Events for 02/02/2015 from all calendars

Groups and Dynamics Seminar

Time: 11:30AM - 12:30PM

Location: BLOC 220

Speaker: Jean Bellissard, Georgia Institute of Technology

Title: Describing Aperiodic Media: Delone sets and the Anderson-Putnam Complex

Abstract: Lecture 1, Mon, Delone sets and the Anderson-Putnam Complex Lecture 2, Tue, C*-algebras and Noncommutative Geometry Approach Lecture 3, Wed, Periodic Approximants, an spectral properties, the 1D-case Lecture 4, Thu, Feb 5, Prospect and discussions

Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Jeanne Clelland, U Colorado

Title: Isometric embedding via strongly symmetric positive systems

Abstract: In this talk, I will give an outline of our new proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated microlocal analysis used in earlier proofs by Bryant-Griffiths-Yang and Nakamura-Maeda; instead, it is based on a new local existence theorem for a class of nonlinear, first-order PDE systems that we call "strongly symmetric positive." These are a subclass of the symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not fall under one of the standard types (elliptic, hyperbolic, and parabolic). As in earlier proofs, we construct solutions via the Nash-Moser implicit function theorem, which requires showing that the linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies "smooth tame estimates." We accomplish this in two steps: (1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive after a carefully chosen change of variables. (2) Show that any such system has local solutions that satisfy smooth tame estimates. The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE systems used in prior proofs, while step (1) requires only linear algebra. The talk will focus on the main ideas of the proof; technical details will be kept to a minimum. This is joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang.

Colloquium - Kyungyong Lee

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Kyungyong Lee, Wayne State University

Description:

Title : Positivity for Cluster Algebras

Abstract :
Cluster algebras were discovered by Fomin and Zelevinsky in 2001. Since then, they have been shown to be related to diverse areas of mathematics and physics such as Total positivity, Quiver representations, String theory, Statistical physics, Non-commutative geometry, Teichm\"uller theory, Hyperbolic geometry, Tropical geometry, KP solitons, Integrable systems, Quantum mechanics, Lie theory, Algebraic combinatorics, Number theory and Poisson geometry.
We introduce cluster algebras and their remarkable properties. Positivity is a central theme in this field. In joint work with Ralf Schiffler, we prove the positivity conjecture. In this talk we briefly sketch an idea of the proof.