Georgia Institute of Technology College of Sciences

Wiring diagrams are classical objects of combinatorics. Plabic graphs were defined by Postnikov, to study the total positivity of the Grassmannian. We will show how to generalize several definitions and properties of wiring diagrams to Plabic graphs, proving a conjecture by Leclerc-Zelevinsky and Scott on the way. We will begin with a brief introduction to total positivity and end with connection to cluster algebras. Major part of the talk comes from a joint work with Alexander Postnikov and David Speyer.

The cohomology ring of the absolute Galois group Gal(kbar/k) of a field k controls interesting arithmetic properties of k. The Milnor conjecture, proven by Voevodsky, identifies the cohomology ring H^*(Gal(kbar/k), Z/2) with the tensor algebra of k* mod the ideal generated by x otimes 1-x for x in k - {0,1} mod 2, and the Bloch-Kato theorem, also proven by Voevodsky, generalizes the coefficient ring Z/2. In particular, the cohomology ring of Gal(kbar/k) can be expressed in terms of addition and multiplication in the field k, despite the fact that it is difficult even to list specific elements of Gal(kbar/k). The cohomology ring is a coarser invariant than the differential graded algebra of cochains, and one can ask for an analogous description of this finer invariant, controlled by and controlling higher order cohomology operations. We show that order n Massey products of n-1 factors of x and one factor of 1-x vanish, generalizing the relation x otimes 1-x. This is done by embedding P^1 - {0,1,infinity} into its Picard variety and constructing Gal(kbar/k) equivariant maps from pi_1^et applied to this embedding to unipotent matrix groups. This also identifies Massey products of the form <1-x, x, … , x , 1-x> with f cup 1-x, where f is a certain cohomology class which arises in the description of the action of Gal(kbar/k) on pi_1^et(P^1 - {0,1,infinity}). The first part of this talk will not assume knowledge of Galois cohomology or Massey products.

A polytope is a convex hull of a finite set of points in a vector space. The set of polytopes in a fixed vector space generate an algebra where addition is formal and multiplication is the Minkowski sum, modulo some relations. The algebra of polytopes were used to solve some variations of Hilbert's third problem about subdivision of polytopes and to give a combinatorial proof of Stanley's g-Theorem that characterizes face numbers of simplicial polytopes. In this talk, we will introduce McMullen's version of polytope algebra and show that it is isomorphic to the algebra of tropical cycles which are balanced weighted polyhedral fans. The tropical cycles can be used to do explicit computations and examples in polytope algebra.

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.