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## An old and a new topological tool for nonlinear analysis

### Robert F. Brown

University of California, Los Angeles

###
Thursday, September 29, 2005

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** In the first part of the talk, we will see how
one of the earliest fruits of algebraic topology, the Brouwer Fixed
Point Theorem, leads to the Schauder Fixed Point Theorem, an old
(1930) but still widely used tool for proving the existence of
solutions to problems involving nonlinear differential equations.
Application to a forced pendulum problem and an equilibrium heat
distribution problem will demonstrate the use of Schauder's theorem.
The second part of the talk will discuss how, starting in the late
1980's, the Nielsen number of topological fixed point theory has
been used to establish the existence of multiple solutions to
nonlinear analytic problems. Multiple controllability results in
control theory will illustrate the use of this new topological tool.

This talk will be accessible to graduate students.