MAAGC 2014 ABSTRACTS:
Sam Payne (Yale): Brill-Noether theory on graphs
Riemann-Roch theory for graphs, as developed by Baker and Norine, points to the existence of deep combinatorial structures related to the classical algebraic geometry of linear series on algebraic curves. Subsequent research in this area includes tropical (essentially graph-theoretic) proofs of the Brill-Noether and Gieseker-Petri theorems, which describe the dimension and local structure of the moduli spaces parametrizing linear series of given degree and rank on a general curve, as well as deepening connections to classical algebraic geometry, including the Eisenbud-Harris theory of limit linear series.
In this talk, I will focus primarily on the combinatorial aspects of recent work in this area, including joint work with Filip Cools, Jan Draisma, and Elina Robeva, and joint work with Dave Jensen. I will also highlight open problems on the combinatorial side of this story, such as the Brill-Noether existence problem for finite graphs.
John Stembridge (Michigan): Generalized stability of Kronecker coefficients
Kronecker coefficients are tensor product multiplicities for symmetric group representations. It is well-known that we do not know much about them. In this talk we plan to discuss some new theorems and conjectures about how these coefficients stabilize or de-stabilize in various limiting cases. A typical (easy) case is the classical result of Murnaghan that corresponds to incrementing the first rows of a triple of Young diagrams.
Julianna Tymoczko (Smith): Splines in Geometry and Combinatorics
Splines are a well-known algebraic/combinatorial construction developed for engineering applications and now used widely in computer graphics, differential equations, numerical analysis, and other fields. Billera and others pioneered an algebraic approach to splines, using techniques from commutative and homological algebra, among others. Over the last fifteen years, geometers and topologists independently developed a way to construct equivariant cohomology rings for large classes of varieties; their construction turns out to coincide with the ring of splines.
In this talk, we describe how to generalize the construction of splines to a more natural geometric and combinatorial setting, starting from a commutative ring and a graph G. We'll also show how powerful this construction can be, building bases for families of generalized splines and even doing Schubert calculus in some instances. We'll end with a number of open questions with connections to Schubert calculus, geometric representation theory, and approximation theory
Mike Zabrocki (York):
The non-commutative symmetric functions and quasi-symmetric functions are the second and third examples of a combinatorial Hopf algebra that one encounters (the first being the symmetric functions). In recent years there have been at least two bases proposed as analogues of the Schur functions and they are in addition to the "ribbon=fundamental^*" basis. I´ll list properties that we would want these bases to have as analogues of the Schur functions and then explain some computational results that tell us what is possible (surprisingly, it is not possible to have it all!). I will also discuss some symmetric function positivity open problems that we hope these bases will resolve.
This is joint work with Laura Colmenarejo.