Suggested Courses for Preparation for Candidacy, Mathematics and
Statistics Ph.D. Programs
(Applicable to students who have passed the Ph.D. written
examination)
A student who wishes to pursue thesis research in algebra/number
theory should complete Math 600, 601, and either 606 or 620.
In addition, the student should take at least one course chosen from
among Math 603, 607, 608, and 621. A typical candidacy oral
examination in algebra consists of a two-lecture
oral report on a research paper or advanced
monograph chosen in consultation with the student's advisor. The report
will typically take the form of two lectures,
each lasting approximately one
hour.
A student who wishes to work in partial differential equations is
advised to take the two basic one-year sequences in differential
equations, MATH 673/674 (PDE) and MATH 670/671 (ODE). Some background
in areas where partial differential equations are applied (e.g.,
physics, differential geometry) and/or other areas of analysis
(especially complex and functional analysis) is also helpful.
A student wishing to work in geometry or topology is advised
to take at least 15 credit hours of course work in topology or
geometry, including at least three of the courses 730, 734, 740, 742
and at least one special topics course. It is also advisable to
attend the weekly Geometry/Topology Seminar.
Representation theory (of real and p-adic Lie groups, quantum
groups, Lie algebras, etc.) is a somewhat interdisciplinary field,
involving a combination of algebra, analysis, and geometry. Students
interested in working in this area should have a basic preparation in
these three areas (at the level of the written qualifying exams), and
are also advised to take MATH 744/745 (Lie groups) and MATH 636
(representation theory). Courses in number theory, algebraic
geometry, or functional analysis may also be useful.
To prepare for work in numerical analysis, a student should
take AMSC 666/667 and at least two of the courses
AMSC 600, 607, 610, 612, 614, or 660/661.
A student who wishes to pursue thesis research in
real/functional analysis is advised to complete MATH 630, 631, and 660, or
the equivalent, plus two of the courses MATH 632 (functional
analysis), MATH 634 (harmonic analysis), and MATH 636 (representation
theory).
The basic probability courses are STAT 600/601 and 650.
Depending on the area of probability one is interested in, courses in
analysis, geometry, or differential equations may also be useful.
To prepare to do research in mathematical logic, a student is advised to
complete MATH 712/713, MATH 710, and MATH 715, and to participate
regularly in the Logic Seminar.
A student who wishes to pursue thesis research in ODE/Dynamical
systems should complete MATH 630, 631 and 670 or the
equivalent, plus two of the
following: MATH 671, 642, 643, 730. Such a student is also advised
to attend the Dynamics Seminar and/or Applied Dynamics Seminar on a regular
basis.
Students wishing to specialize in complex analysis
should pass the written qualifying exam at the PhD level
in analysis, and should also complete MATH 660/661 and
at least one reading course or special topics course in
complex analysis. Complex analysis is not so much a single
field as a point of view impinging on many areas of mathematics
(dynamics, geometry, number theory, PDE, harmonic analysis,
probability, functional analysis, etc.), and it is perfectly
possible to use complex analysis in diverse areas of mathematics
without being a specialist in complex analysis.
Students interested in work in statistics (this especially
applies to students in the STAT program) are advised to begin with the
basic sequences STAT 600/601, STAT 700/701, and STAT 740/741,
STAT 440, and STAT 650, and are
also urged to attend the Statistics Workshop and Statistics Seminar.
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