# Graduate Colloquium Fall 2007

**October 25, 2007**

**When are two real numbers equal?**

**When are two real numbers equal?**

**Marty Weissman **

How can you tell if two real numbers are equal to each other? There are many ways in which the same real number can arise -- by algebraic operations and by computing definite integrals, for example. So, given two such real numbers, can one algorithmically decide whether they are equal? We'll discuss this problem in the context of algebra and calculus, and a related conjecture of Kontsevich and Zagier.

**November 8, 2007**

**Geometrical and Topological methods in N-body problem(s)**

**Geometrical and Topological methods in N-body problem(s)**

**Richard Montgomery **

I will try to describe how and why I do mathematics by focussing on three problems which I've made some contributions to: How much does a rigid body rotate? How does a cat twist in order to land on its feet? Are stutters important in the three-body problem? My contributions to better understanding the first two problems used small bits of Lie group theory, and the idea of a connection for a principal bundle. Motivating the third problem uses a bit of homotopy theory. The underlying viewpoint connecting the three problems is a differential geometric one, hence `geometric mechanics'.

**November 15, 2007**

**Number Theory and Physics**

**Number Theory and Physics**

**Geoff Mason**

While they may seem to be at opposite ends of the mathematical spectrum, parts of number theory and theoretical physics have come to significantly influence each other in recent years. This talk will be a modest introduction to this trend. No specific knowledge of either number theory or physics will be assumed.

**November 29, 2007**

**Are Partial Differential Equations fun?**

**Are Partial Differential Equations fun?**

**Maria Schonbek **

If PDE's are fun or not, is a very personal question. In my case I think they are. I will try to tell you why.

The most attractive feature about PDE's is that many of them model real physical phenomena. It is true that when we look at these models, many times we have to water them down to get results :-). Nevertheless many of the results obtained are the basis of progress in other sciences.

The models I want to talk about come from Fluid Dynamics. I will discuss a bit about Conservation Laws, use them to introduce the concept of weak solutions to PDE's. I will show you how to handle some very simple ones. I will also try to explain what differences will appear in the solutions when we go from linear to non-linear equations.

The main system I want to consider are the incompressible Navier-Stokes equations, they model the evolution of motion for the velocity vector of a Newtonian fluid. These are nonlinear equations, which have as underlying linear part the heat equations. I will show that the solutions to this system of equations at large time will behave close to the solutions without the non-linearity. The main tool I will be using for this last part is the Fourier Transform.

What I hope is that you will interrupt with lots of questions so that we might all learn something.