The following topics courses have been suggested for the upcoming academic year. Please take a look at the list and indicate which course you would be interested in taking. Based on your feedback a decision will be made as to which course will be offered.
Big P Little N Problems in Statistics
A recurrent issue in Math Finance is the difficulty of detecting changing market conditions in close-to-real time. One aspect of this is the problem of having a large number p of time series of relatively small length N. This is called the “Big P little N” problem. An example of this is the daily returns on the S&P 500 index. For one quarter each of the 500 return time series have length 63. Even for 10 years the time series only have length 2520. This introduces significant bias in estimating important invariants such as the dimension, the Singular Value Decomposition or the mean/variance efficient frontier. The first part of the course is aimed at understanding the Marchenko-Pastur distribution that describes the distribution bias. The second part of the course will study the techniques Foreman has developed that repeatedly sample small numbers of time series in order to get unbiased estimates of statistics involving the whole collection of time series.
Cancer Modeling
This course is based on Natalia Komarova's 2015 book on modeling of cancer. It talks about evolutionary principles that are at place in tumor development, and it involves dynamical systems and stochastic methods.
Dynamical Systems
Basic examples of topological and smooth dynamics: linear maps, translations on the torus, gradient flows, expanding maps, symbolic dynamical systems. Fundamental concepts of dynamical systems: conjugacy, equivalence, classification, invariants, structural stability. Low-dimensional dynamics, homeomorphisms of the circle, rotation number. Local analysis in smooth dynamics: hyperbolic periodic orbits, Hadamard-Perron theorem, Hartman-Grobman theorem, local structural stability, normal forms. Symbolic dynamics, coding, horseshoes, attractors. Hyperbolic dynamics: horseshoes, Anosov diffeomorphisms, DA maps, Smale-Williams solenoid, general hyperbolic sets, Markov partitions, coding, local product structure, stability, spectral decomposition. Fractals in dynamics. Dynamically defined Cantor sets. Topological entropy. Calculation of a topological entropy for topological Markov shifts, hyperbolic automorphisms of the torus, solenoid. Applications of hyperbolic dynamics in celestial mechanics and spectral theory.
Ergodic Theory
A very short intro/reminder of some elementary measure theory. Measure preserving transformations. Poincare's Recurrence Theorem. Ergodicity, basic examples (circle rotations, the doubling map, Bernoulli schemes and Markov chains, cat map). Mixing and weak mixing. Birkhoff and von Neumann Ergodic Theorems. Metric entropy. The Shannon-McMillan-Breiman Theorem. Topological dynamics, existence of invariant measures, variational principle. Smooth ergodic theory, Lyapunov exponents, Furstenberg-Kesten Theorem, Oseledets Theorem. Applications to number theory (the Gauss transformation and Continued Fractions, Szemeredi's Theorem). Applications to spectral theory (ergodic Schrodinger operators).
Introduction to Random Matrix Theory
This course will cover the basics of random matrix theory (RMT). Connections to statistics, numerical analysis, and physics will be given. This will provide students in probability, applied math, and mathematical physics with additional background.