Professor David Blecher
Information and Instructions
for end of semester project in Real Variables II (Math 6321))
Here are some project choices for your April--May project. (In addition there were a few suggestions for projects
that were made in class/in the notes.) Tell me
your choice by April 10. You will be given some class days off in
April--May to
work on these. You will have to turn in a write-up of your project,
which should probably be 5--10 typed pages depending on difficulty or
what spacing or fonts you use. There is not a strict page limit upper bound but if it is longer that 10 pages I may not read the extra pages. Please have a bibliography, and if you omit any proof add the precise cross reference (e.g. Theorem XX.x p. X in [F]).
Usually you are required to turn in to me a scan of the pages of the original text or source you used.
You will also do a small presentation in class (10 minutes)
on your project as part of the final exam. You can choose from the list below,
or choose your own project (if you do this you will need to discuss
this with me by April 10). For example something related to your research,
or some sections skipped in the classnotes. I have usually indicated a few recommended texts
for each item (if not please ask), although in some cases there may be other better texts).
You can work on projects others have taken, alone or as a group
(but if as a group, it should be much longer and you must ensure everyones
contributions are equal, and you write it in your own words)
Project due date May 10, or 14th latest by special request.
The grade will be 30 points (the `discussion points'
on the syllabus) plus 10 presentation points to be included in final exam score.
Probability (Folland chapter 10, Bogachev?)
Conditional expectations (and Lp spaces, stochastics, or martingalesi; see e.g. Bogachev?)
Ergodic theory (texts: “Invitation to Ergodic Theory” by Silva,
“Ergodic Theory and Dynamical Systems” by Coudene, Karl Petersen ``Ergodic Theory'' (More sophisticated: Walters, “Foundations of Ergodic Theory” by Viana and Oliveira; Glassner)). Websites: Terry Tao: http://terrytao.wordpress.com/tag/254a-ergodic-theory/, Sarig https://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/ergodicnotes.pdf , McMullen).
Integration theory on locally compact spaces (eg. Fremlin Chapter 7, Cohn Measure Theory Chapter 7,
Salomon Chapter 3)
Fourier analysis, e.g. Fourier series or more on
the Fourier transform in Rn (Jones Chapter 13 and 14/Folland Sections 8.4-8.7, Sections 7.4 and 7.5 in Class notes)
Maharam's theorem (the classification of finite measure spaces, and maybe the classification of localizable
measure spaces, see e.g. Fremlin volume 3 in the context of measure algebras, or his shorter
expositions on that topic)
The classification of standard Borel spaces (Kuratowski's theorem, see e.g.
Srivastava ``A course on Borel sets", Jesse Peterson's Notes on operator algebras
Section 3.9, or Cohn's Measure Theory Chapter 8)
The lifting theorem (see e.g. Handbook of Measure Theory, or Fremlin volume 3)
The full Arzela-Ascoli theorem (see eg. Munkres section 43, 45-47,
if you did not do this in topology)
Choquet theory (Phelps text)
Abelian von Neumann algebras and localizable measure spaces
(Notes, Jesse Peterson's Notes on operator algebras)
Abstract harmonic analysis (Haar measure and the Fourier transform on a
locally compact group (Folland 11.1 plus some things from e.g. Rudin "Fourier
analysis on groups",
Loomis "Abstract harmonic analysis", Folland "A course in Abstract harmonic
analysis")
Hausdorff measure and dimension (Folland 11.2-3, Edgar,
Brockner-Brockner-Thomson)/geometric measure theory
Distributions and Sobolev spaces (Folland chapter 9)
Infinite product measures (Classnotes, Folland, Halmos, other texts such as Fremlin or Bogachev.)
Vector valued measures, and/or the Henstock-Kurzweil integral. (An old classic text by Diestel and Uhl, a stretch in Ryan "Introduction to Tensor Products of Banach spaces", or the chapter in The Handbook of Measure Theory.)
Measure theoretic set theory. (e.g.\ Fremlin Volume 5 or Jech "Set theory", Springer.)
The Banach-Tarski paradox and amenability (e.g. Stromberg)
Potential theory in the plane (Ransford book)
Please email me if you have any questions.