Seminar

Mike Stillman: "Kuramoto oscillators: dynamical systems meet algebra"

Abstract: Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields.  They appear in nature: fireflies flashing in sync with each other is one fun situation.



In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneos Kuramoto systems (we will define these notions!).  They are determined from a finite graph.  In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations.  We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems.  We will see how

computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions.



Note: we assume essentially NO dynamical systems in this talk!  The parts of the talk that are new represent joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend.

Craig Huneke: "Number of Generators of Licci Ideals"

Abstract: This talk will discuss a somewhat surprising conjectured bound on the number of generators of a licci (in the linkage class of a complete intersection) ideal, namely that the number of generators of a homogeneous licci ideal is bounded above by the greatest last twist in a minimal graded free resolution of the ideal. This is continuing joint work with Claudia Polini and Bernd Ulrich.



We will give a brief introduction to licci ideals, discuss where this conjectured bound comes from, why it is useful, and then describe various cases in which we have been able to prove the bound, as time permits. These cases include monomial licci ideals of finite colength, ideals with a maximal regular sequence of quadrics, and licci ideals with nearly pure resolutions.  In the non-homogeneous case, we state a related conjecture and introduce new classes of licci ideals.



The techniques used are varied, including Golod rings, Boij-Soederberg theory, and the Eisenbud-Green-Harris conjecture.