Research

Abstract: Recent work of Lam, Bychkov-Gorbounov-Kazakov-Talalaev, and Chepuri-George-Speyer gave a stratification of the totally nonnegative Lagrangian Grassmannian into electroid cells parameterized by cactus networks, paralleling Postnikov's stratification of the totally nonnegative Grassmannian by positroid cells. Electroid varieties arise as an algebro-geometric extension of electroid cells. The combinatorics of these varieties was studied by Lam in 2018. We build on this work and study the geometric properties of electroid varieties. In analogy to results of Knutson, Lam, and Speyer on positroid varieties, we show that electroid varieties are reduced, irreducible, regular in codimension one, compatibly Frobenius split, and form a stratification. We also show a decomposition of certain electroid varieties as a product of two electroid varieties. As a consequence, the grove measurement map that embeds electroid cells can be extended algebraically to embed an algebraic torus. 

Abstract:The ABCT variety V(3,n) is the image closure of the rational Veronese map from the Grassmannian Gr(2,n) to the Grassmannian Gr(3,n). It was studied by Arkani-Hamed--Bourjaily--Cachazo--Trnka in the context of tree-level scattering amplitudes arising in planar N=4 supersymmetric Yang-Mills theory and Witten's twistor string theory. From this perspective, V(3,n) is conjectured to be a positive geometry by Lam. In this paper, we study the combinatorial and algebraic geometry aspects of 

V(3,n) and its subvarieties induced by iteratively taking analytic boundaries of the totally nonnegative part. We interpret these subvarieties as point configurations on the projective plane by the Gelfand-MacPherson correspondence. We construct a top-degree meromorphic form on V(3,n) and show that it is a positive geometry, proving Lam's conjecture.

Abstract: In this paper, we investigate the relationship between Temperley-Lieb immanants, which were introduced by Rhoades and Skandera, and %-immanants, an immanant based on a concept introduced by Chepuri and Sherman-Bennett. Our main result is a classification of when a Temperley-Lieb immanant can be written as a linear combination of %-immanants. This result uses a formula by Rhoades and Skandera to compute Temperley-Lieb immanants in terms of complementary minors. Using this formula, we also derive an explicit expression for the coefficients of a Temperley-Lieb immanant coming from a 321 and 1324 avoiding permutation containing the pattern 2143, which we use to derive our main result. 

Abstract: The ABCT variety is defined as the closure of the image of G(2,n) under the Veronese map. We realize the ABCT variety V(3,n) as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of V(3,n). As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way tothis, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler. 

Abstract: The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang.  It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex.

It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders.  Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.

Abstract: In this paper, we study prime graphs of finite groups. The prime graph of a finite group G, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing |G|} and an edge p-q if and only if there exists an element of order pq in G. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any natural number n, solvable groups of n^th-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or A_5  and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.