
Current
Course
Offerings
Fall
2004
MATH
4315:
GRAPH
THEORY
WITH
APPLICATIONS
(Section
10139) 
Time: 
1:002:00
pm,
MWF,
154F

Instructor: 
S.
Fajtlowicz 
Prerequisites: 
Discrete
Mathematics. 
Text(s): 
The
course
will
be
based
on
the
instructor's
notes. 
Description: 
Planar
graphs
and
the
FourColor
Theorem.
Trivalent
planar
graphs
with
applications
to
fullerness

new
forms
of
carbon.
Algorithms
for
Eulerian
and
Hamiltonian
tours.
Erdos'
probabilistic
method
with
applications
to
Ramsey
Theory
and
,
if
time
permits,
network
flows
algorithms
with
applications
to
transportation
and
job
assigning
problems,
or
selected
problems
about
trees. 
MATH
4331:
INTRODUCTION
TO
REAL
ANALYSIS
(Section
10140) 
Time: 
9:0010:00
am,
MWF,
347
PHG

Instructor: 
S.
Ji 
Prerequisites: 
Math
3333.

Text(s): 
Principles
of
Mathematical
Analysis,
Walter
Rudin,
McGrawHill,
Latest
Edition
(required).

Description: 
Elements
of
topology;
sequences
and
series;
continuity,
differentiability
and
integrations
of
functions
of
one
and
several
variables;
the
inverse
function
theorem
and
other
fundamental
results.
Rigorous
proofs
are
an
essential
part
of
this
course.

MATH
4350:
DIFFERENTIAL
GEOMETRY
(Section
12122) 
Time: 
10:0011:00
am,
MWF,
345
PGH

Instructor: 
M.
Ru 
Prerequisites: 
Math
2433
(Calculus
of
Functions
of
Several
Variables)
and
Math
2431
(Linear
Algebra).

Text(s): 
Differential
Geometry
of
Curves
and
Surfaces
by
Manfredo
Do
Carmo
(publisher:
Prentice
Hall)

Description: 
This
yearlong
course
will
introduce
the
theory
of
the
geometry
of
courves
and
surfaces
in
threedimensional
space
using
calculus
techniques,
exhibiting
the
interplay
between
local
and
global
quantities.
Topics
include:
curves
in
the
plane
and
in
space,
global
properties
of
curves,
surfaces
in
three
dimensions,
the
first
fundamental
form,
curvature
of
surfaces,
Gaussian
curvature
and
the
Gaussean
map,
geodesics,
minimal
surfaces,
Gauss'
Theorem
Egrigium.
We
hope
that
we
can
cover
up
to
Chapter
3
in
the
first
semester.
In
the
second
semester,
we
will
cover
Chapter
4
and
Chapter
5.

MATH
4364:
NUMERICAL
ANALYSIS
(Section
10142) 
Time: 
4:005:30
pm,
MW,
345
PGH

Instructor: 
T.
Pan 
Prerequisites: 
Math
2331
(Linear
Algebra),
Math
3331
(Differential
Equations).
Ability
to
do
computer
assignments
in
one
of
the
following:
FORTRAN,
C,
Matlab,
Mathematica,
and
Maple.
But
the
publisher
provides
programs
in
Matlab.

Text(s): 
Elementary
Numerical
Analysis
(Third
edition),
K.E.
Atkinson
and
Weimin
Han,
Wiley,
2003.

Description: 
We
will
develop
and
analyze
numerical
methods
for
approximating
the
solutions
of
common
mathematical
problems.
The
emphasis
this
semester
will
be
on
solving
nonlinear
equations,
interpolation,
numerical
integration,
initial
value
problems
of
ordinary
differential
equations,
and
numerical
methods
for
solving
linear
systems
of
algebraic
equations.
This
is
an
introductory
course
and
will
be
a
mix
of
mathematics
and
computing.
Remarks:
This
is
a
first
semester
of
a
two
semester
course.

MATH
4370:
MATHEMATICS
OF
FINANCIAL
DERIVATIVES
(Section
10143) 
Time: 
10:0011:30
TTH,
350
PGH

Instructor: 
E.
Kao 
Prerequisites: 
Math
3338,
3339,
or
equivalent
background
in
probability
and
statistics.

