MATH 565, Advanced Real Analysis II, Winter 2010

Course web page:

  • MWF 9:30-10:30, Burnside 1120
  • Lectures start on january 4 and end on april 14
  • There will be no lecture on january 15
  • Make-up lecture: thursday, february 4, 17:30-19:30, burnside 1120
  • There will be no lecture on wednesday, march 10
  • Nikolay Dimitrov will lecture during the week of march 15
  • There will be no classes on april 2 and april 5

  • Instructor: D. Jakobson
    Office: BH1212, Office Hours: Time TBA or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 564
    Marker: Dylan Attwell-Duval

  • Required: E. Lieb and M. Loss. Analysis. 2nd edition. Graduate Studies in Mathematics, 14. AMS, 2001.
  • Recommended: W. Rudin, Real and Complex Analysis, McGraw-Hill.

  • Syllabus: Continuation of Math 564.
  • There will be a midterm. Date: Thrusday, February 18, 18:00-19:30pm, Burnside 1120.
  • You can write a take-home midterm, in-class midterm or both. Your grade will be the maximum of the 2 scores.
  • Take-home midterm: pdf and ps. Due: March 1, 2010 (after the winter break).
  • The final exam is on April 16, at 2pm.
  • You can write a take-home final, in-class final or both. Your grade will be the maximum of the 2 scores.
  • The take-home final exam will be given out on April 15 and will be due back on April 27. Take-home final: pdf and ps
  • List of things to review for the final and the quals: pdf and ps.

  • Oral Presentation
    Students will have an option of giving a 30-minute oral presentation in class on a topic chosen by the student in consultation with the instructor. If a student chooses that option, the presentation will be evaluated, and will count for 15% of the grade.
    List of possible topics:
  • Rearrangement ineqaultities
  • Steiner symmetrization
  • Sobolev inequalities
  • Poincare's inequalities
  • Banach-Tarski paradox
  • Ruziewicz problem
  • Lie groups, Haar measure
  • Ergodic theorem
  • Isoperimetric inequalities
  • Rayleigh quotient, eigenvalues
  • Co-area formula
  • Cheeger constant
  • Infinite-dimensional spaces
  • Other topics to be announced

  • Grading: Your final mark will be the largest of the following: [25% Assignments + 30% Midterm + 45% Final]; OR [25% Assignments + 75% Final]; OR [15% Presentation + 25% Assignments + 25% Midterm + 35% Final]; OR [15% Presentation + 25% Assignments + 60% Final]
    WebCT: Your scores on assignments, presentation, midterm, final, and your final mark will be posted on WebCT
    Supplemental: There will be a supplemental exam, counting for 100% of the supplemental grade. No additional work will be accepted for D, F or J.
    Homework will be assigned in class and will be due by 5pm by the specified deadline.


  • Problem Set 1, due February 10: ps and pdf.
  • Problem Set 2, due February 15: ps and pdf.
  • Problem Set 3, due date TBA ps and pdf.
  • Problem Set 4, due date TBA ps and pdf.

  • Handouts

  • A short course on rearrangement inequalities, written by Prof. Almut Burchard at the University of Toronto [from Math 564]
  • Steiner symmetrization slides by Andrejs Treibergs at the University of Utah [from math 564].
  • A 2003 paper by V. Milman and B. Klartag about Minkowski symmetrization [from Math 564]
  • A 2002 paper by B. Klartag, titled: 5n Minkowski Symmetrizations Suffice to Arrive at an Approximate Euclidean Ball; its arxiv version [from Math 564]
  • A 2004 paper by B. Klartag about the rate of convergence of sequences of Steiner and Minkowski symmetrizations; its arxiv version [from Math 564]
  • A 1990 paper by E. Lieb, containing the proof of Hausdorff-Young inequality with sharp constants and characterizing the minimizers.

  • Course material from previous courses at McGill:
  • Web page for Math 564, D. Jakobson, Fall 2009.
  • Web page for Math 565, D. Jakobson, Winter 2009.
  • Web page for Math 564, D. Jakobson, Fall 2008.
  • Old Math 354 web page, D. Jakobson, Fall 2006
  • Sam Drury's lecture notes for MATH 354 and MATH 355
  • Old Math 366 web page, D. Jakobson, Fall 2007
  • Vojkan Jaksic's Lecture Notes in Spectral Theory, ps and pdf.

  • Lecture notes in Measure Theory
  • S. Sternberg's Theory of functions of a real variable lecture notes.

  • Web links in Analysis
  • Metric space, Topology glossary, Functional analysis in Wikipedia
  • Norm, Holder's inequality, Minkowski inequality, Lp space, Hilbert space, Banach space, Cantor set, p-adic numbers in Wikipedia
  • Notes on differentiation of functions of several variables, implicit function theorem
  • Companion notes to Rudin's (undergraduate!) book
  • Harmonic Analysis page by Terry Tao (there is a lot of advanced stuff there)

  • HELPDESK and their email:
    NOTICE: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).