Below is a list of topics courses that could be offered next year. Please take a look at the list and give us feedback about your interest in taking any of these by answering the question at the bottom of the page. You can indicate interest in more than one couse.

 

The course is devoted to more advanced applications of the forcing method that are beyond the scope of Math 281, but are crucial for active research in the area. The following types of forcing will be discussed:

a. Finite support iteration. Application: Martin Axiom.

b. Easton support iteration. Application: Laver preparation, Kunen preparation, construction of satuarted ideals.

c. Countable support iteration. Application: Proper Forcing Axiom, Borel's conjecture.

d. Iterations with flat conditions. Application: Iterated club shooting and properties of the non-stationary ideal.

e. Prikry type forcings. Applications: Singular cardinal combinatorics. 

 

The course will introduce the students to a number of topics in mathematical modeling at the interface between mathematical and life sciences, including modeling of cancer and modeling of  language. The emphasis will be on exposing the students to a wide variety of mathematical tools which can be used in a great variety of research areas. The course will combine elements of mathematical modeling, dynamical systems and applied stochastic processes. The instructor will give introduction to all these topics. We will talk about cancer initiation and progression,  Darwinean selection acting upon cells, stem cells, and treatment of cancer. We will also discuss how the Darwinean approach together with game theory can be applied to the modeling of natural languages.

 

- the Monte Carlo method or Metropolis algorithm, devised by John von Neumann, 

   Stanislaw Ulam, and Nicholas Metropolis;

- the simplex method of linear programming, developed by George Dantzig;

- the Krylov Subspace Iteration method, developed by Magnus Hestenes, Eduard Stiefel, 

   and Cornelius Lanczos;

- the Householder matrix decomposition, developed by Alston Householder;

- the Fortran compiler, developed by a team lead by John Backus;

- the QR algorithm for eigenvalue calculation, developed by J Francis;

- the Quicksort algorithm, developed by Anthony Hoare;

- the Fast Fourier Transform, developed by James Cooley and John Tukey;

- the Integer Relation Detection Algorithm, developed by Helaman Ferguson and Rodney Forcade; (given N real values XI, is there a nontrivial set of integer coefficients AI so that sum ( 1 <= I <= N ) AI * XI = 0? the fast Multipole algorithm, developed by Leslie Greengard and Vladimir Rokhlin; (to calculate gravitational forces in an N-body problem normally requires N^2 calculations. The fast multipole method uses order N calculations, by approximating the effects of groups of distant particles using multipole expansions)

 

The course will be on classical harmonic analysis, useful for all students in analysis, PDE, and geometry.

 

In the first part we will present a brisk review of stochastic processes and Ito calculus. We will next discuss techniques of rare event simulation and quantile estimation, moreover, associated approaches to variance reduction. Then we will present modern techniques associated with uncertainty quantification in stochastic systems. In the second part we will give an introduction of credit and insurance risk markets and techniques for portfolio optimization and evaluation. Prerequisites : Multivariable calculus and some exposure to differential equations and probability.

 

Description: Introduction to optimization: convex sets/functions, minimization of convex functions, variational inequalities, duality. Basic computational algorithms: linear programming, simplex method, line search, augmented Lagrangian methods, projected gradient methods. Applications: image processing, data analysis, machine learning.