Text(s): 
Derivative
Securities,
the
second
edition,
by
Robert
Jarrow
and
Stuart
Turnbull,
SouthWestern
College
Publishing.

Description: 
The
course
is
an
introduction
to
financial
derivatives.
We
study
the
roles
played
by
options,
futures,
forwards,
and
swaps
in
risk
management.
We
introduce
the
notions
of
geometric
Brownian
motion,
riskneutral
pricing,
binomial
models,
and
martingales.
We
will
also
also
examine
interest
rate
contracts,
the
HJM
Model,and
nonstandard
options.

MATH
4377:
ADVANCED
LINEAR
ALGEBRA
(Section
10144) 
Time: 
2:304:00
pm,
TTH,
347
PGH

Instructor: 
J.
Johnson 
Prerequisites: 
Math
2431
and
minimum
3
hours
of
3000
level
math. 
Text(s): 
Linear
Algebra,
K.
Hoffman
and
R.
Kunze,
2nd
Edition,
PrenticeHall. 
Description: 
Syllabus:
Topics
to
be
covered
in
this
course
include
linear
equations,
vector
spaces,
polynomials,
linear
transformations
and
matrices.

MATH
4383:
Number
Theory
(Section
12123) 
Time: 
10:0011:30
am,
TTH,
127
SR

Instructor: 
J.
Hardy 
Prerequisites: 
Math
3330.
or
equivalent.

Text(s): 
To
be
determined.

Description: 
This
course
covers
most
of
the
material
on
classical
number
theory
that
a
mathematics
major/minor
ought
to
know.
Topics
will
include
divisibility
and
factorization,
congruences,
arithmetic
functions,
primitive
roots,
quadratic
residues
and
the
Law
of
Quadratic
Reciprocity,
Diophantine
equations,
and
other
topics
as
time
permits.

MATH
4385:
MATHEMATICAL
STATISTICS
(Section
12124
)

Time: 
4:005:30
pm,
TTH,
309
PGH

Instructor: 
C.
Peters 
Prerequisites: 
Math
3339,
or
the
equivalent.

Text(s): 
Introduction
to
Linear
Regression
Analysis,
3rd
Ed.,
by
Montgomery,
Vining,
and
Peck,
Wiley
2001.

Description: 
Multiple
linear
regression,
and
linear
models,
regression
diagnostics,
model
selections,
other
topics
as
time
permits.
Projects
using
data
analysis
software.

Math
5397:
Analysis
(OnLine
course)
(Section
12222) 
Time: 
On
Line 
Instructor: 
G.
Etgen 
Prerequisites: 
Consent
of
instructor.

Text(s): 
Calculus,
Michael
Spivak,
Publisher:
Pulish
or
Perish.

Description: 
A
survey
of
the
concepts
of
limit,
continuity,
differentiation
and
integration
for
functions
of
one
variable
and
functions
of
several
variables;
selected
applications
are
used
to
motivate
and
to
illustrate
the
concepts.

Math
5397:
PROBABILITY
(OnLine
course)
(Section
12221) 
Time: 
On
Line 
Instructor: 
C.
Peters 
Prerequisites: 
Math
5331
or
consent
of
instructor.

Text(s): 
Concepts
in
Probability
and
Stochastic
Modeling
,
by
James
J.
Higgins
&
Sallie
KellerMcNulty,
Duxbury
1995.

Description: 
Probability,
random
variables,
distributions,
Markov
chains,
counting
processes,
continuous
time
processes.

Math
5397:
Abstract
Algebra
(OnLine
course)
(Section
12223) 
Time: 
On
Line 
Instructor: 
K.
Kaiser 
Prerequisites: 
3330
or
consent
of
instructor.

Text(s): 
Abstract
Algebra:
A
First
Course
by
Dan
Saracino,
Waveland
Press,
Incorporated,
Hardcover,
ISBN:
0881336653
/
0881336653

Description: 
The
basic
elements
of
groups,
rings
and
fields
will
be
covered
with
special
emphasis
on
divisibility
theory
for
rings.

Math
5397:
Graph
Theory
with
Application
(OnLine
course)
(Section
12324) 
Time: 
On
Line 
Instructor: 
S.
Fajtlowicz 
Prerequisites: 
Graduate
standing
or
consent
of
the
instructor.
Graduate
standing
in
engineering
departments
is
also
enough.

