Spring 2012 Course Syllabus - Mathematics 9420.001

Course: Mathematics 9420.001.
Course Title: Topics in Differential Equations: Solvability of PDEs.
Time: 12:30 pm-1:50 pm.
Place: Wachman 617.
Instructor: Mendoza, Gerardo A.
Instructor Office: Wachman 618.
Instructor Email: gmendoza@temple.edu
Instructor Phone: (215) 204 5053.
Course Web Page: http://math.temple.edu/~gmendoza/
Office Hours: by appointment.
Prerequisites: Math 8141 or equivalent or permission from instructor.
Textbook: Notes by the instructor and research articles from the literature.
Course Goals: To gain a research, state-of-the-art level understanding of the issue of solvability of differential (and pseudodifferential operators.
Topics Covered: The topic for this course will be solvability of linear partial (pseudo-)differential operators. Contrary to ordinary differential equations, where there is a very general simple theorem asserting the existence and uniqueness of solutions of such equations under rather mild conditions, the theory for partial differential equations is much more involved, and even rather tame-looking operators (for instance $\partial/\partial x+i x\partial /\partial y$) may not be solvable. The course will cover, in more or less historical order, the Ehrenpreis-Malgrange theorem, Lewy's example (the first example of a non-solvable operator), H\"ormander's theorem on solvability (the first general result), Conditions ($P$) and ($\Psi$) of Nirenberg-Treves, solvability for operators of principal type, some cases of operators with characteristics of varying multiplicity, and special differential complexes. Concepts of microlocal analysis and basic symplectic geometry will be introduced as needed. This course is a gateway to many important open problems in an exciting research area.
Course Grading: Expositions of particular topics chosen from the research literature on the topic.
Exam Dates: n/a.
Attendance Policy: Attendance is required. Missing four or more classes will result in a failing grade.

Any student who has a need for accommodation based on the impact of a disability should contact me privately to discuss the specific situation as soon as possible. Contact Disability Resources and Services at (215) 204-1280, 100 Ritter Annex, to coordinate reasonable accommodations for students with documented disabilities.

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• The first day of classes is Tuesday, January 17.
• The last day to drop/add (tuition refund available) is Monday, January 30.
• Spring recess is the week of Sunday, March 4.
• The last day to withdraw (no refund) is Monday, March 20.
• The last day of classes is Monday, April 30.

During the first two weeks of the fall or spring semester or summer sessions, students may withdraw from a course with no record of the class appearing on the transcript. In weeks three through nine of the fall or spring semester, or during weeks three and four of summer sessions, the student may withdraw with the advisor's permission. The course will be recorded on the transcript with the instructor's notation of "W," indicating that the student withdrew. After week nine of the fall or spring semester, or week four of summer sessions, students may not withdraw from courses. No student may withdraw from more than five courses during the duration of his/her studies to earn a bachelor's degree. A student may not withdraw from the same course more than once. Students who miss the final exam and do not make alternative arrangements before the grades are turned in will be graded F.

The grade I (an "incomplete") is reserved for extreme circumstances. It is necessary to have completed almost all of the course with a passing average and to file an incomplete contract specifying what is left for you to do. To be eligible for an I grade you need a good reason and you should have missed not more than 25% of the first nine weeks of classes. If approved by the Mathematics Department chair and the CST Dean's office, the incomplete contract must include a default grade that will be used in case the I grade is not resolved within 12 months.