster. In this paper, we show that the Newton polytopes of these Laurent polynomials are saturated in the case of cluster algebras of types A and D with frozen variables corresponding to the boundary segments of the appropriate marked surface. For type A, we additionally show that the cluster variable Newton polytopes have no non-vertex lattice points. Our main tool is the snake graph expansion formula of Musiker--Schiffler--Williams for cluster algebras from surfaces.

We show that each positroid variety has many cluster structures, with seeds given by relabeled plabic graphs (i.e. plabic graphs whose boundary vertices have been relabeled). We conjecture that the seeds in all of these cluster structures are related by mutations and rescaling by Laurent monomials in frozen variables, and in particular have the same cluster monomials. We establish this conjecture for (open) Schubert and opposite Schubert varieties. We also show a special case of a conjecture by Muller--Speyer on the "source" and "target" cluster structures. In the course of our proofs, we also provide nontrivial twist isomorphisms between many positroid varieties.

Rhoades and Skandera defined Kazhdan-Lusztig immanants, which are indexed by permutations and involve q=1 specializations of Type A Kazhdan-Lusztig polynomials. We investigate signs of Kazhdan-Lusztig immanants evaluated on k-positive matrices (matrices whose minors of size k×k and smaller are positive). We show that the Kazhdan-Lusztig immanant indexed by v is positive on k-positive matrices when v avoids 1324 and 2143 and for all non-inversions i<j of v, either j−i≤k or v(j)−v(i)≤k. Our main tool is the Desnanot-Jacobi identity.

We establish a long-standing folklore conjecture: Postnikov's plabic graphs for open Schubert varieties endow their coordinate rings with a cluster algebra structure. Results of Leclerc on Richardson varieties in the full flag variety imply that the coordinate ring of (the affine cone over) an open Schubert variety is a cluster algebra. However, Leclerc's methods are categorical, and explicit computations of seeds require understanding irreducible morphisms in the module category of the preprojective algebra of type A. We use a construction of Karpman to show that plabic graphs give seeds in Leclerc's cluster algebra. 

We count the number of X-variables (also called "y-hat" variables) in finite type cluster algebras with full rank exchange matrices over the universal semifield. For classical types, we show X-variables are in bijection with quadrilaterals in tagged triangulations of the corresponding marked surface. For exceptional types, we provide Mathematica code to count X-variables (here as a Mathematica notebook, here as a txt file). A root-theoretic interpretation formula for these numbers is still unknown.

Written after the Oregon State University REU in 2015.

On research done at Budapest Semesters in Mathematics in Spring 2015.

My undergraduate senior thesis, focusing on some connections between finite groups and their Cayley graphs. Includes results on colorings of Cayley graphs that respect certain subgroups of their automorphism groups.

Collaborators

I've had the pleasure of collaborating with: Cora Borradaile, Roger Casals, Sunita Chepuri, Chaim Even-Zohar, Chris Fraser, Pavel Galashin, Andras Gyarfas, Tsviqa Lakrec, Thomas Lam, Ian Le, Amal Mattoo, Matteo Parisi, Alex Riasanovsky, Khrystyna Serhiyenko, David Speyer, Ran Tessler, Hung Viet Le, Daping Weng, Lauren Williams.