Text(s): 
No
textbook

Description: 
Participants
of
this
course
will
study
Texas
style
the
basics
of
graph
theory
by
exclusively
working
on
conjectures
of
the
computer
program
Graffiti.
A
version
of
Graffiti
will
be
available
for
individual
use,
so
that
students
can
learn
or
expand
their
knowledge
of
several
subjects
of
their
own
choice,
including:
trees,
planar
graphs,
independence
and
matching
theory,
network
flows,
chemical
graphs,
Ramsey
Theory
and
eigenvalues
of
graphs.
More
information
about
the
program
is
available
on
the
web
pages
of
the
instructor,
and
Craig
Larson.
One
significant
difference
between
the
Texas
(the
method
developed
by
the
UT
Professor
R.
L.
Moore)
style,
and
what
we
refer
to
as
the
Red
Burton
style,
is
that
rather
than
to
be
led
to
the
rediscovery
of
known
results,
the
participants
will
work
exclusively
on
conjectures
of
selected
versions
of
Graffiti,
without
getting
any
hints
whether
these
conjectures
are
true
or
false.
Another
difference
is
that
unlike
in
traditional
Texas
style
courses
the
participants
will
be
allowed,
to
read
textbooks
and
even
solutions
of
previous
conjectures
of
Graffiti,
because
the
problems
they
will
encounter
are
unlikely
to
be
found
in
textbooks
anyway.
This
will
create
a
more
realistic
setting
for
acquisition
of
research
experience.
Active
participants
will
have
an
opportunity
to
discover
new
original
results.
That
does
not
mean
that
the
course
will
be
more
difficult
than
other
math
classes.
The
only
prerequisites
are
graduate
standing
in
the
College
of
Natural
Sciences
and
Mathematics
or
consent
of
the
instructor.
One
advantage
of
running
Graffiti
individually,
is
that
the
difficulty
of
conjectures
can
be
tailored
to
a
preferred
level
of
users,
presumably
making
the
class
actually
easier.
The
course
will
be
conducted
by
email
and
a
discussion
list.

MATH
6302:
MODERN
ALGEBRA
(Section
10160) 
Time: 
10:0011:30
am,
TTH,
345
PGH

Instructor: 
J.
Hausen 
Prerequisites: 
MATH
3330
(Abstract
Algebra)
or
equivalent.

Text(s): 
W.
J.
Wickless,
A
FIRST
GRADUATE
COURSE
IN
ABSTRACT
ALGEBRA,
Marcel
Dekker,
Inc.,
New

Description: 
This
is
a
twosemester
course
on
Abstract
Algebra.
It
is
anticipated
that
most
of
the
first
four
chapters
(Groups,
Rings,
Modules,
(infinite
dimensional)
Vector
Spaces)
will
be
covered
in
the
fall
and
chapters
five
and
six
(Fields
and
Galois
Theory,
Topics
in
Noncommutative
Rings)
in
the
spring.
Additional
topics
as
time
permits.
Homework
will
be
an
integral
part
of
the
course.

MATH
6304:
THEORY
OF
MATRICES
(Section
12125) 
Time: 
10:0011:00
am,
MWF,
314
PGH

Instructor: 
V.
Paulsen 
Prerequisites: 
Math
4377
and
4331
or
Math
6377.

Text(s): 
Matrix
Analysis,
Horn
and
Johnson,
Cambridge
University
Press
NOTE:
This
book
is
available
in
paperback.

Description: 
We
will
present
topics
in
linear
algebra
and
matrix
theory
that
have
proven
to
be
important
in
analysis
and
applied
mathematics.
We
assume
that
the
student
is
familiar
with
standard
concepts
and
results
from
linear
algebra
and
basic
analysis.
We
will
study
canonical
factorizations
of
matrices,
including
the
QR,
triangular
and
Cholesky
factorizations.
We
will
develop
ways
to
acheive
the
Jordan
canonical
form.
We
will
study
eigenvalue
perturbation
and
estimation
results
and
we
will
study
special
families
of
matrices
such
as
positive
definite,
Hermitian,
Hankel,
Toeplitz.
Matrix
analysis
is
in
a
sense
an
approach
to
linear
algebra
that
is
willing
to
use
concepts
from
analysis,
such
as
limits,
continuity
and
power
series
to
get
results
in
linear
algebra.

