MATHEMATICS
Exponential Sums
Exponential sums have important applications in many areas of mathematics, including Diophantine equations and Goppa codes. The principal aim of this research is to study properties of exponential sums over finite fields, particularly those properties that are reflected in the L-function of the exponential sum and hence have a cohomological interpretation. Both l-adic and p-adic methods will be used. A key component of the project will be to study the p-adic differential equation that describes the variation of p-adic cohomology in a parametrized family of exponential sums.
National Security Agency
Alan Adolphson
Graphing Calculators: Teachers and Students Learning Together
This project responds directly to the needs expressed by mathematics teachers to know more about graphing calculators and how to integrate them into their teaching. The objectives are to develop a user-friendly Student Learning System (SLS), including written materials, programs, and video supplements for grades 9-12; develop a user-friendly independent study Teacher Learning System (TLS), including written materials, programs, and video supplements for grades 9-12; and disseminate the SLS and TLS to the state Professional Development Center (PDC) directors and mathematics coordinators through a one-day conference. The state-of-the-art TI-85 graphing calculator has been selected for the project. The plan for accomplishing the project objectives utilizes close collaboration with four Summer Academy projects funded by the Oklahoma State Regents for Higher Education, the Woodrow Wilson National Fellowship Foundation Summer Institute program for teachers, Texas Instruments, Incorporated, and the state PDCs.
Oklahoma State Regents for Higher Education
Douglas Aichele
Summer Institute in Functions
This summer institute will present new directions in the content and teaching of the college-preparatory curriculumÑfirst-year algebra through mathematics courses preceding the study of calculus. Participants will use graphing calculators and the most powerful computer software available to explore new, as well as traditional, topics in the school mathematics curriculum. Visualizing functions, developing mathematical models, and solving real-world problems provide a basis for rethinking the curriculum.
Woodrow Wilson National Fellowship Foundation
Douglas Aichele
Geometry of Classical Banach Spaces
Geometrical questions concerning both finite and infinite dimensional Banach spaces are the subject of this research. One goal is to classify up to isomorphism the complemented subspaces of continuous functions and spaces of functions with integrable pth powers. Another goal is to understand the relationship between convex bodies and associated objects and affine invariants such as the convex floating body, projection bodies and the affine surface area.
National Science Foundation
Dale Alspach and Carsten SchŸtt
Equity 2000 Awareness Workshop
An agreement to provide consultant services regarding Equity 2000 awareness.
Oklahoma State Department of Education
James Choike
Analytic Theory of L-Functions
The theory of L-functions is central in the study of number theoretical questions, especially in the realm of analytic number theory. For most applications, the relevant questions usually involve the analytic continuation of the L-functions and estimates for their growth in vertical strips, their functional equations, and the distribution of their zeros. This project deals with both very general questions (the classification of Dirichlet series with functional equations and Euler products) and particular questions (L-functions, the evaluation of their zero-free regions, and the distribution of their zeros in the critical strip).
National Security Agency
Brian Conrey and Amit Ghosh
Paraconformal Geometry, Twistor Theory, and Lie Groups
Research in the overlap of complex geometry, twistor theory, and the representation theory of Lie groups. In particular, the application of techniques and ideas originally used in twistor theory to construct new representations of Lie groups. The search for the singular representations has been one of the most important driving forces in representation theory for the last twenty years.
Dean's Incentive Grant
Edward Dunne
Topics in Automorphic Forms
The principal aim of this research is to continue work on three projects in automorphic forms. All three projects are joint work with Piatetski-Shapiro. The first is a book project on L-functions for GLn. The aim of this project is to collect data and make sure all necessary results on L-functions for GLn are in place in order to apply Piatetski-Shapiro's Converse Theorem to the problem of Langlands' Lifting of automorphic representations from the classical groups to GLn. The second project is to determine whether the knowledge of the L-function of a representation of GLn over a finite or local field, twisted by representations of GL[n/2], is enough to determine the representation uniquely. These are the first steps toward proving a global Converse Theorem for automorphic forms on GLn involving only twists by GLm for 1 ² m ² [n/2]. The last project is unrelated to the first two. It is to investigate the possibility of interesting phenomenon in Base Change for certain orthogonal groups coming from the strange behavior of Base Change for the metaplectic group SL2.
National Security Agency
James Cogdell
21st Century Mathematics Classrooms for the Oklahoma Higher Education System
This grant provided funding to place microcomputers in mathematics classrooms. Computers, used properly in the classroom, can serve to brighten up mathematics and may serve to improve student retention rates. The project will develop materials and software that can be distributed to all Oklahoma colleges and universities.
Oklahoma State Regents for Higher Education
Benny Evans and Jerry Johnson
Summer Academic Futures in Science Young Scholars Program
A three-week summer session designed for rising junior and senior high school achieving, high-potential students. Those who attend better understand the ways that scientists think and work, learn of the many opportunities in science, and experience the excitement of interacting with scholar/mentors and other outstanding students in an enriched academic environment. Morning sessions deal with the multidisciplinarity of problem solving. Afternoon classes focus on specific disciplines of mathematics, biochemistry, and chemistry.
Oklahoma State Regents for Higher Education
Joel Haack and Wayne Powell
RCMS: Conference for Oklahoma Native American Mathematics Educators and Students
The purpose of this project was to design, develop, promote, conduct, and evaluate a conference for Oklahoma Native American educators and students. The objectives of the conference were to prioritize mathematics education issues for Oklahoma Native American students and their mathematics teachers and to determine a general plan of action to solve these issues. The overall goal is to help more Oklahoma Native American students (grades 6-12) to successfully take more mathematics courses and, as a result of this, improve the high school graduation rate of Oklahoma Native American students, in particular, those sufficiently prepared in mathematics to enter a mathematics-dependent college plan of study.
