## Seminar:

Fall 2009, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** October 15 **

**" Mini-Talk "**

* Prof. Andrew Comech

* Prof. Matthew Papanikolas

* Prof. Andrew Comech

* Prof. Matthew Papanikolas

__Abstract__ * A. Comech*: **"Nonlinear evolution equations: solitary waves and attractors"Abstract**: We will discuss properties of nonlinear solitary waves on the example of a Klein-Gordon (relativistic Schroedinger) equation.

There are solitary wave solutions, sometimes quite easy to construct. Yet, there are some natural questions which are in many cases still not answered:

1. Are these solitary waves stable?

2. Does any finite-energy solution look like a solitary wave for long times ("is attractor formed by solitary waves")?

Instead of taking a true nonlinear equation, we will couple a linear equation to a nonlinear spring and will watch how this mechanical system evolves.

Keywords: Infinite-dimensional Hamiltonian systems, nonlinear dispersive equations, U(1) invariance, solitary waves, orbital stability, global attractors.

*M. Papanikolas*:

**" Modular forms and number theory "**

Abstract: Modular forms are a particular class of complex analytic functions that are remarkably linked to various problems in number theory. This talk will survey some of these interesting connections, including relations among modular forms, partition functions, representations of integers by quadratic forms, and solutions of polynomial equations over finite fields.

Abstract