Graduate Analysis: 18.155 -- Fall 2017

• Lectures in 2-139, Tuesdays and Thursdays 1-2:30, Room 2-139.
• Instructor: Richard Melrose
• Grader: Vishesh Jain
• My office hour in 2-480, Wednesday (probably) 3PM (or by arrangement).
• Please ask questions!
• Comments and remarks:
  • Don't forget to evaluate the course.
  • Grades should be up by the end of the week.
  • Brief notes for L25
  • Brief notes for L24
  • Brief notes for L23
• 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
  • Elliptic regularity
• Spring 2018:- I will be teaching 18.966. Here is the preliminary blurb:

  Riemann Moduli spaces, from the ground up.

  I am planning on going through the background (Riemann surfaces and all the necessary manifold theory and differential topology) to define and examine the basic moduli spaces, $M_{g,n},$ of complex structures on an $n$-pointed Riemann surface of genus $g.$ This will take much of the semester -- I will take my time and cover as much of the background as is needed by the audience. There will be no homework as such but each student will be responsible for reading, correcting and expanding as necessary a section, or sections, of the notes that I will initially write.
• Old Notes from an earlier version of this course.
  1. Chapter 1 Measure and Integration -- I will not cover this in lectures but assume some knowledge of it.
  2. Chapter 2 Hilbert space and operators -- later
  3. Chapter 3 Distributions -- I will begin here.
  4. Chapter 4 Elliptic regularity
  5. Chapter 5 Coordinate invariance and manifolds
  6. Chapter 6 Invertibility of elliptic operators
  7. Chapter 7 Suspended families and the resolvent -- I do not expect to cover this.
  8. Topologies
  9. 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 -- this is only approximate more than a day or two from the current lecture.
  1. 7 September: Lecture 1 Schwartz space
  2. 12 September: Lecture 2 Distributions/Fourier transform
  3. 14 September: Lecture 3 Fourier inversion/Plancherel
  4. 19 September: Lecture 4 Convolution, Sobolev spaces, Sobolev embedding started, v2, 19 September, 2017.
  5. 21 September: Lecture 5 Sobolev embedding, duality, fractional order, Schwartz structure theorem
  6. 26 September: Lecture 6 Supports and maybe sheaf property
  7. 28 September: Lecture 7 Convolution and supports

     v2 28 September, 2017

  8. 3 October: Lecture 8 Fundamental solutions and Ellipticity
  9. 5 October: Lecture 9 Singular supports and symbols.
  10. 12 October: Lecture 10 Local elliptic regularity.
  11. 17 October: Lecture 11 SKT and Hilbert space
      
  12. 19 October: Lecture 12 Hilbert space continued
  13. 24 October: No lecture
  14. 26 October: Lecture 13 Spectral theorem
  15. 31 October: Lecture 14 Trace and Hilbert-Schimdt ideals
  16. 2 November: Lecture 15 Fredholm operators, index.
  17. 7 November: Lecture 16 Harmonic oscillator, densities
  18. 9 November: Lecture 17 Distributions on manifolds
  19. 14 November: Lecture 18 Pseudodifferential operators
  20. 16 November: Lecture 19 Elliptic regularity
  21. 21 November: Lecture 20 Borel's lemma and pseudodifferential operators
  22. 28 November: Lecture 21 Manifolds and differential operators
  23. 30 December: Lecture 22 Elliptic operators on compact manifolds are Fredholm
  24. 5 December: Lecture 23 Hodge theory
  25. 7 December: Lecture 24 Dirac operators and Clifford modules
  26. 12 December: Lecture 25 Riemann Moduli space and Hodge theory for the Weil-Petersson metric.
• 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 should be submitted through Stellar. If you have any trouble you can email it to me directly.
• Eight problem sets in all; note versions and date last modified:-
  1. Problems 1 -- solutions was due 15 Sept.
     18 September, 2017
• Problems 2 due 22 Sept.
  v1 16 September, 2017.
• Problems 3 due 29 Sept.
  v1 23 September, 2017.
• Problems 4 due 13 October.
• Problems 5 due 27 October.
• Problems 6 due 3 November.
• Problems 7 due 11 November; v2 with hints restored.
• Problems 8 due 1 December -- delayed to Dec 6 since I forgot to put it on stellar.
  Problems 8s same with supplement.