National Science Foundation
John Jobe
A Learning System for Intermediate Algebra
The purpose of this project is to improve the learning and the instructional quality of Intermediate Algebra at all of Oklahoma's institutions of higher education. This will be accomplished by designing, developing, and disseminating a coordinated package of learning materials for Intermediate Algebra. The package is called "A Learning System for Intermediate Algebra" and will serve as a supplement to a textbook.
Oklahoma State Regents for Higher Education
John Jobe and James Choike
Implementation of Harvard Consortium Materials in Oklahoma
Funding to fully implement the Harvard Consortium materials in OSU's calculus curriculum beginning in the Fall of 1993. Faculty will disseminate the experience to regional institutions during the remaining life of the grant.
Arizona University
Jerry Johnson
Fresh Perspectives on Old Topics: A Workshop and Forum for College Faculty
It is difficult for an instructor to maintain a sense of enthusiasm and vitality in calculus and college algebra when he or she teaches the same problems and topics year after year. This project involved holding two summer workshops for fifty participants to introduce them to fresh perspectives and to provide a forum for them to share their own ideas. Many of the topics involved hands-on use of microcomputers as a tool. The project also included the publication of workshop materials for national dissemination.
National Science Foundation
Jerry Johnson and Benny Evans
A Computerized Classroom for Undergraduate Mathematics
Funding to create a computerized classroom to serve primarily Calculus, but also Linear Algebra and Differential Equations. It will contain 40 IBM compatible PCs networked to a file server with the latest software, such as Derive, GyroGraphics, and MicroCalc. The creation of the computerized classroom allows students to explore mathematics at a level of depth and excitement not possible before.
National Science Foundation
Jerry Johnson and Benny Evans
Projective Geometry and Threefolds
This research includes three projects in algebraic geometry. The first is to find the largest geometric genus among codimension two projective varieties of degree d. This will be done by generalizing work of Gruson and Peskine, Harris, Hartshorne, and Hirschowitz. Techniques to be used include postulation of hyperplane sections, liaison, and reflexive sheaves. The second project is the investigation of which cDV threefold singularities admit small resolutions. This research will closely follow a method previously used by the investigator to give the complete answer in the cAn case. The third project is a collaboration with Dr. Bruce Crauder and will build on previous work on the classification of all birational automorphisms of projective space having a smooth and irreducible fundamental locus by completing more of the classification, assuming the validity of Hartshorne's conjecture on complete intersections.
National Security Agency
Sheldon Katz
Computational Explorations in Geometry and Analysis
Funding to host a Research Experiences for Undergraduates (REU) site at the Department of Mathematics during the summers of 1993 and 1994. Applications will be solicited for students to engage in one of two research projectsÑone in algebraic geometry, commutative algebra, and computer algebra; the other in analysis and representation theory. The common focus will be on discovery through examples and computation.
National Science Foundation
Lisa Mantini
Topics in the Cohomology of Arithmetic Groups
Projects dealing with the cohomology of arithmetic subgroups of GL(3,Z) and Galois representations, classical projective geometry and cohomology, and a retract for Sp(4).
National Science Foundation
Dean's Incentive Grant
Mark McConnell
Futures in Science and Mathematics: A Summer Academy
A three-week academy was sponsored for rising juniors and seniors in high school who have special talents in science and mathematics. A separate two-week academy was conducted for freshmen and sophomores. Each student worked on problem solving techniques along with computer implementation skills. The students were split into three groups for concentrated research projects. These research projects involved topics in mathematical games, biochemistry DNA investigations, and chemistry studies of corrosion. Specific attention was also given to communication skills, both written and oral.
Oklahoma State Regents for Higher Education
Wayne Powell, Dennis Bertholf, and Douglas Aichele
Oklahoma Principals' Science Scholars
This program was designed to enhance the mathematical and scientific skills of minority students who are rising juniors and seniors in high school. The students were selected from nominations made by high school principals. A three-week summer academy gave concentrated study in problem solving skills, computer science, botany, statistics and zoology.
Oklahoma State Regents for Higher Education
Wayne Powell and John Wolfe
Office of Multicultural Affairs--Gregory Washington
Early Placement Evaluation in Mathematics: Linking High School and College Mathematics
The twofold goal of this project is to carry out a process to clarify and define the competencies in mathematics necessary for success in college and university work, and to execute a program to articulate these mathematical needs directly to high school students, to high school mathematics teachers and counselors, and to the general public.
Oklahoma State Regents for Higher Education
John Wolfe
Number Fields and Prehomogeneous Vector Spaces
The principal aim of this project is to compute the density of discriminants of quartic and quintic extensions of global fields by means of the Shintani-Sato theory of zeta functions associated with prehomogeneous vector spaces. The basic arrangement for this calculation has been discovered in the past two years in joint work with Akihiko Yukie. There are many technical local calculations to be carried out before the final answer will be known. We expect to be able to use observations of Heilbronn to deduce mean-value theorems for the 2-class-numbers of cubic fields from the quartic disciminant densities.
National Security Agency
National Science Foundation
David Wright and Akihiko Yukie
Unitary Representations in Dolbeault Cohomology
This project is concerned with the representation theory of semisimple Lie groups. An important problem is to realize unitary representations in a natural geometric way; this includes a natural formula for the invariant inner product giving the unitary structure. In this project, we will study the realization of singular unitary representations associated to elliptic coadjoint orbits, that is, representations on Dolbeault cohomology. The point is to unitarize these representations using L2 harmonic forms and the invariant indefinite metric on the orbit.
National Science Foundation
Roger Zierau