The course is thought to address spectral aspects of linear operators on Banach spaces.We will start with brief preliminaries about Banach spaces and bounded linear operators including the classical theorems of Hahn-Banach, Banach-Steinhaus, Open Mapping Theorem, and Closed Graph Theorem. The basic facts of spectral theory will be developed in the more general context of Banach algebras. A classification of elements of the spectrum will be provided, resolvents and Riesz functional calculus will be treated at length. A detailed description of spectral properties of compact operators will be given.  Continuity and analytic properties of spectral sets when operators are treadted as variables will be studied in detail culminating at the full version of Kato-Rellich Perturbation Theorem. Ionescu-Tulcea and Marinescu Theorem will be proved. Apart from being interesting itself it can also serve as a good tool for verifying the assumptions of Kato-Rellich Theorem and creating a so called spectral gap. This will also have striking consequences for the asymptotic behavior of iterates of operators under consideration.  I then plan to deal with positive operators on Banach lattices. First, their general properties, in particular their spectra, then Garret Birkhoff’s machinery of Hilbert metric and (called after him) Birkhoff’s cones. This method also can be applied to verifying Kato-Rellich assumptions as well as to create spectral gaps. Secondly, Markov operators, as special cases of positive operators, with some application to Markov processes and iterated function systems. Third, more special Markov operators called Perron-Frobenius operators and some of their applications to dynamical systems, eg. Lasota-Yorke mappings.  If time permits we would also deal with Grothendieck’s nuclear operators and/or spectral theory of normal (including self-adjoint and unitary) operators on Hilbert spaces.  I would possible like to cover both bounded and unbounded case. A third option would  be to cover, somewhat shorter, theory of almost-periodic operators.

There will be no single textbook. Good sources for the material of the course include:

[A] B. Aupetit, A primer on spectral theory, Springer-Verlag 1991.

[C] J. B. Conway, A course in functional analysis, 2nd edition, Springer-Verlag 1990.

[D] E. B. Davies, Linear operators and their spectra, Cambridge University Press 2007.

[S] H. Schaefer, Banach lattices and positive operators, Springer-Verlag 1974.

[RS] M. Reed, B. Simon Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press 1978.

Professor Su Gao

Professor Joseph Kung

This course will be about the algebra and combinatorics of symmetric functions.

permuted. For example, x1+x2 and x1x2 are symmetric functions in two variables.

Professor Kai-Sheng Song

Nearly all scientific disciplines grapple with the challenge of analyzing and making sense of increasingly vast amounts of experimental/observational data that may be censored, missing, or sparse and noisy. Data mining such highly complex data across many scales for knowledge discovery and prediction requires increasingly sophisticated statistical methods and techniques. The aim of this 6000-level course is to present several special topics in statistics, which cover some major statistical procedures that are widely-used in many areas such as physical sciences, biomedical and life sciences, computer science and engineering, as well as business, finance and economics.

These special topics include survial analysis that is also called reliability analysis in engineering and duration analysis in economics/sociology (censoring and truncation, proportional hazards model, product limit estimator, competing risks, additive hazards model), parametric/nonparametric regression (logistic and Poisson regression, quasi-likelihood, kernel, local polynomial and smoothing spline regression), Bayesian methods for machine learning as well as other statistical methods in bioinformatics. This course should be highly useful and valuable to graduate students with a wide variety of research interests and career goals. Mastering the key topics would certainly provide graduate students with the competitive edge they need to succeed in finding satisfying careers in academia and industry.

Prerequisite: Some background in statistics and probability (for example, Math 4610/5810) or permission of the instructor.

Statistics is a common subject for which mathematicians are often consulted. Also, proven expertise with statistical analysis and software packages is a valuable skill to have for anyone seeking a job in industry.  In this class, we will learn some fundamental techniques of statistical analysis, including hypothesis testing and confidence intervals using the normal curve, the Student t distribution, and the chi-squared distribution. Computation will be emphasized throughout the course, including applications of statistics to various areas of the sciences.  We will also study enough probability theory to understand why these statistical techniques work.

Chaotic Dynamics MATH 5900.766

Spring 2008

MW 3:30-4:50 p.m. (meets in CURY 110)

Professor Mrinal Roychowdhury

This course will be a beginning course in Chaotic Dynamical Systems.

Chaotic dynamics studies the behavior of nonlinear dynamical systems

that are sensitive to initial conditions. Observations of chaotic

behavior in nature include the dynamics of satellites in the solar

system, population growth in ecology, the dynamics of the action

potentials in neurons, and many others examples. Everyday examples of

chaotic systems include weather and climate.

The course will be at the beginning graduate level. The required

background consists of calculus, elementary linear algebra, and the

desire to learn about dynamical systems and what "chaotic" means to a

mathematician (it's a lot more understandable than it might seem!). The

text for the course will be “An Introduction to Chaotic Dynamical

Systems (second edition)” by Robert L. Devaney.

Undergraduates are welcome to enroll in the course, but will need to

contact the instructor (mrinal@unt.edu) before doing so.

GRADING POLICY: Grades will be based on the total number of points accrued from a mixture of homeworks and possible in class presentations. It is difficult a priori to determine break points for the final grades. However, the golden rule in determining the final assigned grade is that if the number of points earned by person A is ≥ to the number of points earned by person B, then person A has a grade which is ≥ to the grade of person B.

NOTES: The basic theme for this year long course will be the spectral theorem for self-adjoint operators in Hilbert space and the extensive background needed to present this theorem and its many applications and ramifications from the correct, general, modern measure theoretic point of view. Many possible dissertation research topics should be presented throughtout the year.

Algebraic Topology, Math 6620

Spring 2008

TR 12:30-1:50 p.m.

When most people think of algebraic topology, they think of homology and homotopy groups. Both homology and homotopy measure "holes" in topological spaces, but they have slightly different ways of distinguishing what a hole is. In the case of homotopy, a hole is defined to be an embedded sphere that cannot be made to collapse by perturbing the embedding. For example, the unit circle around the origin of the plane can be perturbed or shrunk to a point, whereas this cannot happen in the punctured plane. In homology objects more general than spheres are allowed to "surround" holes. For example, a torus has isomorphic 1-dimentional homology and 1-dimentional homotopy groups, but in dimension two the homotopy group is trivial (meaning that any 2-dimentional sphere mapped to the torus can be shrunk to a point) while the homology group is not. Interestingly, homology groups are a little harder to define, but generally much easier to compute than homotopy groups. In fact, the homology groups of spheres are very easy to compute, while the homotopy groups of spheres are not known.

I intend to cover the traditional topics of the fundamental group (dimension 1 homotopy group) and singular homology (and cohomology) in the fall. Most of the semester will be dedicated to defining the concepts and developing the basic properties of each. I also hope to define the higher homotopy groups. In the process, we will see some applications such as the Brower Fixed Point Theorem and some results on tangent vectors on spheres (you can’t comb the hair on a billiard ball).

If all goes well, I hope to do a few other topics in the spring that use the basic algebraic topology covered in the fall. For example, I hope to cover some topics such as characteristic classes, topological K-theory, and an introduction to spectral sequences used in algebraic topology.