Course Information

Course Booklet Information Page

The course meets in the J+ block, from 3:00 - 4:15pm on Tuesdays and Thursdays, in Room BP-5. My office hours are from 11:00 AM - 12:30 PM on Tuesdays and Fridays, in Room BP-212; I am also available by appointment.

There are two textbooks for this course. The required book is An introduction to partial differential equations by Pinchover and Rubinstein, published by Cambridge University Press. The recommended book is Applied Functional Analysis by Griffel, published by Dover. Both are available from the bookstore or, more cheaply, from amazon.

There will be (roughly) weekly homework assignments for this course, a midterm, and a final exam.

Homework will account for 40% of your final grade, with each assignment weighted equally. Typically, homework assignments will be distributed in Tuesday's class, and collected the following Thursday (i.e., 9 days later) in class. Late homework will be penalized by 10% per day late - the only exception to this is that you are permitted two "freebies" in this system, that can be used to excuse a single day's lateness on a single assignment each. (These may be compounded to hand in one assignment two days late.) You are permitted (and, generally, expected) to discuss the assignments with others in the class, but must hand in your own work individually.

Both the midterm and the final will consist of two components, a "take-home" exam and an oral examination. The take-home components will be directly based on the material covered in class and on homework assignments. Written solutions will be due in class for the midterm and at a fixed time for the final, with no late submissions accepted. The subsequent oral examinations will be up to one hour in length, scheduled at a mutually acceptable time, where we will discuss your answers to the problems from the take-home component. No notes will be allowed for the oral examinations. The midterm exam will be worth 20% of your final grade (10% written and 10% oral), while the final exam will be worth 40% of your final grade (20% written and 20% oral). Again, you are permitted (and expected) to discuss the solutions with take-home components with others in the class, but must hand in your own work individually.

Academic Integrity
While there are no in-class exams in this course, students are required to abide by the university guidelines on academic integrity. For full details, see this link.

Disability Services
If you are requesting an accommodation due to a documented disability, you must register with the Disability Services Office at the beginning of the semester. To do so, call the Student Services Desk at 617-627-2000 to arrange an appointment with Linda Sullivan, Program Director of Disability Services.

Learning Objectives
This course addresses the following learning objectives of the Ph.D. Program in Mathematics

  • 1.b. Clear understanding of key hypotheses and conclusions
  • 1.c. Synthesis of formal theory into a comprehensive picture of mathematical phenomena
  • 1.d. Application of general theory to specific examples
  • 1.e. Sharpening of intuition through appropriate counterexamples
  • 3.a. Explanation of key ideas and general strategies
  • 3.b. Motivation of underlying issues
  • 3.c. Clear oral presentation of arguments
  • 3.f. Thinking on one's feet; fielding questions
  • Schedule

  • 9/4: Introduction
  • 9/6: Wave Equation, Conservation Laws
  • 9/11: Diffusion Processes; HW1 distributed
  • 9/13: Initial and Boundary Conditions
  • 9/18: Separation of Variables; HW2 distributed
  • 9/20: Separation of Variables, Part 2
  • 9/25: Energy arguments; HW3 distributed
  • 9/27: Normed vector spaces and L2
  • 10/2: Inner product spaces, orthogonality; HW4 distributed
  • 10/4: Completeness, Hilbert spaces
  • 10/11: Basis of L2, Gram-Schmidt
  • 10/16: Bessel, Parseval, Take-home midterm distributed
  • 10/18: Riesz-Fischer, Orthogonal Decompositions
  • 10/23: Riesz-Representation, Lax-Milgram, Weak Forms, Galerkin; HW5 distributed
  • 10/25: Weak convergence
  • 10/30: Operators, Adjoints; HW6 distributed
  • 11/1: Self-adjoint operators
  • 11/6: Eigenvalue problems, Sturm-Liouville systems; HW7 distributed
  • 11/8: Distributions
  • 11/13: Fundamental solutions, Green's functions
  • 11/15: Compact Operators; HW8 distributed
  • 11/20: The Spectral Theorem
  • 11/27: Spectral Theory for Sturm-Liouville Problems
  • 11/29: Green's Functions for PDEs