APPLIED MATHEMATICS AND COMPUTATIONAL
SCIENCE FIRST YEAR GRAD COURSES
{AMCS}
AMCS 601-602. Algebraic Techniques for Applied
Mathematics and Computational Science.
Staff. First semester: We begin with an introduction
to group
theory. The emphasis is on groups as symmetries and transformations of
space. After an introduction to abstract groups and the basic facts
about finite groups, we turn our attention to compact Lie groups and
their representations. In the latter connection we explore the
connections between orthogonal polynomials, classical
transcendental
functions and group representations. This unit is completed with
a
discussion of finite groups and their applications in coding
theory.
Second semester: We turn to linear algebra and the structural
properties of linear systems of equations relevant to their numerical
solution. In this context we introduce eigenvalues and the
spectral
theory of matrices. Methods appropriate to the numerical solution
of
very large systems are discussed. We then turn to the problem of
solving systems of polynomial equations, introducing basic properties
of rings, ideals and modules. This allows us to define Grobner bases
and their use in the numerical solution of algebraic equations.
The
theoretical content of this course is illustrated and supplemented
throughout the year with substantial computational examples and
assignments.
STAT 531-531/MATH 546-547. Probability,
and Stochastic Processes for Applied Mathematics and Computational
Science.
Staff. Probability and statistics are the language most
appropriate to
the discussion of experimental data. This course introduces and places
on a firm foundation, methods and ideas arising in probability theory
and the theory of stochastic processes, and their applications to
problems of empirical science. First semester: Basic concepts of
measure theory, as measurement theory, probability theory, random
variables, expectation, and independence, the weak and strong laws of
large numbers, the central and Poisson limit theorems. Second semester:
Measure theory revisited, conditional expectations, Martingales,
Brownian motion, diffusion processes and stochastic integration and
differential equations. Applications to the analysis of partial
differential equations.
AMCS 608-609. Analytic
techniques for Applied Mathematics and Computational Science.
Staff. This course covers topics from real analysis, complex analysis,
and functional analysis and their applications to the analysis of
solutions of ODES and PDES. First semester: We begin with a review of
methods from real analysis with an emphasis on Fourier analysis,
approximation theory, the solution of ordinary differential equations,
the method of stationary phase, and other asymptotic analytic
techniques. We then consider tools from complex analysis, with an
emphasis on analytic continuation, and the saddle point method.
We
then turn to analysis in infinite dimensional spaces. After covering
the basic results on complete normed linear spaces, we discuss the
Riesz-Fredholm theory of integral equations. Second semester: We
consider Hilbert space, unbounded operators, self adjointness and the
spectral theorem, with applications to ordinary differential
equations. After a discussion of the basic Sobolev and Holder
spaces
we show how to use classical and functional analytic techniques to
solve partial differential equations. The course concludes with a
discussion of numerical techniques for the solution of ODES and PDES
and their mathematical foundations.
CIS 502: Analysis
of Alogorithms
Prerequisite:
CIT 594 or equivalent. An investigation of several major algorithms
and their uses in areas including list manipulation, sorting,
searching, and graph manipulation. Efficiency and complexity of
algorithms. Complexity classes.