# Fall 2018 MATH 278 002 LEC

Section | Days/Times | Location | Instructor | Class |
---|---|---|---|---|

002 LEC | TuTh 02:00PM - 03:29PM | Evans 891 | Fraydoun Rezakhanlou | 24672 |

Units | Enrollment Status |
---|---|

4 | Open |

**Prerequisites:** Consent of instructor

**Description:**

In this course we will explore the connection between Hamilton-Jacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to Kolmogorov-Arnold-Moser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. Aubry-Mather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated Hamilton-Jacobi PDE to construct Aubry-Mather invariant measures. Recently there have been several interesting works to understand the connection between Aubry-Mather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with non-convex Hamiltonian functions. In this course, we also explore the connection between Aubry-Mather Theory and the homogenization phenomena for Hamilton-Jacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.

**Office:** Evans 803

**Office Hours:** TT 3:30-4:30

**Required Text:**

**Recommended Reading:**

**Grading:** Letter grade.

**Homework:**

**Course Webpage:**