Convexity and Fixed Point Algorithms in Hilbert Space

My aim is to present a largely self-contained 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.

Integral geometry of convex bodies and polyhedra

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, Brunn-Minkowski and related inequalities.
6. The Minkowski problem, Blashcke sums.
7. The Euler characteristic, Dehn-Sommerville 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.
The Mathematics of Finance

It is intended to cover the following topics :

- Primary and derivative assets; self-financing 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 discrete-time (multiperiod) models and then carried over to continuos-time 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 Feynman-Kac formula)
- Absolutely continuous transformations of probability measures (Girsanov transformation) and the methodology of the "change of numeraire"

Mathematical Statistics

Required textbook: Bickel, P. and Doksum, K. Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I, Second Edition, Prentice Hall.

The pre-requisites 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.

  1. Appendices A1-A14, B1-B3. A review of some basic ideas from probability, including transformations of random variables, calculations of moments, and properties of some common distributions.
  2. 1.1-1.3, 1.5,1.6. Statistical models, parameters, sufficiency and exponential families.
  3. 2.1-2.3. Methods of estimation, including method of moments, maximum likelihood.
  4. 3.1-3.4. Bayes, minimax and unbiased estmators. Information bound.
  5. 4.1-4.5. Hypothesis testing and confidence intervals. NP lemma and LRT tests. The relationship between testing and confidence regions.
  6. 5.1-5.4. Basic ideas in asymptotics. Large sample distribution of the MLE.