MATH
6320:
FUNCTIONS
OF
A
REAL
VARIABLE
(Section
10191) 
Time: 
11:0012:00
am,
MWF,
315
PGH

Instructor: 
M.
Friedberg

Prerequisites: 
Math
4331;
4332
or
consent
of
instructor

Text(s): 
Real
Analysis,
3nd
Ed.,
H.L.
Royden,
Prentice
Hall.

Description: 
Lebesgue
Measure
and
Integration,
functions
of
bounded
variabtion,
obsolute
continuity,
the
classical
L_{p}
spaces,
general
measure
theory.

MATH
6324:
ORDINARY
DIFFERENTIAL
EQUATION
(Section
12126) 
Time:

12:001:00
pm,
MWF,
345
PGH

Instructor: 
J.
Morgan

Prerequisites: 
Math
4331
and
a
first
course
in
linear
algebra.

Text(s): 
Differential
Equations,
Dynamical
Systems
and
Linear
Algebra
,
2nd
Edition,
by
Dr.
Morris
W.
Hirsch
,
University
of
Wisconsin,
Madison,
USA
Dr.
Stephen
Smale
,
University
of
California,
Berkeley,
USA,
and
Dr.
Robert
Devaney
,
Boston
University,
Massachussetts,
USA.
ISBN
0123497035
.
Hardback
.
400
Pages
Academic
Press
.
Published
December
2003.

Description: 
This
is
the
first
semester
of
a
two
semester
sequence.
The
topics
from
the
fall
semester
will
include:
 A
review
of
linear
algebra.
 Autonomous
first
order
linear
systems,
steady
states
and
stability.
 An
introduction
to
function
spaces
and
the
contraction
mapping
theorem.
 Well
posedness
for
general
first
order
nonlinear
systems.
 Continuous
dependence
on
initial
data
and
parameters.
 Stability
theory,
and
linearized
stability.
 The
implicit
function
theorem.
 The
stable
manifold
theorem.
 Elementary
bifurcation
theory.

MATH
6342:
Topology
(Section
?) 
Time:

9:0010:00
am,
MWF,
350
PGH

Instructor: 
D.
Blecher

Prerequisites: 
Math
4331
and
Math
4337
or
consent
of
instructor.

Text(s): 
Topology,
A
First
Course,
J.
R.
Munkres,
Second
Edition,
PrenticeHall
Publishers
(not
absolutely
required).

Description: 
This
is
the
first
semester
of
a
twosemester
introductory
graduate
course
in
topology
(the
second
semester
is
largely
devoted
to
algebraic
topology).
This
is
a
central
and
fundamental
course
and
one
which
graduate
students
usually
enjoy
very
much!
This
semester
we
discuss
a
little
set
theory,
the
basic
definitions
of
topology
and
basis,
separation
properties,
compactness,
connectedness,
nets,
continuity,
local
compactness,
Urysohn's
lemma,
Tietze,
the
characterization
of
separable
metric
spaces,
and
basic
constructions
such
as
subspaces,
quotients,
and
products.
The
final
grade
is
aproximately
based
on
a
total
score
of
400
points
consisting
of
homework
(100
points),
a
semester
test
(100
points),
and
a
final
exam
(200
points).
The
instructor
may
change
this
at
his
discretion.

MATH
6360:
APPLICABLE
ANALYSIS
(Section
10195) 
Time: 
11:301:00
TTH,
345
PGH

Instructor: 
R.
Glowinski 
Prerequisites: 
Math
4331
and
4332.
Real
Analysis,
Mathematics
for
Engineers 
Text(s): 
Suggested
Textbook:
K.E.
Atkinson
and
W.Han,
Theoretical
Numerical
Analysis,
SpringerVerlag,
2001
(this
book
contains
a
large
section
on
Applicable
Functional
Analysis).

