Professor David Blecher
Topics course:
Convexity & Choquet Theory (Math 6397)
Time and place : MWF 12-1pm in AH 204
Office Hours MW: 2-3 pm (or by
appointment--email or call 713-743-3451)
Email: dblecher@math.uh.edu
Final exam: Date to be announced.
Prerequisites:
Graduate Standing, consent of instructor. Math 4331-4332 or equivalent. A little topology
and metric spaces, a little Real Variables (6321), or a little functional analysis (Math 7320
or some basic knowledge of Banach spaces, and Hilbert spaces)
would also be helpful.
Texts/Recommended reading:
R. R. Phelps, ``Lectures on Choquet's Theorem'' (2nd Edition, Springer Lecture Notes in Mathematics). Instructor will provide typed notes, drawn in part from the following
texts: Alexander Barvinok, ``A Course in Convexity (AMS Graduate Studies in Mathematics, V. 54), Lay "Convex Sets and ...",
"Infinite Dimensional Analysis: A Hitchhiker's Guide" by
Aliprantis and Border, "Compact Convex Sets and Boundary Integrals" (Springer), by E.M. Alfsen.
Infinite dimensional convexity, by V.V. Fonf et al, in Handbook of the Geometry of Banach Spaces, Volume 2
edited by W. B. Johnson, J. Lindenstrauss.
"Convexity" (Oxford Science Publications), by Roger Webster,
and "Integral Representation Theory: Applications to Convexity, Banach Spaces and
Potential theory" by Lukes et al.
Course description:
Convexity is a simple idea that is used in very many parts of mathematics, sometimes in surprising ways. The field has a very rich structure and theory, with numerous powerful applications. We will be touching on several topics, namely a subset of the list below (this list will be pruned to fit the needs and mathematical maturity of the class). We also note that some topics in this list may be out of order. For each of these we will develop the basic theory, and illustrate it with selected applications. We will begin with convex sets in finite dimensional spaces, and the theorems of Caratheodary, Radon, and Helly.
Supporting and separating hyperplanes. Faces and facial structure in convex sets.
Polyhedra and Polytopes. Applications: Schur-Horn and Birkhoff-von Neumann theorems. [Blaschke selection theorem. Duality.] Krein-Milman and Milman theorems.
[The Bishop-Phelps theorem.]
Convex functions. The affine function space A(K). Cones and ordered spaces. Fixed point theorems. Integral representation,
representing measures and maximal measures. The barycenter formula.
Choquet's theorem. The Choquet-Bishop-deLeeuw theorem and applications. The Choquet and Shilov boundary. Peak sets.
Convex and concave envelopes and `interpolation'.
Possible applications as time permits: Positive definite functions and Bochner's theorem. Solution of the one-dimensional moment problem.
The noncommutative Choquet boundary. Korovkin and Saskin theorems, and applications. Choquet (and other) simplexes. Applications e.g. to measures and ergodic theory. Noncompact convex sets, cones, and caps. Bauer's
abstract potential theory.
Final grade is aproximately based on a total score
of 300 points consisting of homework (100 points), possibly a semester test
(100 points), and a project/final exam (100 points).
The instructor may change this at his discretion.
Please send me an email (or give me your email address)
so that you can get the regular course emailings.
Almost all course communication and instructions, assignments, keys
etc. will be sent by email.