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.

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.