Peter Wong

Bates College (USA)

Program

During the first half of the twentieth century, the foundations of algebraic topology were laid and the subject dominated the development of modern mathematics. Many basic notions and tools of algebraic topology were developed, partially due to the need for new techniques to solve problems in fixed point theory. In the early 1920's, S. Lefschetz established the field of topological fixed point theory by proving his celebrated Fixed Point Theorem. Subsequently, J. Nielsen, H. Hopf, K. Reidemeister, F. Wecken, W. Franz and others further advanced the subject. Nearly a century later, topological fixed point theory continues to play an important role in applications to dynamics and non-linear analysis, among others. Furthermore, other branches of mathematics have found connections with topological fixed point theory.

These lectures present a brief introduction to topological fixed point theory. First, we present the elements of fixed point theory leading up to the celebrated Lefschetz-Hopf Fixed Point Theorem. The classical Borsuk-Ulam Theorem and its many variations will be discussed. The theory of fixed point classes, also known as Nielsen fixed point theory will be presented, followed by some specific calculations of the Nielsen number. We conclude the mini-course with several natural generalizations of fixed point theory, including coincidence theory, root theory, equivariant fixed point and root theories, and their applications.

O curso será oferecido em português.