Description: 
The
main
objective
of
this
course
is
to
provide
the
students
with
mathematical
tools,
which
have
proved
useful
when
addressing
the
solution
of
applied
problems
from
Science
and
Engineering.
Among
the
topics
to
be
addressed
let
us
mention:
 1.
Functional
Spaces
with
a
particular
emphasis
on
Hilbert
spaces
and
the
projector
theorem.
Weak
convergence.
 2.
Minimization
of
functional
in
Hilbert
spaces.
 3.
Iterative
solution
of
linear
and
nonlinear
problems
in
Hilbert
spaces.
 4.
The
LaxMilgram
theorem
and
Galerkin
methods
in
Hilbert
spaces.
 5.
Some
notions
on
the
Theory
of
Distributions.
 6.
Application
to
the
solution
of
variational
problems
from
Mechanics
and
Physics.
 7.
Time
dependent
problems
and
operatorsplitting.
 8.
Constructive
methods
for
linear
and
nonlinear
eigenvalue
problems.
 9.
Boundary
value
problems
and
their
approximation.

MATH
6366:
OPTIMIZATION
(Section
10196) 
Time: 
5:307:00
pm,
MW,
309
PGH

Instructor: 
G.
Auchmuty 
Prerequisites: 
M4332
and
M4377
or
consent
of
instructor.

Text(s): 
Convexity
and
Optimization
in
R^n
.
Leonard
D.
Berkowitz,
Wiley,
2002.

Description: 
This
course
will
cover
the
major
issues
in
the
theory
of
unconstrained
finite
dimensional
optimization,
and
of
nonlinear
and
convex
programming.
We
will
develop
the
theory
of
quadratic
programming
and
the
analysis
of
finitedimensional
convex
sets
and
functions.
Steepest
descent
and
conjugate
gradient
algorithms
will
be
described
and
analyzed.
Lagrangian
methods,
duality
theory
and
saddle
point
methods
will
also
be
treated.

MATH
6370:
NUMERICAL
ANALYSIS
(Section
10197) 
Time: 
4:005:30
pm,
MW,
348
PGH

Instructor: 
J.
He 
Prerequisites: 
Graduate
standing
or
consent
of
instructor.
Students
should
have
had
a
course
in
Linear
Algebra
and
an
introductory
course
in
analysis.
Familiarity
with
Matlab
is
also
required.

Text(s): 
Numerical
Linear
Algebra,
Lloyd
N
.
Trefethen
and
David
Bau,
SIAM,
1997,
ISBN:
0898713617

Description: 
This
is
the
first
semester
of
a
twosemester
course.
The
focus
in
this
semester
will
be
on
numerical
linear
algebra.
A
short
introduction
to
iterative
solution
of
nonlinear
systems
and
numerical
optimization
will
also
be
given.

MATH
6376:
NUMERICAL
LINEAR
ALGEBRA
(Section
12127) 
Time: 
5:307:00
TTH,
348
PGH

Instructor: 
E.
Dean 
Prerequisites: 
Graduate
standing
or
consent
of
the
instructor.

Text(s): 
Iterative
Methods
for
Sparse
Linear
Systems
,
by
Y.
Saad,
(2nd
edition).

Description: 
This
semester
we
will
develop
and
analyze
iterative
methods
for
the
solution
of
large
systems
of
linear
equations.
Some
of
the
topics
to
be
covered
include:
basic
iterative
methods,
conjugate
gradient,
Krylov
subspace
methods
for
nonsymmetric
problems,
and
multigrid
methods.
We
will
also
look
at
eigenvalue
problems
including
the
QR
algorithm,
divideandconquer
technique,
Lanczos
and
Arnoldi
procedures.

MATH
6377:
BASIC
TOOLS
FOR
APPLIED
MATHEMATICS
(Section
10198) 
Time: 
4:005:30
pm,
TTH,
345
PGH

Instructor: 
R.
Sanders 
Prerequisites: 
Second
year
Calculus.
Elementary
Matrix
Theory.
Graduate
standing
or
consent
of
instructor.

Text(s): 
Lecture
notes
will
be
supplied
by
the
instructor.

Description: 
Finite
dimensional
vector
spaces,
linear
operators,
inner
products,
eigenvalues,
metric
spaces
and
norms,
continuity,
differentiation,
integration
of
continuous
functions,
sequences
and
limits,
compactness,
fixedpoint
theorems,
applications
to
initial
value
problems.

MATH
6382:
PROBABILITY
MODELS
AND
MATHEMATICAL
STATISTICS
(Section
10199
) 
Time: 
2:304:00
pm,
TTH,
350
PGH

Instructor: 
M.
Nicol

Prerequisites: 
Math
3334,
Math
3338
and
Math
4377,
or
consent
of
instructor.

