| MATH 343 |
|
SPRING 2000
|
Instructor: Christophe (Chris) Golé
Office: Burton 316
Telephone: (585) 3875
Email & www: cgole@math.smith.edu & www.math.smith.edu/~cgole
Office hours: to be set in class
Class meeting: MWF 10 - 10:50, Burton 307
Text: Elementary Classical Analysis, J. Marsden
& M. Hoffman.
Also recommended: Intuitive Topology, V.V. Prasolov (both at
the bookstore)
Prerequisite: MTH 243 or permission of the instructor
Topics covered: The main topic of the class is Topology, as it applies to Analysis. In a first part of the class, we will cover the fundamental aspects of topology (distance; open, close, compact, connected spaces; continuity, limit points and convergence). We will go over examples in both finite (e.g. Euclidean space and manifolds) and infinite dimensions (function spaces). The second part of the class will be organized around the theme of fixed-point theorems. The contraction mapping theorem, in particular, will lead us to proofs of the inverse and implicit function theorems, as well as that of existence of solutions of differential equations. We will also look at the notion of index of (planar) vector fields, which yields proofs of the fundamental theorem of algebra, the Brouwer fixed point theorem, and can be used to estimate the number of rest points of a differential equation on a surface. You may consult the (very) tentative schedule of lectures for more details.
Your in-class participation: An important aspect of the class will be your active role in reading, writing and explaining proofs. Topology is an ideal setting for honing these skills. You will be assigned some proofs from the book to read especially carefully. I will call for volunteers to present these proofs on the board the next class. These should be short presentations of about 5-10 minutes. Allowing for variations depending on enrollment, I will ask that each student give a minimum of 3 presentations in the semester, at a minimum of 1 per month.There will be a couple of quizzes during the semester on these proofs as well, which will count towards the oral participation grade.
Homework: You are strongly encouraged to form groups and submit group homework. However you should make clear arrangements so that each member contributes to each assignment. Assignments will be posted on my website.
Exams: There will be one midterm and one final exam, both take-home. The midterm will consist of two stages: the first stage showing your personal work, the second, a version improved with your group. The grade for the midterm will be some average of both versions. The final will be an individual take home.
Grading:
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Attendance: Attendance to all the classes is very
important. I would appreciate your writing me an email message or calling
me if you cannot make it to a class.
MATH 343 TOPICS IN ANALYSIS SPRING 2000 Instructor: Christophe (Chris) Gol* Office: Burton 316 Telephone: (585) 3875 Email & www: cgole@math.smith.edu & www.math.smith.edu/~cgole Office hours: to be set in class Class meeting: MWF 10 - 10:50, Burton 307 Text: Elementary Classical Analysis, J. Marsden & M. Hoffman. Also recommended: Intuitive Topology, V.V. Prasolov (both at the bookstore) Prerequisite: MTH 243 or permission of the instructor Topics covered: The main topic of the class is Topology, as it applies to Analysis. In a first part of the class, we will cover the fundamental aspects of topology (distance; open, close, compact, connected spaces; continuity, limit points and convergence). We will go over examples in both finite (e.g. Euclidean space and manifolds) and infinite dimensions (function spaces). The second part of the class will be organized around the theme of fixed-point theorems. The contraction mapping theorem, in particular, will lead us to proofs of the inverse and implicit function theorems, as well as that of existence of solutions of differential equations. We will also look at the notion of index of (planar) vector fields, which yields proofs of the fundamental theorem of algebra, the Brouwer fixed point theorem, and can be used to estimate the number of rest points of a differential equation on a surface. You may consult the (very) tentative schedule of lectures for more details. Your in-class participation: An important aspect of the class will be your active role in reading, writing and explaining proofs. Topology is an ideal setting for honing these skills. You will be assigned some proofs to read in the book, or, sometimes, to write yourselves (e.g. as part of your homework assignment). I will call for volunteers to present these proofs on the board the next class. These should be short presentations of about 10 minutes. Allowing for variations depending on enrollment, I will ask that each student give a minimum of 3 presentations, at a minimum of 1 per month.There will be a couple of quizzes during the semester on these proofs as well, which will count towards the oral participation grade. Homework: You are strongly encouraged to form groups and submit group homework. However you should make clear arrangements so that each member contributes to each assignment. Assignments will be posted on www.math.smith.edu/~cgole/343/hw343.html. Exams: There will be one midterm and one final exam, both take-home. The midterm will consist of two stages: the first stage showing your personal work, the second, a version improved with your group. The grade for the midterm will be some average of both versions. The final will be an individual take home. Grading: What When % Oral In class 20 participation Homework Weekly 30 Midterm March 20 - 27 30 Final Due May 5 20 Attendance: Attendance to all the classes is very important. I would appreciate your writing me an email message or calling me if you cannot make it to a class.