The
PhD
degree
is a
research
degree
and
is granted
only
as a
consequence
of original
research
presented
in the
form
of a
formal
dissertation.
No
specified
course
plan
nor
any
particular
number
of credit
hours
is given
as a
partial
requirement
of the
degree.
It is
the
student's
responsibility
to be
informed
of the
current
regulations
as set
forth
in the
Graduate
Studies
Bulletin
and
the
Bulletin
of the
College
of Natural
Sciences
and
Mathematics.
This
includes
a residency
requirement
of enrollment
in 9
semestercredithours
each
in two
consecutive
long
semesters,
and
a completion
of at
least
24 semestercredithours
in the
PhD
program.
Other
basic
requirements
must
be met
as follows:
 The
student
is
required
to
pass
a
comprehensive
preliminary
examination.
The
examining
committee
shall
consist
of
at
least
four
members,
three
of
whom,
including
the
faculty
advisor,
shall
be
from
the
Department
of
Mathematics.
The
committee
shall
be
appointed
by
the
student's
faculty
advisor
and
the
department
chairman.
The
student
must
attempt
the
examination
within
three
long
semesters
(Fall
and
Spring)
of
fulltime
study
from
the
time
of
admission
to
the
Ph.D.
program
with
the
M.S.
degree
or
its
equivalent.
The
examination
may
be
taken
only
twice.
If
a
student
fails
the
examination
on
the
first
attempt,
a
second
attempt
must
be
made
within
two
long
semesters
of
the
first.
Further
details
are
given
below
in
the
Preliminary
Examinations
section.
 The
student
must
complete
a
minimum
of
one
full
year
of
continuous,
fulltime
residence
after
obtaining
a
Master's
degree
in
mathematics
or
the
equivalent.
Currently,
registration
for
nine
semester
hours'
work,
each
semester
for
two
consecutive
semesters
implies
fulltime
residence.
 As
a
primary
requirement
for
the
degree,
the
student
must
produce,
under
the
guidance
of
a
faculty
advisor,
original,
independent
research
in
mathematics
in
the
form
of
a
formal
dissertation.
The
results
should
be
of
a
sufficiently
scholarly
nature
to
be
considered
publishable
in
the
mathematical
literature.
 The
student
must
defend
the
dissertation
in
a
final
examination
before
a
reading
committee
which
shall
consist
of
at
least
four
members;
three
of
whom,
including
the
faculty
advisor,
shall
be
from
the
Mathematics
Department,
and
one
from
outside
the
Mathematics
Department.
This
committee
must
be
approved
by
the
Chairman
of
the
Mathematics
Department
and
the
Dean
of
the
College
and
the
names
of
the
committee
members
must
be
on
file
in
the
Office
of
the
Dean
at
least
one
semester
prior
to
graduation.
Copies
of
the
dissertation
must
be
in
the
hands
of
the
committee
at
least
three
weeks
prior
to
the
scheduled
date
of
the
final
examination.
One
copy
of
the
dissertation
is
to
be
placed
(at
departmental
expense)
in
the
Mathematics
Department
office
at
the
time
it
is
distributed
to
the
committee.
The
dissertation
committee
will
not
be
assigned
until
the
language
and
preliminary
examinations
have
been
passed.
 The
student
must
complete
twelve
hours
of
Doctoral
Dissertation
(MATH
8399).
The
University
requires
that
a
student
who
registers
for
Doctoral
Dissertation
be
enrolled
for
dissertation
courses
continuously
(Fall
and
Spring)
up
to
and
including
the
semester
in
which
the
dissertation
is
accepted
by
the
University.
 PLEASE
NOTE:
The
Coordinating
Board
has
put
a
uniform
cap
on
doctoral
hours
at
Texas
institutions.
All
resident
doctoral
students
admitted
(starting
Fall
1993
and
after)
will
be
transferred
to
nonresident
tuition
status
as
of
the
semester
following
the
one
in
which
they
exceed
the
99
doctoral
hours
limit.
The
Preliminary
Examination
Background
and
Purpose
The
preliminary
exam
is a
part
of the
PhD
requirement.
There
are
no specific
course
requirements
for
the
PhD
degree.
A strong
background
in a
variety
of subjects
is assured
by the
requirement
that
PhD
applicants
have
the
equivalent
of a
master's
degree
in mathematics,
and
by the
preliminary
examination
requirement.
A student
who
obtains
a MS
in Mathematics
form
UH may,
in fact,
be ready
to take
the
exam
immediately.
Timing
The
student
must
attempt
the
examination
within
three
long
semesters
of fulltime
study
from
the
time
of admission
to the
PhD
program
with
the
MS degree
or its
equivalent.
It may
be taken
only
twice;
if the
student
fails
on the
first
attempt,
a second
attempt
must
be made
within
two
long
semesters
of the
first.
Note
that
for
students
who
continue
into
the
PhD
program
after
obtaining
a MS
here,
the
clock
starts
with
their
first
semester
of enrollment
after
completing
the
work
for
their
MS degree;
students
who
are
admitted
to UH
directly
into
the
PhD
program
begin
counting
with
their
first
semester
here.
Fulltime
teaching
fellows
are
expected
to enroll
for
nine
hours
of bona
fide
coursework
each
long
semester
until
they
take
the
exam.
Parttime
students
will
be expected
to take
the
exam
by the
end
of the
semester
in which
they
complete
27 hours
of coursework
(after
admission
to the
PhD
program).
Administration
of the
Exam
The
examining
committee
consists
of a
chair
(who
may
be the
student's
dissertation
supervisor),
and
at least
three
others;
at least
three
members,
including
the
chair,
must
be faculty
members
of the
Mathematics
department.
The
committee
may
be chosen
by the
student,
but
must
be approved
by the
graduate
studies
committee.
By tradition,
the
format
is oral,
and
consists
of approximately
two
hours
or questioning.
Students
may
request
a written
exam.
Other
members
of the
faculty
may
be present
at the
exam.
Topics
must
include
a yearlong
core
course
from
three
of the
following
five
areas:
algebra,
topology,
analysis,
applied
mathematics,
and
computational
mathematics;
each
topic
is represented
by one
examiner
(not
to include
the
committee
chair);
the
chair
may,
optionally,
question
the
candidate
on an
advanced
or other
agreedupon
topic.
The
level
and
extent
of knowledge
tested
in the
exam
corresponds
to that
acquired
in a
oneyear
graduate
course
in these
areas.
The
following
is a
list
of what
are
generally
accepted
in the
department
at this
time
as suitable
core
courses.
 Algebra
MATH
6302;6303:
Modern
Algebra
 Topology
MATH
6342;6343:
Point
Set
Topology
MATH
6340;6341:
Algebraic
Topology
MATH
7350;7351:
Geometry
of
Manifolds
(Differential
Topology)
 Analysis
MATH
6320;6321:
Theory
of
Functions
of
a
Real
Variable
MATH
6322;6323:
Theory
of
functions
of
a
Complex
Variable
MATH
7320;7321:
Functional
Analysis
 Applied
Mathematics
MATH
6324;6325:
Differential
Equations
MATH
6326;6327:
Partial
Differential
Equations
MATH
6382;6383:
Probability
Models
and
Mathematical
Statistics
 Computational
Mathematics
MATH
6366;6367:
Optimization
and
Variational
Methods
MATH
6370;6371:
Numerical
Analysis
MATH
6374
and
MATH
7374:
Numerical
Partial
Differential
Equations,
and
Mathematical
Theory
of
Finite
Element
Methods.
