This is the first semester of the 5041-42 Ph.D. qualifying exam sequence in differential geometry/differential topology. The first semester will be entirely differential geometry, the second semester mostly algebraic and differential topology.
Time and Place: Monday, Wednesday, Friday, 11:00-12:00 noon, in Room 111, Cupples I
Instructor: Edward
N. Wilson
Office: Cupples I, Room 18 (in the basement)
Tentative Office Hours: MWF 1:00-2:00 and by appointment
Office Tel: 935-6729 (has voice-mail)
E-mail: enwilson@math.wustl.edu
Prerequisites: Hard and fast requirements are courses closely equivalent to Math 4111, Math 429, and Math 417. Very desirable is some knowledge of Lebesgue integration theory, e.g., a course like Math 4121. Those who haven't seen rigorous proofs of the Inverse/Implicit Function Theorem on R^n and the Picard Theorem on existence and uniqueness of solutions of systems of ordinary differential equations are strongly advised to read up on these topics in any good undergraduate analysis text. These two theorems are the basis for many tools and theorems in differential geometry. We'll use them often.
Textbook: Differential Forms and Applications, Manfredo do Carmo
Topic Outline: We'll
go through do Carmo's book fairly sytematically, pausing at one point to
discuss some topics not covered by do Carmo, namely: integral curves and
flows of vector fields (ordinary differential equations on manifolds),
the Frobenius Theorem, and an introduction to Lie groups and Lie algebras.
The hope will be to get through essentially the entire book by the end
of the semester.
Exams/Homework: There will be two take-home exams, one around the middle of the semester and the other during finals week at the end of the semester. Homework (mostly problems from do Carmo) will usually be assigned weekly.
Grading: Final
averages will be determined by the following formula:
Final average = .30E1 + .40E2 + .30
Thus, the first exam
will be 30% of the final average, homework another 30%, and the second
exam 40%. That said, a different formula giving the final exam higher
weight will be used for those who do poorly on the first exam but improve
considerably on the last exam and a revised formula for those who struggle
a bit with exams but excel on homework. Also, the process of converting
final averages to letter grades won't, from a student's point of view,
be any worse than the traditional 90-100 A,
80-90 B, 70-80 C scale
but might well be better, i.e., more generous. There won't be any
such generousity for those who elect to do very little homework.
Academic Integrity:
As
with all Washington University courses, cheating on exams will be
taken very seriously
with evidence supporting a cheating allegation forwarded to the Arts
and Sciences Integrity
Committee for adjudication. When the Committee concludes that a
student cheated on an
exam, it normally directs the instructor to assign the student a
failing grade for the
course.
Cheating on homework consists of either blindly copying off someone else's
solutions
or not acknowledging
the receipt of assistance from others in completing the assignment.
It's not anticipated
that students will work in isolation on homework problems. To the
contrary, discussing
problems with others is often a way to avoid frustration and gain
useful insight. However,
all students are expected to write up their own assignments and
to indicate in a short
note at the top of the first page the names of any people (other
than the instructor)
with whom they discussed the problems or from whom they received some
hints. Violation
of these requests will result in an instructor-imposed penalty (e.g.,
something like half
credit for the assignment) but won't be treated as a "hanging"
offense--in particular,
won't be brought to the attention of the Arts and Sciences
Integrity Committee.