University of Connecticut

Course Info

MATH 5040: Topics in Applied Analysis I

Description: Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory. With a change of content, this course is repeatable to a maximum of twelve credits.

Credits: 3

MATH 5040 - Section 2: Indefinite Inner Product Spaces

Description: An indefinite inner product is defined like the usual inner (or scalar) product, but we drop the positivity requirement, so that can be negative. A rich and elegant theory emerges, going back to Krein, Gelfand, Lidskii and Levinson, and it originates in the study of stability of canonical differential equations with periodic coefficients, in mechanical and electrical engineering, in the theory of matrix polynomials and in Ricatti equations, to mention just a few.

Though this theory is a very active area of current research, and there is a vast literature on the subject, until recently it was sort of difficult to present an elegant introductory graduate course. The reason was in that such a course would be based on various journal publications that use quite different notations, terminology, and approaches. Fortunately, a recent excellent graduate textbook by Gohberg, Lancaster and Rodman makes such a course feasible and enjoyable. In the first part of this course we will cover various topics, including the geometry of inner product spaces, and the theory of matrices that are self-adjoint, unitary or normal with respect to an indefinite inner product. Several years ago I taught this course here at UConn. Solving one of the homework problem led to further research resulting in a paper published by two students (Tom Bella and Upendra Prasad) and myself in a first-rate scholarly journal. In the second part of this course we will study the results of Bella and Prasad, and maybe at the end we shall address some interesting open questions as well.

Offered: Fall

Credits: 3

Sections: Fall 2015 on Storrs Campus

PSCourseID Course Sec Comp Time Room Instructor
13644 5040 002 Lecture MWF 09:05:00 AM-09:55:00 AM MSB219 Olshevsky, Vadim