Each semester, faculty propose some 600-level courses to possibly be offered to graduate students, primarily doctoral students. Some of these courses follow up on 500-level courses, and some are primarily focused on current research. Depending on several variables, including student interest, these courses may be offered as regular courses, or they may be offered by faculty as independent study courses, or the offered course may be withdrawn (i.e., cancelled).
The following 600-level courses were offered in the Fall 1999 semester:
Prerequisite: MATH 531 or MATH 550, or consent of department.
In this course the theory of Calculus is extended to infinite dimensions. Whenever possible the theory is applied to the `classical' optimization problems in the calculus of variations.
I. The relationship between the Gataux derivative and the Frechet derivative is examined. Other less known derivatives are also introduced. The second order derivative is defined. The second derivative test is shown to be less useful in infinite dimensions. It is however possible to extract stability information from a knowledge about the location of the conjugate points.
II. To deal with smooth constraints the concept of a Banach manifold is introduced. In the special case of Hilbert manifolds smooth functions generate gradient vector fields and an associated flow. Projections onto the tangent planes give a geometric meaning to Lagrange multipliers. Constrained stability is introduced and related to the `bordered' Hessian in finite dimensions.
III. The purpose of Morse theory is to explain the presence and the stability of critical points in terms of the topology of the underlying manifold. Morse theory is initially explored in finite dimensions. Tromba's extension to Banach manifolds is also covered.
IV. The question of the existence of critical points, conjugate points and global optimizers is the essential theme of the course. Compactness is established in the Eberlein theorem. The Palais-Smale compactness condition is also discussed and illustrated. In conjunction with this some natural open problems presented.
Text: Lecture Notes.
References:
Instructor: Greg Ammar
Prerequisites: Math 562, or MATH 423 and MATH 434, or consent of department.
Many problems that arise in digital signal processing involve solving linear-algebraic problems of special structure. The importance of these problems in applications stems from the use of efficient algorithms designed to take advantage of these structures. In fact, the existence of fast algorithms, such as fast Fourier transforms and fast Toeplitz solvers, provides the foundation for the practical use of ideas from signal processing.
After introducing some background concepts concerning digital signals, digital filters, and time series, we will focus on the design and analysis of efficient algorithms for solving various problems involving structured matrices that arise in some fundamental digital signal processing applications. Our goal will be to obtain an understanding of some central mathematical ideas in signal processing applications and of some efficient algorithms for solving the resulting problems involving structured matrices.
Tentative outline:
References: