2011-2012 Bulletin – PDF - SEAS Bulletin - Columbia University

squares method, pseudo-inverses, singular
value decomposition. Adjoint operators,
Hermitian and unitary operators, Fredholm
Alternative Theorem. Fourier series and
eigenfunction expansions. Introduction to the
theory of distributions and the Fourier Integral
Transform. Green’s functions. Application to
Partial Differential Equations.
APMA E4101y Introduction to dynamical
systems
3 pts. Lect: 3. Professor Wiggins.
Prerequisites: APMA E2101 (or MATH V1210)
and APMA E3101 or their equivalents, or
permission of instructor. An introduction to the
analytic and geometric theory of dynamical
systems; basic existence, uniqueness and
parameter dependence of solutions to ordinary
differential equations; constant coefficient and
parametrically forced systems; Fundamental
solutions; resonance; limit points, limit cycles
and classification of flows in the plane (Poincare-
Bendixson Theorem); conservative and
dissipative systems; linear and nonlinear stability
analysis of equilibria and periodic solutions;
stable and unstable manifolds; bifurcations,
e.g., Andronov-Hopf; sensitive dependence and
chaotic dynamics; selected applications.
APMA E4150x Applied functional analysis
3 pts. Lect: 3. Professor Bal.
Prerequisites: Advanced calculus and course
in basic analysis, or instructor’s approval.
Introduction to modern tools in functional
analysis that are used in the analysis of
deterministic and stochastic partial differential
equations and in the analysis of numerical
methods: metric and normed spaces, Banach
space of continuous functions, measurable
spaces, the contraction mapping theorem,
Banach and Hilbert spaces bounded linear
operators on Hilbert spaces and their
spectral decomposition, and time permitting
distributions and Fourier transforms.
APMA E4200x Partial differential equations
3 pts. Lect: 3. Professor Bal.
Prerequisite: Course in ordinary differential
equations. Techniques of solution of partial
differential equations. Separation of the variables.
Orthogonality and characteristic functions,
nonhomogeneous boundary value problems.
Solutions in orthogonal curvilinear coordinate
systems. Applications of Fourier integrals, Fourier
and Laplace transforms. Problems from the fields
of vibrations, heat conduction, electricity, fluid
dynamics, and wave propagation are considered.
APMA E4204x Functions of a complex
variable
3 pts. Lect. 3. Instructor to be announced.
Prerequisite: MATH V1202 or equivalent.
Complex numbers, functions of a complex
variable, differentiation and integration in the
complex plane. Analytic functions, Cauchy
integral theorem and formula, Taylor and Laurent
series, poles and residues, branch points,
evaluation of contour integrals. Conformal
mapping. Schwarz-Christoffel transformation.
Applications to physical problems.
APMA E4300y Introduction to numerical
methods
3 pts. Lect: 3. Instructor to be announced.
Prerequisites: MATH V1201, MATH E1210,
and APMA E3101 or their equivalents. Some
programming experience and MATLAB will be
extremely useful. Introduction to fundamental
algorithms and analysis of numerical methods
commonly used by scientists, mathematicians
and engineers. This course is designed to give
a fundamental understanding of the building
blocks of scientific computing that will be used in
more advanced courses in scientific computing
and numerical methods for PDEs. Topics include
numerical solutions of algebraic systems, linear
least-squares, eigenvalue problems, solution of
non-linear systems, optimization, interpolation,
numerical integration and differentiation, initial
value problems and boundary value problems for
systems of ODEs. All programming exercises will
be in MATLAB.
APMA E4301x Numerical methods for partial
differential equations
3 pts. Lect: 3. Professor Spiegelman.
Prerequisites: APMA E4300 and E3102 or
E4200 or equivalents. Numerical solution of
partial differential equations (PDE) arising
in various physical fields of application.
Finite difference, finite element, and spectral
methods. Elementary finite volume methods
for conservation laws. Time stepping,
method of lines, and simultaneous spacetime
discretization. Direct and iterative
methods for boundary-value problems.
