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The goal of this course is to give a modern introduction to mathematical methods for solving hard mathematics problems that arise in the sciences. The main focus will be to explain the process of applied mathematics, namely how to take a hard problem, of the type ordinarily encountered in applications, and gain insight into its important features. Applied Mathematics is a no-holds-barred competition, in which one uses all available tools to understand a problem as much as possible. The approach requires a combination of (a) “real” mathematics, comprised of theorems and exact results; (b) courage and skill in making legitimate approximations; and (c) intelligent use of computers to both verify and extend the validity of the approximations. Theory, Approximate techniques, and Numerical methods will be taught as needed to solve the problems at hand. We will discuss these methods in the context of mathematics problems that arise in a variety of fields, ranging from pure mathematics (e.g... the zeros of the Riemann zeta function), to optics (e.g.. the colors of the rainbow), to quantum mechanics (e.g.. the semi-classical limit), to fluid mechanics. We will start with simple problems (polynomial equations, simple integrals and simple differential equations) and end the quarter with a study of nonlinear partial differential equations. We will try to convince you that one can understand quantitative features of arbitrarily hard mathematics problems, by intelligently combining all of the resources (computational and analytical) that you have at your disposal.
