My aim is to present a largely selfcontained account of convex
analysis and optmization in Hilbert space, and to provide a concise
exposition of the large body of related constructive fixed point
theory that has become indispensible in applications.
The central theme is the interplay among convexity,
monotonicity, and nonexpansivity. Topics covered (in varying
levels of detail) are: convex sets and cones; convex functions;
Fenchel conjugates and duality; subdifferentiability; convex
optimization; monotone operators; nonexpansive operators and
generalizations; algorithms for convex feasibility and best
approximation problems.
Prerequisites: undergraduate level real analysis and linear algebra
We will cover the following topics, as time permits:
 1. Convex bodies, convex functions, support functions, Minkowski sums.
 2. Hausdorff metric, continuity, Blaschke selection theorem.
 3. Polytopes, volume, surface area, projections, Cauchy surface area formula.
 4. Mixed volumes, projection bodies, quermassintegrals.
 5. Isoperimetry, BrunnMinkowski and related inequalities.
 6. The Minkowski problem, Blashcke sums.
 7. The Euler characteristic, DehnSommerville Equations, Helly's Theorem.
 8. Buffon needle problem, Crofton formulas, valuations.
 9. Hilbert's Third Problem and other applications of valuations.
 10. Continuous valuations, Hadwiger's characterization theorem.
 11. Convexity in spherical and hyperbolic space.
It is intended to cover the following topics :
  Primary and derivative assets; selffinancing portfolios
  Pricing and hedging in complete and incomplete markets; the economic
principle of "Absence of Arbitrage" and its
mathematical counterpart : equivalent martingale measures
  Models for the term structure of interest rates and their calibration
to market data
  Interest rate derivatives
  Introduction to portfolio optimization
 Note 1 : The basic notions concerning pricing and hedging will first be
discussed for discretetime (multiperiod) models and then carried over
to continuostime models
 Note 2 : In case the students do not possess sufficient "working
knowledge" fom stochastic processes and stochastic analysis, the
following notions will be recalled in a first preliminary part of the
course (perhaps dropping in exchange one of the topics outlined above) :
  Ito calculus (stochastic integrals, Ito's formula, stochastic
differential equations); Markov diffusion processes
  Connections between stochastic differential equations and partial
differential equations (Kolmogorov equations and the
FeynmanKac formula)
  Absolutely continuous transformations of probability measures
(Girsanov transformation) and the methodology of the "change
of numeraire"
Required textbook: Bickel, P. and Doksum, K. Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I,
Second Edition, Prentice Hall.
The prerequisites for this course are a knowledge of multivariable calculus, and
linear algebra.
The course will provide an introduction to the methods of mathematical statistics.
The following is an outline of topics to be covered, and associated sections from the book.
 Appendices A1A14, B1B3. A review of some basic ideas from probability, including
transformations of random variables, calculations of moments, and properties of some
common distributions.

1.11.3, 1.5,1.6. Statistical models, parameters, sufficiency and exponential families.

2.12.3. Methods of estimation, including method of moments, maximum likelihood.

3.13.4. Bayes, minimax and unbiased estmators. Information bound.

4.14.5. Hypothesis testing and confidence intervals. NP lemma and LRT tests. The relationship
between testing and confidence regions.

5.15.4. Basic ideas in asymptotics. Large sample distribution of the MLE.