Fixed Points and Iterative Solutions of Equations

Abstract
This lecture is designed as part of a first-year calculus course and immediately follows the mean value theorem. Situated immediately after the mean value theorem, this lecture begins by exploring repeated cosine numerically and graphically. This motivates fixed points and a 1D version of the Banach Fixed-Point Theorem. I will prove the theorem with an error estimate, examine a simple counterexample, and conclude with its broader
significance as a general method for solving equations by iteration.

Biography
Dr Shao received his Ph.D. degree in mathematics from MIT under the supervision of Gigliola Staffilani. He then served as L.E. Dickson instructor at the mathematics department, University of Chicago. He is currently a postdoc at HES. His research works focus on harmonic analysis and its application to evolutionary partial
differential equations arising from physical backgrounds, including Hamiltonian dynamics and fluid mechanics.