University of Washington

Since Szemeredi's Theorem and Furstenberg's proof thereof using ergodic theory, dynamical methods have been used to show the existence of numerous patterns in sets of positive upper density. These tools have led to uncovering new patterns that occur...

The Grassmannian is studied very differently in pure and applied mathematics. From studying these two embeddings of the Grassmannian, a new variety called the squared Grassmannian arises naturally as the image of the Grassmannian in its Plücker...

For a finite connected graph, the following objects have the same cardinality: the set of spanning trees, the Jacobian group, the set of equivalence classes of orientations up to cycle-cocycle reversal, the set of break divisors, the set of reduced...

The q , t -Catalan numbers can be described elegantly in terms of pairs of statistics on Dyck paths: area and bounce, or area and dinv. Using bijective and recursive methods, we prove new expressions of the q , t -Catalan numbers in terms of pairs of...

Anosov flows are a standard model for strongly chaotic behavior in dynamical systems. A classical example is the geodesic flow on a compact negatively curved Riemannian manifold. The chaotic behavior of an Anosov flow manifests in decay of...

We prove a new signed elementary symmetric function expansion of the chromatic symmetric function. We then use sign-reversing involutions to prove e -positivity for graphs formed by joining cycles or cliques at single vertices. By considering...

Richard Stanley asked in 1995 whether a tree is determined up to isomorphism by its chromatic symmetric function. This question remains unanswered and frequently keeps the speaker awake at night. Our approach to understanding the strength of the...

The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still...

We consider the Grassmannian G r C ( k , n ) , a manifold parameterized by k -dimensional planes in complex n -dimensional space. The Schubert decomposition of the Grassmannian is perhaps the classic way to split this space into smaller pieces...

Quivers and their mutations play a fundamental role in the theory of cluster algebras. We focus on the problem of deciding whether two given quivers are mutation equivalent to each other. Our approach is based on introducing an additional structure...