Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

Differential Equations and Projective Geometry

Robert R. Bruner

After a quick introduction to the projective plane, we show
that extending a differential equation to the projective plane
is a quick and effective way to study the asymptotic behavior
of its solutions. The simplest approach leads to equations
with ugly singularities, so we also show how to use the method
of rescaling time to desingularize them.

The main theorem is as follows:

It is simple to apply and gives information that would otherwise be difficult to extract.

The only real prerequisite will be differentiation and integration of functions of one variable. The approach to differential equations will be qualitative and intuitive, so the talk could also serve as a good introduction to the geometric point of view on differential equations.

The audience will be left with an open ended list of examples and applications to explore.

The main theorem is as follows:

- there are at most
*N+1*possible slopes "at infinity" for the unbounded trajectories, or - there are at most
*N-1*slopes which are omitted: all other slopes "at infinity" actually occur.

It is simple to apply and gives information that would otherwise be difficult to extract.

The only real prerequisite will be differentiation and integration of functions of one variable. The approach to differential equations will be qualitative and intuitive, so the talk could also serve as a good introduction to the geometric point of view on differential equations.

The audience will be left with an open ended list of examples and applications to explore.