Brief Syllabus for MATH 648M: Advanced Analytic Methods with Applications

  • Prerequisite:

    • None. The course is largely self-contained. Some knowledge of complex-variable theory and differential equations is assumed. (The UMD courses MATH 414, MATH 462, or MATH 463, for example, would suffice, but they are not required. Handouts with reviews will be given in class.) Ask the instructor if in doubt !

  • Focus:

    • Concepts and analytic techniques that permeate various scientific disciplines; and therefore are useful for formulating and solving problems in the physical sciences and mathematics. Emphasis will be placed on:
    (i) asymptotics, especially ODEs and PDEs (boundary layers, homogenization): approximate solutions of problems
    that have small (or large) parameters or variables; and
    (ii) stochastic methods, especially aspects of Brownian motion and model reduction/renormalization.
    The course material will be based on both theory and worked examples from various disciplines, including: fluid dynamics, condensed matter physics, materials science, biology, astrophysics, electrical and mechanical engineering, probability.

  • Tentative list of topics:

    • PART I:

    ASYMPTOTICS: THEORY: Introduction to approximations; Laplace's method (briefly). Solutions to differential equations: regular and singular perturbations; boundary layers (extensively); multiple scales; WKB method;
    homogenization method (extensively). Asymptotics for discrete problems and difference equations.
    APPLICATIONS: What is the calcium density in a cell? What is the mass density of a planetary ring? How can one estimate the air flow past an airplane wing? the level of impurities in a transistor? How fast do nanostructures decay on crystal surfaces? What is the motion of an ion in a trap? What is the shape of a laser pulse in a fiber? What can be shown about the zeros of the Riemann zeta function?

    PART II:

    STOCHASTIC METHODS: THEORY: Introduction to probability (briefly); Monte Carlo methods. Brownian motion (extensively); random walks; Langevin and Fokker-Planck equations; stochastic differential equations. Stationary stochastic processes. Methods of statistical mechanics; the renormalization group; Mori-Zwanzig formalism (extensively); fluctuation-dissipation theorem(s); model reduction.
    APPLICATIONS: How do line defects fluctuate on the surface of crystals? How can one solve the heat equation with potential by random walks? What is the inertial spectrum in turbulence? How fast does fuel burn in a nuclear reactor? Can billiard balls have memory?