Graduate Analysis: 18.155 -- Fall 2016
- Lectures in 2-135, Tuesdays and Thursdays 11-12:30.
- Instructor: Richard Melrose
- Grader: Ethan Jaffe, Office hour Friday 2PM -- in the space outside 2-231.
- My office hour in 2-480, Wednesday 3PM (or by arrangement).
- Please ask questions!
- Comments and remarks:
- That's it folks, grades are in and should appear soon. I hope to have a little time to fix up the notes a bit in January. Any new version will be posted here. Feel free to ask questions as always.
- Material to be covered:
- Tempered distributions and Fourier transform
- Constant coefficient differential operators
- Operators on Hilbert space and spectral theorem
- Kernel theorem and fundamental solutions
- Hyperbolic operators
- Old Notes from an earlier version of this course.
- Chapter 1 Measure and Integration -- I will not cover this in lectures but assume some knowledge of it.
- Chapter 2 Hilbert space and operators -- later
- Chapter 3 Distributions -- I will begin here.
- Chapter 4 Elliptic regularity
- Chapter 5 Coordinate invariance and manifolds
- Chapter 6 Invertibility of elliptic operators
- Chapter 7 Suspended families and the resolvent -- I do not expect to cover this.
- Pseudodifferential operators
- References (besides the notes). For the early part of the course I suggest:
- Volume 1 of Hörmander's 4 volume work,
- The little book of Friedlander and Joshi (Cambridge University Press) is also good.
- There is a book by Duistermaat and Kalka which is quite good.
- Lecture contents
- 8 September: Lecture 1 Schwartz space
- 13 September: Lecture 2 Fourier transform
- 15 September: Lecture 3 Sobolev spaces
- 20 September: Lecture 4 Sobolev embedding
- 22 September: Lecture 5 Homogeneity in 1-D
v2: 23 September, 2016, some typos fixed.
- 27 September: Lecture 6 Supports and sheaf property
- 29 September: Lecture 7 Convolution and supports
v2 minor changes
- 4 October: Lecture 8 Ellipticity and symbols
v2 9 October, 2016: A little more on sheaves.
v3 12 October, 2016: Eqn (31) corrected.
- 6 October: Lecture 9 Local elliptic regularity.
- 13 October: Lecture 10 Hilbert space
- 18 October: Lecture 11 Operators
V2 with proofs of Baire-related theorems included (and corrected).
- 20 October: Lecture 12 Trace and Fredholm operators
- 25 October: Lecture 13 Spectral theorem
- 27 October: Lecture 14 Spectral projections
- 1 November: Lecture 15 Schwartz kernel theorem
- 3 November: Lecture 16 Variable coefficients
- 8 November: Lecture 17 Elliptic regularity
- 9 November: Lecture 18 3-4:30, 2-449. Harmonic oscillator and pseudodifferential operators (partial notes so far)
- 10 November: Lecture 19 Paley-Wiener Theorems
- 15 November: Lecture 20 Hyperbolic polynomials
- 17 November: Lecture 21 Ethan Jaffe: Cauchy problem
- 1 December: Lecture 22 Manifolds and diffeomorphisms
- 6 December: Lecture 23 Hodge theorem
- 8 December: Lecture 24 Cancelled, sorry about that.
- No tests or exams. Grades: Will be based on the problem sets. Warning. I plan to give the grade of B to graduate students who do insufficent homework and A to those who score over half marks. For undergraduates I will have a more standard grading scheme and will give progressive grades.
- Problem sets. I want all problem sets to be submitted to me electronically at the email address above. It is fine if you hand-write the solutions (clearly) and scan them. All homework will be due on Fridays but the truth is that I will not look at them until Saturday morning, and will accept anything that has arrived by then; after that I make no promises. Of course you should write out your own answers to all problems but are permitted, indeed encouraged, to discuss it with others, ask me and/or read whatever you want -- but please then think it all through for yourself and formulate your own solution.
- Ten problem sets in all; note versions and date last modified:-
- Problems 1 due 16 Sept.
v2: 10 September, 2016, due date corrected
- Problems 2 due 23 Sept.
v2: 16 September, 2016, typos
v3: 21 September, 2016, more annoying typos squashed. Thank you Minjae
- Problems 3 due 30 Sept.
- Problems 4 due 14 Oct.
v2 Missing `u' in Q4 corrected.
- Problems 5 due 21 Oct.
v2 A stray `if' removed from Q4. Thank you Jingwen.
- Problems 6 due 28 Oct.
v2 Q3 corrected (`semi-Fredholm)' and hint expanded a little.
v3 Additional wording in Q2.
- Problems 7 due 4 Nov.
v2 Correction to definition of symmetry.
v3 Simplified Q4 and lots of hints.
v4 Further revised Q4 and a couple of typos fixed.
v2 of partial solutions P7-Partial-Solutions
- Problems 8 due 18 Nov.
v2 13 November, 2016: Small changes, indexing fixed.
v3 19 November, 2016 (I am in Australia ..). Typo in Q4 (thank you Sammy) and hint as well.
- Problems 9 due 9 Dec.
v2 wording clarified (harmonic rather than eigenfunction) in Q5
V3 9 December, 2016: Hints added. Let me know if you need more!
- Problems 10 There is no problem set 10