I will review integration by the method of Darboux for second-order hyperbolic PDE in the plane, and its natural extension to hyperbolic exterior differential systems (EDS). I will then discuss the extension of this technique to decomposable EDS, as studied by Anderson, Fels and Vassiliou in their groundbreaking paper "Superposition Formulas for Exterior Differential Systems."
Peter Vassiliou (University of Canberra) will be visiting the department this week and next, and this week's Ulam seminar is designed to provide background for Vassiliou's seminar talks next week.
We will discuss two varieties of permutation patterns: classical patterns and the more generally (and more recently) defined mesh patterns. We will analyze completely when a mesh pattern is equivalent to a finite collection of classical patterns, in a sense declaring that the additional data of the mesh was superfluous for these patterns. We will also describe the permutations having the fewest superfluous meshes, and the permutations having the most, enumerating the superfluous meshes in each case.
(Mesh) Patterns Sponsored by the Meyer Fund
Dec. 03, 2013 3pm (MATH 350)
PDE/Analysis
Thomas Ivey
X
Austere submanifolds are a special sub-class of minimal submanifolds in Euclidean space. Harvey and Lawson defined this class, and showed that the conormal bundle of a submanifold M is special Lagrangian if and only if M is austere. Later, Bryant classified austere 3-folds in Euclidean space, as well as linear-algebraic models for austere 4-folds. More classification results in dimension four were obtained by Marianty Ionel and myself, and we have recently generalized the austere condition to submanifolds in complex projective space.
Generalizations of Darboux Integrability