Christopher Judge - US grants

Proposal: DMS-9972425

PI: Christopher Judge

Abstract: We consider the asymptotic behavior of the geodesic flow and Laplace eigenvalues on manifolds of nonpositive curvature with an emphasis on aspects related to metric variation. For example, the billiard ball flow of a rational polygon belongs to a natural moduli space of flows, and the system's periodic asymptotic behavior can be deduced from knowledge of the moduli space. Although, the geodesic flow and the Laplace spectrum are known to be related via trace formulae, physicists have long believed in a more direct relationship provided by the `correspondence principle'. This relationship will be considered in the context of the moduli space theory.

The objects under study are models for physical systems. In particular, we consider here the asymptotic behavior of particles, both classical and quantum, under varying conditions. The systems we consider exhibit a high degree of chaotic behavior as is often the case in the real world. Although these systems are simple abstractions, they represent a testing ground for further understanding of real world systems. Eventual 'downstream' applications could include high energy physics and cryptography.

Abstract

Award: DMS-0107018
Principal Investigator: Christopher M. Judge

The Bloomington Geometry Workshop is the first of an intended
series of regional conferences on geometry. The main purpose of
the Bloomington Geometry Workshop, to be held at the University
of Indiana, is to promote scientific cooperation and
communication between geometers and geometric analysts in a
region including Indiana, Illinois, Kentucky, Michigan, Ohio, and
Tennessee.

Modern geometry addresses problems such as understanding the ways
in which typical spaces are curved and how far from the flat
Euclidean model a space might be, the influence of curvature upon
analysis in a particular geometry, interactions of large-scale
geometry with probability theory and the algebraic properties of
the fundamental group, hyperbolic geometry and the portion of
low-dimensional topology that increasingly depends upon
hyperbolic structures, and gauge field theories and other ideas
from theoretical physics.

Abstract

Award: DMS-0503596
Principal Investigator: Christopher M. Judge and Bruce Solomon

The Bloomington Geometry Workshop brings together leading
geometers to discuss and disseminate recent advances in geometry
and related fields. This year's list of speakers includes Ian
Agol, Francois Ledrappier, John Lott, Gang Tian, and Fangyang
Zheng. These speakers will inform participants of work in
hyperbolic manifolds, rigidity, Kahler geometry, and the Ricci
flow.

The purpose of the Workshop is to help foster cooperation and
collaboration among researchers in the midwest region as well as
help educate graduate students and young researchers about the
latest developments in geometry. Most of the funding will be used
to support the travel of graduate students and recent PhD's.
These young researchers will come away from the workshop better
prepared to take part in current research in geometry, a field of
basic research that is important to computer graphics, quantum
physics, material science, and many other scientific disciplines.

Research Experiences for Undergraduates in Mathematics at Indiana University will provide twelve undergraduate mathematics majors with a research topic and a research environment so that they have an active eight week experience involving mathematical research. The students will face the intellectual challenges of learning more about a mathematical topic and developing approaches and solutions for unsolved
questions at a research level. This research experience will enhance the students' perspective on mathematics and provide impressions of how mathematical knowledge is evolving. It will also be a valuable
experience in actively finding results and communicating their work to their peers.

The grant provides funds to support the Bloomington Geometry Workshop. It is an annual meeting with the 2014 edition taking place over the weekend of April 26-27. The meeting starts with a Friday afternoon lecture and then there are seven to nine presentations on Saturday and Sunday by well-known active geometers primarily from the Midwestern region.

The Bloomington Geometry Workshop aims to promote scientific cooperation between geometers in Indiana and its neighboring states. Graduate student participation has continued to increase, and the organizers encourage this by asking speakers to make at least a portion of their talk accessible to graduate students. The Workshop organizers also invite at least one graduate student speaker and solicit graduate student participation from individual departments, especially from among under-represented groups.

More information about the workshop can be found on the website

http://www.math.indiana.edu/bgw/