Applied numerical analysis of PDE, including
sources of numerical error and notions of
convergence and stability, to an extent
necessary for successful numerical modeling
of physical phenomena. Applications will
include the Poisson equation, heat equation,
wave equation, and nonlinear equations of
fluid, solid, and gas dynamics. Homework
assignments will involve substantial
programming.
AMCS E4302x Parallel scientific computing
3 pts. Lect: 3. Instructor to be announced.
Prerequisites: APMA E3101, E3102, and
E4300, or their equivalents. Corequisites: APMA
E4301, and programming ability in C/C++ or
FORTRAN/F90. An introduction to the concepts,
the hardware and software environments, and
selected algorithms and applications of parallel
scientific computing, with an emphasis on tightly
coupled computations that are capable of scaling
to thousands of processors. Includes high-level
descriptions of motivating applications and
low-level details of implementation, in order to
expose the algorithmic kernels and the shifting
balances of computation and communication
between them. Students run demonstration
codes provided on a Linux cluster. Modest
programming assignments using MPI and PETSc
culminate in an independent project leading to an
in-class report.
APMA E4400y Introduction to biophysical
modeling
3 pts. Lect: 3. Not offered in 2011–2012.
Prerequisites: PHYS W1401 or equivalent, and
APMA E2101 or MATH E1210 or equivalent.
Introduction to physical and mathematical
models of cellular and molecular biology. Physics
at the cellular scale (viscosity, heat, diffusion,
statistical mechanics). RNA transcription and
regulation of genetic expression. Genetic and
biochemical networks. Bioinformatics as applied
to reverse-engineering of naturally-occurring
networks and to forward-engineering of synthetic
biological networks. Mathematical and physical
aspects of functional genomics.
APMA E4901x Seminar: problems in applied
mathematics
0 pts. Lect: 1. Professor Wiggins.
This course is required for, and can be taken
only by, all applied mathematics majors in
the junior year. Prerequisites or corequisites:
APMA E4200 and E4204 or their equivalents.
Introductory seminars on problems and
techniques in applied mathematics. Typical
topics are nonlinear dynamics, scientific
computation, economics, operations research,
etc.
APMA E4903x Seminar: problems in applied
mathematics
3–4 pts. Lect: 1. Tutorial: 2. Professor Wiggins.
This course is required for all applied
mathematics majors in the senior year.
Prerequisites or corequisites: APMA E4200 and
E4204 or their equivalents. For 4 pts. credit, term
paper required. Examples of problem areas are
nonlinear dynamics, asymptotics, approximation
theory, numerical methods, etc. Approximately
three problem areas are studied per term.
APMA E4990x and y Special topics in applied
mathematics
1–3 pts. Lect: 3. Instructor to be announced.
Prerequisites: Advanced calculus and junior
year applied mathematics, or their equivalents.
This course may be repeated for credit. Topics
and instructors from the Applied Mathematics
Committee and the staff change from year to
year. For advanced undergraduate students
and graduate students in engineering, physical
sciences, biological sciences, and other fields.
APMA E6209x Approximation theory
3 pts. Lect: 2. Not offered in 2011–2012.
Prerequisite: MATH W4061 or some knowledge
of modern analysis. Theory and application
of approximate methods of analysis from the
viewpoint of functional analysis. Approximate
numerical and analytical treatment of linear and
nonlinear algebraic, differential, and integral
equations. Topics include function spaces,
operators in normed and metric spaces, fixed
point theorems and their applications.
APMA E6301y Analytic methods for partial
differential equations
3 pts. Lect: 2. Professor Weinstein.
Prerequisites: Advanced calculus, basic
concepts in analysis, APMA E3101, and
E4200 or their equivalents, or permission of
the instructor. Introduction to analytic theory of
PDEs of fundamental and applied science; wave
(hyperbolic), Laplace and Poisson equations
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engineering 2011–2012