Jeremy Teitelbaum - US grants

This award supports the research in arithmetic algebraic geometry of Professor Jeremy Teitelbaum of the University of Illinois at Chicago. Dr. Teitelbaum will pursue his studies of the geometry and arithmetic of the p-adic upper half plane, guided by the Exceptional Zero Conjecture, which relates the values of certain functions, namely p-adic L-functions, to the analytic geometry of certain curves. In addition, he will continue his study of the computational complexity of algorithms from algebraic geometry. Arithmetic algebraic geometry is a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.

Professor Teitelbaum will work on the p-adic analytic theory of modular curves and its arithmetic implications. He will study higher weight invariants of modular forms, the arithmetic of Shimura varieties and the arithmetic of Drinfeld modular curves. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

This award will provide essential computational resources, to several on-going research projects in number theory, algebra, and algebraic geometry. These projects cover the related areas of representation theory; modular forms and elliptic curves; the topology of algebraic varieties and Picard-Fuchs, differential equations; and Arakelov theory, K-theory, and arithmetic geometry. The investigators plan variously to apply symbolic algebra programs, such as REDUCE, Maple, Macaulay, to these problems; to develop packages of procedures in these languages; and to develop programs in lower level languages such as FORTRAN and C. In addition, the equipment will be used to support graduate students working in these areas.

This award provides support for a three-day conference on computational number theory to be held in honor of Professor A.O.L. Atkin. The conference, which will take place on September 14-16, 1995, at the University of Illinois at Chicago, will focus on two fundamental aspects of computation in number theory. The first of these concerns development of computational techniques to conduct experimentation and support conjectures in the pure theory of numbers, especially in areas related to modular forms; the second deals with applications of techniques from the pure theory to algorithmic problems such as those arising in factoring, primality testing, and counting points on algebraic varieties. This research is in the field of number theory. This subject area has its historical roots in the study of the whole numbers, addressing such questions as the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for mainly aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and the design of communication systems.

This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results have implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. Both mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbe rs, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory. This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results hate implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. B oth mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory.

The Department of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago will purchase computational equipment which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects, including in particular research in Geometric Group Theory (Paul R. Brown); Topology of 3-Manifolds (Marc Culler); Algebraic Geometry (Anatoly Libgober); Theory of Finite Groups (Stephen Smith); and Number Theory (Jeremy Teitelbaum). Most of these projects are data-intensive and amenable to distributed computing techniques.

The investigator and his colleagues study the representation theory of p-adic groups in topological vector spaces over p-adic fields, with the goal of relating such representations to arithmetic by means of a "p-adic analytic local Langlands correspondence." Though still in its early stages, work of the investigator, his collaborator Peter Schneider, and others including Robert Coleman and Barry Mazur, and more recently Matthew Emerton and Christophe Breuil, have provided evidence that such a correspondence exists. The investigator has the hope that this approach will provide a conceptual link between p-adic L-functions, p-adic Galois representations, and p-adic automorphic forms, in the same way that the classical Langlands correspondence does for complex L-functions, automorphic forms, and ell-adic representations.

One principal of current research in number theory is the Langlands program, which proposes a deep relationship between certain complex analytic functions related to algebraic groups called "automorphic forms" and the number of integer solutions to classes of polynomial equations, through the medium of functions called "zeta functions." The power of this idea was strikingly demonstrated by Wiles' proof of Fermat's theorem, which proceeded by establishing one very special case of the Langlands conjectures. A second important principal in this field is the idea that a full understanding of number theory requires the study, not only of the geometric behavior of equations over the real and complex numbers, but also over the less widely known fields of p-adic numbers. Indeed, many of the important classical objects of number theory, such as modular forms andL-functions, that are important in the Langlands project, have p-adic versions. These p-adic versions capture important information not accessible from the classical situation. This project proposes to extend our understanding of the p-adic versions of automorphic forms and zeta functions by extending the philosophy of the Langlands program to the setting of p-adic analysis.

The Assuring Stem Credential Expansion through Nurturing Diversity (ASCEND) project is increasing the number of students earning undergraduate degrees in science, technology, engineering, and mathematics (STEM) with special emphasis on retaining and graduating women and minorities. The university has been successful at enrolling students into STEM disciplines but attrition rates are high. ASCEND is addressing this loss of talent through coordinated, mutually reinforcing activities that transform the vulnerable first-year STEM experience.

The intellectual merits of ASCEND include the team's approach to developing and utilizing a dynamic model (one that incorporates national best practices for recruiting and retaining underrepresented STEM students) and a careful and thorough evaluation of its impact. In concert with this, ASCEND is enabling a cadre of highly-committed faculty, advisors, and top administrators to execute four strategies: create an ASCEND Learning Community for STEM freshman; increase the quality and relevance of STEM teaching through a Community of Practice; strengthen STEM support units and Learning Centers and foster collaboration; and build institutional capacity to capture, track, and report on the academic trajectories of STEM students.

ASCEND's broader impacts include improved retention and graduation rates in science, technology, engineering, and mathematics (STEM) for women and minorities. These graduate scientists and engineers are lending their perspectives and talents to the local and national STEM workforce. The ASCEND model also is informing efforts at other comparable institutions, and is having an impact by increasing diversity and gender equity in STEM.