Text(s): 
A
First
Course
in
Probability,
Sixth
Edition
by
Sheldon
Ross,
2002,
Prentice
Hall.

Description: 
This
course
is
intended
to
help
students
build
a
solid
foundation
in
probability.
Emphasis
will
be
placed
on
a
thorough
understanding
of
the
basic
concepts
as
well
as
developing
problem
solving
skills.
Topics
covered
include:
axioms
of
probability;
conditional
probability
and
independence;
discrete
and
continuous
random
variables;
main
discrete
and
continuous
probability
distributions
(Bernoulli,
Binomial,
Poisson,
Exponential
etc);
jointly
distributed
random
variables;
conditional
expectation;
moment
generating
function;
classical
limit
theorems
(strong
and
weak
law
of
large
numbers,
central
limit
theorem
etc);
techniques
of
simulation,
including
Monte
Carlo
simulation.

MATH
6397:
DISCRETETIME
MODELS
IN
FINANCE
(Section
10203
) 
Time: 
2:304:00
pm,
TTH,
301
AH

Instructor: 
E.
Kao 
Prerequisites: 
Math
6382,
or
equivalent
background
in
probability. 
Text(s): 
Introduction
to
Mathematical
Finance:
Discrete
Time
Models
,
by
Stanley
R.
Pliska,
Blackwell,
1997.

Description: 
The
course
an
introduction
to
discretetime
models
in
finance.
We
start
with
singleperiod
securities
markets
and
discuss
arbitrage,
riskneutral
probabilities,
complete
and
incomplete
markets.
We
survey
consumption
investment
problems,
mean
variance
portfolio
analysis,
and
equilibrium
models.
These
ideas
are
then
explored
in
multiperiod
settings.
Valuation
of
options,
futures,
and
other
derivatives
on
equities,
currencies,
commodities,
and
fixedincome
securities
will
be
covered
under
discretetime
paradigms.

MATH
6397:
DYNAMICS
(Section
12120
) 
Time: 
10:0011:30
am,
TTH,
348
PGH

Instructor: 
M.
Golubitsky 
Prerequisites: 
ODEs
(MATH
63246325)
including
some
bifurcation
theory,
or
permission
of
the
instructor

Text(s): 
No
required
text.
Reference
texts:
 The
Symmetry
Perspective
(Golubitsky
and
Stewart,
Birkhauser)
 Singularities
and
Groups
in
Bifurcation
Theory
Vol.
II
(Golubitsky,
Stewart,
and
Schaeffer,
Springer)

Description: 
This
course
will
be
a
combination
lecture
and
seminar
course
with
students
expected
to
read
and
present
research
level
papers
on
theory
and
applications
in
pattern
formation
and/or
coupled
cells
sytems.

MATH
6397:
MATHEMATICAL
HEMODYNAMICS
(Section
12119
) 
Time: 
4:005:30
MW,
315
PGH

Instructor: 
S.
Canic

Prerequisites: 
Multivariable
Calculus,
Real
and
Complex
Analysis.

Text(s): 
None
required.
(Texbooks
that
will
be
used:
W.
Strauss's:
"Partial
Differential
Equations"
,
R.
Glowinski:
"Numerical
Methods
for
Fluids
(Part
3)",
Chorin
and
Marsden:
"Mathematical
Introduction
to
Fluid
Mechanics",
Y.C.
Fung:
"Circulation",
Y.C.
Fung:
"Biomechanics:
Mechanical
properties
of
living
tissues."
R.
LeVeques:
"Conservation
Laws",
Research
Papers)

Description: 
Topics
Covered:
 Review
of
basic
linear
PDEs.
 Analysis
of
quasilinear
PDEs
with
concentration
to
hyperbolic
conservation
laws.
 Introduction
to
fundamentals
of
fluid
mechanics
(basic
equations
of
motion:
continuity,
momentum,
energy,
vorticity).
 Incompressible/compressible
flow
examples
(derivation
of
the
incompressible,
viscous
NavierStokes
equations).
 A
brief
introduction
to
Sobolev
spaces.
Fluidstructure
interaction
arising
in
blood
flow
modeling
(effective
models).
 Energy
estimates.
 Special
topics
related
to
the
study
of
blood
flow
through
compliant
blood
vessels.

