In 2017, Lewis, Reiner, and Stanton conjectured a connection between the modular general linear group and (q,t)-binomial coefficients. This conjecture was an analog to a theorem connecting the representation theory of rational Cherednik algebras for Coxeter groups and Catalan numbers. We will describe a local case - the invariant ring for groups reflecting about a fixed hyperplane acting on a polynomial ring modulo Frobenius powers. When the characteristic of the underlying field divides the order of the group, the subgroup fixing a reflecting hyperplane is a semi-direct product of diagonalizable reflections and transvections. The resulting Hilbert series counts the number of orbits of the group acting on a vector space, solving a special case of the Lewis, Reiner, Stanton conjecture.

A well-quasi-ordering is a reflexive and transitive binary relation such that every infinite sequence has a non-trivial increasing subsequence. The study of well-quasi-ordering can be dated back to two conjectures of Vazsonyi proposed in 1940s stating that trees and subcubic graphs are well-quasi-ordered by the topological minor relation. Both conjectures have been solved, where the second conjecture is particularly difficult in the sense that the only known proof is via Robertson and Seymour's celebrated Graph Minor Theorem stating that the minor relation is a well-quasi-ordering. On the other hand, the topological minor relation is not a well-quasi-ordering in general. Robertson in 1980s conjectured that the known obstruction is the only obstruction. Joint with Robin Thomas, we solved Robertson's conjecture and proved a characterization of well-quasi-ordered topological-minor ideals. We will sketch some ideas in the proof.

Let $a,b$ be a pair of co-prime positive integers. An (a,b)-rational parking function is a sequence (x_1, x_2, ..., x_b) of non-negative integers such that x_{(i)} is less than or equal to ia/b for all i, where x_{(i)} is the i-th smallest term of x_1, x_2, ..., x_b. Rational parking functions are important in the study of rational Catalan combinatorics and representation theory of MacDonald polynomials. In this talk we consider rational parking functions of length n, where n is a multiple of b, and present a counting formula for the number of rational parking functions. The techniques are basic combinatorial tools, including lattice path counting, the cycle lemma, and the inclusion-exclusion principle. This is based on a joint work with Yue Cai.