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as regular courses, or they may be offered by faculty as independent study courses, or the offered course may be withdrawn (i.e., cancelled).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 600-level courses were offered for the fall 1998 semester:
Descriptions follow.
Commutative algebra is mostly the study of commutative rings with an emphasis on the structure of the prime ideals within these rings. As a subject, Commutative algebra mostly grew out of two basic areas of study, algebraic geometry and algebraic number theory. The modern approach to algebraic geometry (due mostly to A. Grothendieck) is an approach based strongly on commutative algebra that encompasses many aspects of algebraic number theory.
This course is meant to be an introduction to commutative algebra as well as an introduction to the algebraic foundations of algebraic geometry. (I intend to offer a follow up course in algebraic geometry if there is interest.) The material to be covered should also provide a good foundation for work in algebraic number theory. This course will require only a basic understanding of commutative rings and of modules from MATH 521. We will develop the subject of commutative algebra using geometric motivations whenever possible. Therefore we will introduce a small amount of algebraic geometry hence students should be able to get a feel for this second subject as well.
After covering some preliminary material (including a quick review of the material needed from MATH 521), we will cover the following topics: commutative localizations, primary decompositions of ideals, integral dependence (including the Going-Up and Going-Down Theorems), filtrations, the Artin-Rees Lemma, the Krull Intersection Theorem, flatness, completions, some dimension theory (including the Krull Principal Ideal Theorem), and discrete valuation rings. Further topics to be covered will depend on time and on the interest of the class.
TEXT: Commutative Algebra with a View Toward Algebraic Geometry, by David Eisenbud, Springer-Verlag, New York-Berlin-Heidelberg, 1995.
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Quasi-measures are objects that generalize the measures of MATH 531. One major difference is that quasi-measures need not be subadditive. Instead, additivity is assumed on disjoint sets. The first example of a quasi-measure that is not a measure was published in 1991, so this is a fairly new area of mathematics. This course will cover the basic theory of quasi-measures with the goal of bringing the student up to the research level in this area. In particular, the Aarnes construction theorem, an investigation of spaces with $g=0$, and a treatment of the completely regular case will be given.
Book: No text is available. We will use lecture notes and research papers.
Prerequisite: Math 531 or consent of instructor. A background in Fourier analysis and some real analysis is suggested.
Wavelets are a mathematical development that may revolutionize the world of information storage and retrieval according to many experts. They are a fairly simple mathematical tool now being applied to the compression of data -- such as fingerprints, weather satellite photographs, and medical x-rays -- that were previously thought to be impossible to condense without losing crucial details. Mathematicians or other scientists and engineers are now interested in the applications (in signal analysis, time-frequency methods, numerical analysis, etc.) of wavelets.
Topics to be discussed will include the following (as time permits):
Texts:
Other References:
The course will deal with several important topics of linear algebra arising in practical applications areas such as control theory, signal and image processing, vibration analysis, statistical and economic modeling, etc.
Three distinct aspects of each topic (problem) will be discussed thoroughly: Theory, practical applications giving rise to the problem, and viable numerical methods. Any student or faculty member who desires to learn how important linear algebra problems arise in practical applications, and how to solve them effectively, is encouraged to take this course.
Course Contents: Nonnegative matrices (Finite Markov Chains, Input-output Analysis in Economics, Generalized Inverses), Inverse Eigenvalue Problems, Matrix Equations (Lyapunov, Sylvester, and Riccati), Inertia and Stability, Least-squares, Generalized Eigenvalue Problems, Singular Value Decomposition, Toeplitz Matrices.
Text: Detailed Lecture Notes will be provided. NO TEXT BOOK.
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