MATH
7320:
FUNCTIONAL
ANALYSIS
(Section
12118
) 
Time: 
1:002:30
pm,
MW,
350
PGH

Instructor: 
A.
Torok

Prerequisites: 
Math
63206321,
or
consent
of
the
instructor

Text(s): 
A
course
in
functional
analysis
by
John
B.
Conway.
2nd
ed,
New
York:
SpringerVerlag,
c1990.
SERIES
of
Graduate
texts
in
mathematics
;
96.
ISBN
0387972455
(alk.
paper)
Also
notes
will
be
handed
out
in
class.

Description: 
We
will
discuss
Hilbert
spaces,
Banach
spaces
and
topological
vector
spaces,
bounded
linear
operators,
the
basic
principles
(HahnBanach,
Uniform
boundedness,
Open
mapping
theorem)
and
their
consequences.
We
will
continue
with
the
spectral
theory
of
compact
and
Fredholm
operators.
We
intend
to
also
discuss
applications
of
Functional
Analysis
to
PDE's,
or
other
topics
of
interest
for
the
students.

Math
7394:
ALGEBRAIC
ITERATIVE
METHODS
(Section
12117) 
Time: 
1:002:30
pm,
MW,
348
PGH 
Instructor: 
y.
Kuznetsov 
Prerequisites: 
Graduate
course
on
linear
algebra
and
matrix
analysis

Text(s): 
None

Description: 
This
is
an
advanced
course
on
the
general
theory
and
applications
of
the
basic
iterative
methods
for
the
numerical
solution
of
large
scale
systems
of
linear
algebraic
equations.
After
a
brief
introduction
to
matrix
analysis
and
the
theory
of
linear
algebraic
equations
we
consider
the
general
convergence
theory
for
the
stationary
iterative
methods
including
those
which
are
applied
to
systems
with
singular
matrices.
The
basic
part
of
the
course
is
devoted
to
the
iterative
methods
based
on
minimization
of
quadratic
functionals
and
orthogonaliza
tion
ideas:generalized
minimal
residual
(GMRES),preconditioned
gradients
(PCG)
and
preconditioned
Lanczos
(PL)methods.
The
major
sources
of
large
scale
systems
are
mesh
discretizations
of
partial
differential
equations.We
shall
use
some
of
them
to
illustrate
applications
of
iterative
methods
and
preconditioning
techniques.

Math
7396:
NUMERICAL
SOLUTION
OF
LARGE
SCALE
NONLINEAR
ALGEBRAIC
SYSTEM
(Section
12128) 
Time: 
1:002:30
pm,
TTH,
345
PGH

Instructor: 
R.
Hoppe 
Prerequisites: 
Calculus,
Linear
Algebra,
Numerical
Analysis.

Text(s): 
P.
Deuflhard;
Newton
Methods
for
Nonlinear
Problems.
Affine
Invariance
and
Adaptive
Algorithms
.
Springer,
BerlinHeidelbergNew
York,
2004
(ISBN
3540210997)

Description: 
Largescale
nonlinear
algebraic
systems
arise,
for
instance,
from
the
discretization
of
differential
and
integral
equations,
in
the
framework
of
inverse
problems
as
nonlinear
leastsquares
problems,
or
as
optimality
conditions
for
nonlinear
optimization
problems.
We
will
consider
local
and
global
Newton
and
GaussNewton
methods
and
variants
thereof.
Emphasis
will
be
put
on
a
thorough
affine
invariant
convergence
analysis
as
well
as
on
appropriate
damping
strategies
and
monotonicity
tests
for
convergence
monitoring.
Compared
to
traditional
approaches,
the
distinguishing
affine
invariance
concept
leads
to
shorter
and
more
transparent
proofs
and
permits
the
construction
of
adaptive
algorithms.
We
will
also
address
parameter
dependent
nonlinear
problems
and
focus
on
pathfollowing
continuation
methods
for
their
numerical
solution.

*NOTE: Teaching
fellows are
required to
register for
three regularly
scheduled
math courses
for a total
of 9 hours.
Ph.D students
who have passed
their prelim
exam are required
to register
for one regularly
scheduled
math course
and 6 hours
of dissertation.
Archive
of Previous
Course Offerings
2004
Spring
2003
Fall
2003
Spring
2002
Fall
2002
Spring
2001
Fall
2001
Spring
2000
Fall
