Graduate Course 22M:313

Functional Analysis

Fall 2006

Some facts and observations:

We will be getting a course email account, a class account, and I will circulate this (with links) and other course material during the semester. Stay tuned!

See also (will be updated) http://www.math.uiowa.edu/grad/gradcourses300fa2004.html.

Course Meeting Schedule:

MWF 3:30 in MLH 205.

Course Books:

The course will start at the beginning, and it will be both pure and applied. We will open with the book by Ron Douglas, and I checked that it is in the bookstores.

Both books for the course are on amazon.com as well. But the second book (by Jorgensen) will arrive (per drop-shipment) in our bookstore here sooner than at amazon.com. More details on Jorgensen's home page.

Relevant Seminars:

FYI: The two seminars, Operator Theory and Math Physics for fall 2006.

One organizational meeting Tuesday 8/22 at 1:30 in VAN 301 (that's the Physics building!)

Two slots: 1:30, and 2:30.

General course info:

What is Functional analysis?

Function spaces, operators in Hilbert space, operator theory/operator algebras, mathematical physics (especially quantum theory), representations of groups, mathematical probability theory.

The answer (to the question "What is Functional analysis?") changes over time, and some answers refer to the mathematical tools, while others to a variety of applications; see e.g., the article by Vershik (handout!)

Topics and names from Vershik's article (P.S. Functional analysis is alive and well!): S. Banach's three theorems: (a) Hahn-Banach (extensions), (b) H. Steinhaus (uniform boundedness), and (c) the inverse mapping theorem (closed graph.)

1. Abstract vs concrete.

Additional references:

Schwartz, Laurent; Some applications of the theory of distributions. 1963 Lectures on Modern Mathematics, Vol. I pp. 23--58 Wiley, New York.

Kuksin, Sergei B.; Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, 237--258, Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI, 2004.

Lax, Peter D. Jürgen Moser, 1928--1999. Ergodic Theory Dynam. Systems 22 (2002), no. 5, 1337--1342.

Raussen, Martin; Skau, Christian; Interview with Peter D. Lax. Reprinted from Eur. Math. Soc. Newslett., September 2005, 24--31. Notices Amer. Math. Soc. 53 (2006), no. 2, 223--229.

2.

(a) Fourier series.

(b) PDE:

(c) Numerical analysis

(d) Calculus of variations

Additional references:

Rudin, Walter; Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1.

Nirenberg, Louis; Laurent Schwartz and some of his mathematical work. Laurent Schwartz (1915--2002). Gaz. Math. No. 98, suppl. (2003), 97--103.

Lax, Peter D.; Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. xx+580 pp. ISBN: 0-471-55604-1

3. Banach-algebras, C*- and W*-algebras. Special topics in algebras of Functional analysis:

(a) KMS

(b) Representations and states:

(c) Tomita-Takesaki.

(d) BDF

Additional references:

Bratteli, Ola; Robinson, Derek W.; Operator algebras and quantum statistical mechanics. Vol. 1. $C\sp{*} $- and $W\sp{*} $-algebras, algebras, symmetry groups, decomposition of states. Texts and Monographs in Physics. Springer-Verlag, New York-Heidelberg, 1979. xii+500 pp. ISBN: 0-387-09187-4.

4. Convexity:

Additional references:

Rudin, Walter; Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8.

Phelps, Robert R.; Lectures on Choquet's theorem. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London 1966 v+130 pp.

5.

(a) Hilbert space,

(b) Banach space:

(c) Algebraic methods:

Additional references:

von Neumann, Johann; Mathematische Grundlagen der Quantenmechanik. (German) Unveränderter Nachdruck der ersten Auflage von 1932. Die Grundlehren der mathematischen Wissenschaften, Band 38 Springer-Verlag, Berlin-New York 1968 v+262 pp.

Sz.-Nagy, Béla; Foia\lfhook s, Ciprian; Harmonic analysis of operators on Hilbert space. Translated from the French and revised North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest 1970 xiii+389 pp.

6. Early quantum theory:

Additional references:

7. Statistical thermodynamics:

Additional references:

Nelson, Edward; Mathematical form and physical reality. (Italian) Stochastic processes in classical and quantum systems (Ascona, 1985), 545--550, Lecture Notes in Phys., 262, Springer, Berlin, 1986.

Nelson, Edward; Construction of quantum fields from Markoff fields. J. Functional Analysis 12 (1973), 97--112.

8. Representations of groups:

Additional references:

Mackey, George W.; Mathematical foundations of quantum mechanics. With a foreword by A. S. Wightman. Reprint of the 1963 original. Dover Publications, Inc., Mineola, NY, 2004. xii+137 pp. ISBN: 0-486-43517-2.

9. Information theory, entropy, quantum computing:

Additional references:

de Leeuw, K.; Moore, E. F.; Shannon, C. E.; Shapiro, N.; Computability by probabilistic machines. Automata studies, pp. 183--212. Annals of mathematics studies, no. 34. Princeton University Press, Princeton, N. J., 1956.

Shannon, Claude E.; Weaver, Warren The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Ill., 1949. vi+117 pp.

10. Wavelets:

Additional references:

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992.

O. Bratteli and P.E.T. Jorgensen, Wavelets through a Looking Glass: The World of the Spectrum, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2002.

11. Mathematical probability theory:

Additional references:

Nelson, Edward; Notes on non-commutative integration. J. Functional Analysis 15 (1974), 103--116.

12. Duality for groups:

Additional references:

Rudin, Walter; Fourier analysis on groups. Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1990. x+285 pp. ISBN: 0-471-52364-X.

Mackey, George W.; The scope and history of commutative and noncommutative harmonic analysis. History of Mathematics, 5. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1992. xii+370 pp. ISBN: 0-8218-9000-X.

13. K-theory:

Additional references:

Brown, L. G.; Douglas, R. G.; Fillmore, P. A.; Extensions of C*-algebras and K-homology. Ann. of Math. (2) 105 (1977), no. 2, 265--324.

14. Symbolic dynamics:

Additional references:

Shannon, Claude E.; Von Neumann's contributions to automata theory. Bull. Amer. Math. Soc. 64 (1958), 123--129.

Four Lists of Books Relevant to 22M:313 at amazon.com:

• Operator Theory
• Functional Analysis
• Wavelets
• Analysis and Probability (Math)

This page was last modified on 24 August 2006 by Brian Treadway.

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