Projects

Erin Delargy, Rylie Harris, Jiachen Kang, Bryan Lu, and Ramanuja Charyulu Telekicherla Kandalam explored two 𝔖_n-equivariant Koszul algebras arising from the Boolean lattice: the Chow ring of the Boolean matroid and the "colorful ring" associated to the barycentric subdivision of a simplex. The dimensions of the graded pieces of both of these algebras correspond to the Eulerian numbers, but the bases and representations for the graded pieces of the two algebras are vastly different. In this talk we compare and contrast the bases and representations for each of these algebras and their Koszul duals. Along the way, we investigate (sometimes noncommutative) Gröbner bases for these algebras and their Koszul duals, and we prove branching rules for representations of the symmetric group which categorify a recursion on the Eulerian numbers.

1. • Mentor: Ayah Almousa; TA: Anastasia Nathanson

   • Final report, final presentation

Isidora Bailly-Hall, Karina Dovgodko, Sean Guan, Sai Sivakumar, and Jishi Sun extended previous work on virtual resolutions in products of projective spaces. Free resolutions, or syzygies, with a graded structure are algebraic objects that encode many geometric properties. This correspondence lies at the heart of classical projective algebraic geometry. Virtual resolutions were recently introduced by Berkesch, Erman, and Smith to produce a similar correspondence for smooth toric varieties. We will describe two methods for producing nice virtual resolutions for a  finite sets of points in $\PP^n\times\PP^m$.

1. • Mentor: Christine Berkesch; TA: Sasha Pevzner

   • Final report, final presentation,

1. Zeus Dantas e Moura, Bryan Lu, and Dora Woodruff investigated how jeu de taquin affects the homeorphism type of order complexes of intervals in Young's lattice. To every finite, closed interval of Young's lattice, there is a naturally associated simplicial complex. A classical theorem of Björner says that these simplicial complexes are all homeomorphic to balls. We explore how jeu de taquin interacts with this theory. In this setting, there is an associated simplicial complex for every dual equivalence class of skew tableaux. We investigate whether these simplicial complexes are also homeomorphic to balls, and whether jeu de taquin induces simplicial isomorphisms. Among other discoveries, we show that not all of these complexes are homeomorphic to balls, but classify several cases in which they are. We also explore a connection to K-jdt, a variant of jeu de taquin for increasing tableaux. 

   • Mentor: Daoji Huang; TA: Carolyn Stephen

   • Final report, final presentation

1. Bjorn Cattell-Ravdal, Erin Delargy, Akash Ganguly, Sean Guan, and Sai Sivakumar examined GL(n,k)-stable ideals and their free resolutions in arbitrary characteristics. Stephen Doty determined the GL(n,k) submodule structure of the degree-d homogeneous component of the polynomial ring in n variables over k, when k is a field of positive characteristic. We build on this work to provide the minimal free resolution for the inclusion-minimal GL(2,k)-stable ideal generated in a single degree in any positive characteristic.

   • Mentor: Michael Perlman; TA: Trevor Karn

   • Final report, final presentation

Zeus Dantas e Moura, Ramanuja Charyulu Telekicherla Kandalam, and Dora Woodruff worked on cluster monomials associated to graph LP algebras. LP algebras are a generalization of cluster algebras, first defined by Lam and Pylyavskyy, and graph LP algebras are a particularly nice class of LP algebras, whose exchange polynomials are defined by a graph. Several well-known properties and conjectures about cluster algebras appear to hold for LP algebras too, and in our project, we extend these properties to graph LP algebras. In particular, we show that cluster monomials form a linear spanning set for all graph LP algebras, and show total positivity for LP algebras arising from certain classes of graphs. 

1. • Mentor: Pasha Pylyavskyy; TA: Robbie Angarone

   • Final report, final presentation

Isidora Bailly-Hall, Karina Dovgodko, Akash Ganguly, Jiachen Kang, and Jishi Sun studied detecting boundary slopes via intersections in the character variety arising from epimorphisms onto certain knots.  The SL(2,C) character variety has long been an important tool in the study of 3-manifolds. In particular, intersection points in character varieties of knot groups detect essential surfaces via SL(2,C)-trees. We describe intersection points in the character varieties of a family of hyperbolic two-bridge knot groups that have epimorphisms onto the trefoil knot. Using the technique of Farey recursion, we show that these intersection points correspond to algebraic non-integral representations. We also determine the boundary slopes detected by these intersection points.

1. • Mentor: Michelle Chu; TAs: Tori Braun & Lilly Webster

   • Final report, final presentation