MA 7920
Differential Galois Theory
Mathematics
MA 792O Home Page
When and Where
ASemester: Spring 2009
Meeting time: TuTh 3.00-4.15
Meeting place: HA 373
Instructor
Course Objectives
The Galois theory of differential
equations was begun in the 19th Century by Picard and Vessiot and
further developed in the 20th Century by Kolchin and
others. This theory allows one to associate to any linear
differential equation a group of matrices. This group of
matrices turns out to be the collection of matrices satisfying a
certain set of polynomial equations in the entries (it is a linear algebraic group).
Properties of solutions of the equation (e.g. solvability in terms of
special functions) are reflected in properties of this group. In
the last several years this theory has been reworked and
generalized so that it now covers the case of difference equations and
parameterized difference and differential equations as well . We
will present these new theories in this course.
We will begin the course by giving an introduction to the
theory
of linear algebraic groups. This should be of independent interest to
those studying lie groups and lie algebras. We will then plunge
into the Galois theory. After the theory has been developed we will
discuss effective methods for calculating Galois groups and
applications to number theory and hamiltonian systems. There are
many open problems suitable for Ph.D. projects.
Prerequisite
A
knowledge of the basic facts about rings and fields as covered in
MA521 and MA721.
Text
The grade will
be
given on the basis of successful completion of
homework assignements (70%) given approximately every two weeks and a
project (30%). The project will involve giving an oral and written
presentation of a recent paper or chapter from a book which will be
selected by the student subject to the approval of the
instructor. Oral presentations of the projects will take place
during the final
week of classes and during the time scheduled for the final exam:
Thursday April 30, 2009, 1.00-4.00 pm
Academic
Integrity Statement
Students with Disabilities
Reasonable accommodations will be made
for students with verifiable
disabilities. In order to take advantage of available
accomodations, students must register with Disability Services for
Students at 1900 Student Health Center, Campus Box 7509,
515-7653. For more information on NC State's policy on working
with students with disabilities, please see the Academic Accommodations
for Students with Disabilities Regulation (REG02.20.1) and visit thiswebsite.