I will introduce a new matroid (graph) invariant: The Arboricity Polynomial.   Arboricity is a numerical invariant first introduced by Nash-Williams, Tutte and Edmonds.  It captures the minimum number of independent sets (forests) needed to decompose the ground set of a matroid (edges of a graph).    The arboricity polynomial enumerates the number of such decompositions.  We examine this counting function in terms of scheduling, Ehrhart theory, quasisymmetric functions, matroid polytopes and the permutohedral fan. 

Consider the conjugation action of \({\rm GL}_2(K)\) on the polynomial ring \(K[X_{2\times 2}]\). When \(K\) is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when \(K\) is a finite field, and show that it is a hypersurface.

Let $X$ be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice $\Gamma$. For instance, $X$ could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let $H$ be a discrete subgroup of isometries of $X$ with critical exponent (exponential growth rate) strictly less than half of the growth rate of $\Gamma$. We show that the injectivity radius of $X/H$ grows linearly along almost every geodesic in $X$ (with respect to the Patterson-Sullivan measure on the Gromov boundary of $X$). The proof will involve an elementary analysis of a novel concept called the "sublinearly horospherical limit set" of $H$ which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups. This talk is based on joint work with Inhyeok Choi and Keivan Mallahi-Kerai.

Khintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms (f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).

In classical algebra, the Picard group of a commutative ring R is invariant under quotient by nilpotent elements. In joint work in progress with Ishan Levy and Guchuan Li, we study Picard groups of some quotient ring spectra. Under a vanishing condition, we prove that Pic(R/v^{n+1}) --> Pic(R/v^n) is injective for a ring spectrum R such that R/v is an E_1-R-algebra. This allows us to show Picard groups of quotients of Morava E-theory by a regular sequence in its π_0 are always ℤ/2. Running the profinite descent spectral sequence from there, we prove the Picard group of any K(n)-local generalized Moore spectrum of type n is finite. At height 1 and all primes p, we compute the Picard group of K(1)-local S^0/p^k when k is not too small.

The Ramsey number R(s,t) is the smallest integer n such that every red-blue coloring of the edges of the complete graph Kn contains a red clique of size s or a blue clique of size t. Ramsey numbers and their variants are some of the most famous numbers in combinatorics, yet computing even small exact values is notoriously difficult. Indeed, Erdős quipped that it would be more difficult for humans to compute the Ramsey number R(6,6) than to fend off an alien invasion. In this talk we highlight recent successes of Boolean satisfiability (SAT) solvers in Ramsey theory in both the arithmetic and graph theoretic settings.

Super-resolution algorithms to learn fine-scale features of a signal beyond the resolution at which the signal was measured. This talk will provide an overview of the mathematical theory of super-resolution, including new results by the presenter and collaborators, and show how these mathematical techniques can also be used to design quantum algorithms for problems in scientific computing. This talk is designed for a broad mathematical audience and assumes no prior knowledge in super-resolution or quantum computation.

The Banach-Tarski paradox states that a ball can be disassembled into finitely many disjoint pieces and reassembled via translations and rotations into two copies of the original ball. It turns out that this "paradoxical decomposition" is precisely characterized by the group theoretic property known as (non)-amenability. Along the way, we will encounter various equivalent definitions of amenable group (of which there are many) and some applications. This talk will be accessible to anyone who knows what a group is.

In this talk, we describe different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. These spaces are constructed via generalizations of the Fulton-MacPherson compactification. For specific weight choices, we show that our moduli problem admits toric compactifications that coincide with the polypermutohedral variety of Crowly-Huh-Larson-Simpson-Wang and with the polystellahedral variety of Eur-Larson. This is joint work with J. Gonzalez-Anaya and J.L. Gonzalez.