SCIENTIFIC PROGRAMS AND ACTIVITIES

September  1, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
FIELDS UNDERGRADUATE SUMMER RESEARCH PROGRAM
July 2 to August 23, 2013
Project 3 - Logic and operator algebras
Supervisors: Ilijas Farah (York University) and Bradd Hart (McMaster University)

Reading list:
(A) Model theory:

(1) Bradd's lecture notes:
http://ms.mcmaster.ca/~bradd/courses/math712/

(2) I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov,
Model theory for metric structures
http://www.math.uiuc.edu/~henson/cfo/mtfms.pdf

(3) Our paper with David Sherman (although it is #2 in the series, #1 is actually more advanced and will not be needed for this project):
http://www.math.yorku.ca/~ifarah/Ftp/2010d-stable-th.pdf
(Note: Chang-Keisler's book `Continuous model theory' is dated and not quite what we will need.)

(B) Functional analysis:
There is a number of excellent books in the subject. Among them are:

Pedersen, Gert Kjærgård. Analysis now. Vol. 2. Springer, 1995.
Conway, John B. A course in functional analysis. Vol. 96. Springer, 1990.
Rudin, Walter. Functional analysis. International series in pure and applied mathematics. McGraw-Hill (1991).
Zimmer, Robert J. Essential results of functional analysis. University of Chicago Press, 1990.

Each of these books contains way more material than we can possibly cover in two months. Central topics from the functional analysis that we will need are the following:
Hilbert spaces, operators on Hilbert spaces, spectral theorem, continuous functional calculus.

(C) C*-algebras.

(1) An introduction is given in Ilijas's Singapore lecture notes:
http://www.math.yorku.ca/~ifarah/Ftp/2012g08-singapore-notes-final.pdf

There are many excellent books on C*-algebras (some of them published by the Fields Institute) but the above will do for now.

(D) Further reading.

(1) Last year's project is presented here
http://www.math.yorku.ca/~ifarah/Ftp/types-23dec12.pdf

(2) Slides from Ilijas's Logic Colloquium 2012 lectures also contain some information on the `big picture' (lecture three is most relevant to our project). Attached here:

Lecture 1
Lecture 2
Lecture 3

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