Math 412, Spring 2010

## SPRING, 2011

Instructor: Konstantina Trivisa:
Office: 3307 Mathematics Bldg
Phone: (301) 405-6865
Office Hours: (Spring 2011) Monday 10:30-12PM or by appointment.
Email: trivisa@math.umd.edu
Course website: http://www-users.math.umd.edu/~trivisa/math648M_S11.html

SCOPE AND TOPICS:
The course encompasses concepts and analytic techniques that:
(i) help formulate, solve and analyze two-scale or multiple-scale problems; and
(ii) permeate various scientific disciplines.
Applications and examples are intended to span fluid dynamics, condensed matter physics, materials science, electrical and mechanical engineering, biology, astrophysics, applied probability, and number theory. There are two main parts/groups of topics:

PART I: Asymptotics.
This part includes methods for problems with large (or small) parameters or variables (in ODEs and PDEs). A tentative list of topics follows.

Theory:

• Introduction and motivation.
• Laplace's method for integrals;definition of asymptotic expansion.
• Approximate solutions of ODEs and PDEs: Singular points.
• WKB method: Zeroth and first orders; turning points; quantum tunneling; diffraction.
• Perturbation theory: Regular perturbation; singular perturbation and boundary layers (extensively): plate in a fluid; low-viscosity flow; shocks; crystal facet evolution.
• Nonlinear oscillations: Renormalized perturbation; introduction to the Renormalization Group; nonlinear oscillations.
• Homogenization method: elliptic PDEs; parabolic PDEs; layered materials; porous media.

Applications I (partly in homeworks):

• 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 tools.
This part includes methods that stem from probability theory and help tackle problems of stochastic nature. A tentative list of topics follows.

Theory:
• Introduction to probability (briefly).
• Monte Carlo methods.
• Brownian motion and random walks (extensively).
• Langevin and Fokker-Planck equations.
• Introduction to stochastic differential equations.
• Stationary stochastic processes.
• Methods of statistical mechanics:
• Application of the Renormalization Group.
• Mori-Zwanzig formalism (extensively); fluctuation-dissipation theorem(s); model reduction; averaging and memory in dynamical systems.

Applications II (partly in homeworks):
• How do line defects fluctuate on crystal surfaces?
• 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?
• How are the zeros of the Riemann zeta function related to Random Matrices and approach to equilibrium?

Prerequisites:
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 whenever appropriate. If you are in any doubt about your background, please consult with the instructor.

Class Times: Tuesday and Thursday: 11:00am - 12:15pm.
Location: MTH 0409

Handouts: Many handouts that were used in the course in previous years can be downloaded from the course webpage. Some of these handouts are summaries of class lectures. Others are supplementary material. These notes were written by Prof. Margetis and by students that were taking the class. They are based on the books listed in the suggested bibliography.

• Handouts

Textbook: There are no required books. A few recommended texts can be found in the suggested Bibliography. At times, we will closely follow the book by Holmes, and the book by Chorin & Hald.

• Bibliography

Homework policy: It is your responsibility to check, download, solve, and submit the homework assignments on time. The due date for each assignment will be listed on the webpage.

• Homework

Grading policy: NO exams or tests. Grades will be based exclusively on problem sets that will be distributed on the web.