Charles Bu
Megan Kerr
Martin A. Magid
Alan Shuchat
Frederic W. Shultz
Alexia Sontag
Ann Trenk
Howard Wilcox
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Charles Bu - Applied Mathematics
Partial Differential Equations, Initial and/or Boundary Value Problems, Nonlinear Wave Phenomena, Applications in other Physical Sciences.
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I am working in differential geometry, specifically the geometry of
submanifolds. A surface in 3-space is an example of a submanifold. I study the properties of the submanifold in relation to the larger space. Recently I have begun to work in affine differential geometry. In this setting I study properties which are invariant under the special linear group, rather than the orthogonal group.
I have supervised several independent studies in logic and one in dynamical systems, even though these are not my areas of research.
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My main areas of interest are functional analysis and operations research. Functional analysis involves using techniques of analysis to study infinite - dimensional vector spaces and their linear transformations. Operations research applies linear algebra, probability and network theory to decision-making problems in government and industry.
Student Projects:
Markov chains, queues, and stochastic processes
Linear algebra and population modeling
Integration theory and probability
Analysis in infinite-dimensional vector spaces
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Area of Interest:
My area of research is functional analysis - a combination of vector spaces and analysis. In general, I enjoy problems involving the interplay of different areas of mathematics. My current research concerns C*-algebras, and involves geometry, topology, algebra, analysis and some applications in physics. Finally, in the last few years I have become very interested in applications of mathematics, in particular, building mathematical models. My thesis work related quantum mechanics and mathematics; recently I have focused on applications of mathematics in the social and life sciences.
Student Projects:
Currently an honors student is working on C*-algebras. Previous student work concerned Lebesque integration, probability, and Fourier series.
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Area of Interest:
My research interests are in complex analysis, especially quasiconformal mappings. These generalizations of conformal or angle-preserving maps arise in many contexts and have fruitful generalizations in higher dimensions.
My work also includes collaboration with C. Constantinescu and K. Weber, of the Federal Institute of Technology, Zurich, in writing Integration Theory, Volume 1: Measure and Integral, Wiley-Interscience, 1985, and Volume 2: Real Integration Theory, in progress.
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My research field is graph theory which is a subfield of combinatorics. The "graphs" in graph theory are not the usual type with x and y-axis. Instead, a graph is an abstract object which consists of a collection of points, some of which are connected by lines. Graphs can be used to model many situations. For example, an airline might be interested in the graph that has one point for every city that airline services, and a line between two cities if and only if the airline runs a direct flight between those cities. This graph could help the airline re-route passengers efficiently in the event of overbooking or delays.
Many of the problems which interest me are characterization problems, that is, I try to classify the set of all graphs with a particular property. This can be easy or hard, depending on the property. I am currently studying a class of graphs called "tolerance graphs". Tolerance graphs model scheduling problems in which some overlap between scheduled events is permitted. I am also interested in a class of graphs called "leveled-planar graphs", whose drawings (on a piece of paper) must satisfy certain geometric restrictions.
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Area of Interest:
Structure of Topological Groups. In particular, structure of compact non-abelian groups.
Applications of Mathematics in Music Theory. In particular, group theoretic applications in 12-tone composition.
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Megan Kerr (e-mail Professor Kerr)
RESEARCH DESCRIPTION:
I work in the area of differential geometry. Geometry is a subject with a long, rich history, and a subject used in such diverse areas as astronomy (the shape of the universe), machinery design, and the study of DNA coiling. Riemannian geometry is the study of the shapes of manifolds (generalized surfaces) endowed with metrics (distance functions). A choice of metric completely determines the shape of a manifold. A manifold can carry infinitely many metrics, since there are infinitely many ways to bend or stretch a manifold without making holes or creases.
My research involves the interplay of group theory and geometry. I consider a special class of manifolds with a high degree of symmetry, called homogeneous spaces. Most recently, I have been working on these two projects:
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The first project is to find an explicit description of all the Einstein solvmanifolds near a particular one, the quaternionic hyperbolic symmetric space HH3. The family of Einstein solvmanifolds near this symmetric space is known to be 12-dimensional, yet there is no explicit description of its neighbors. (A solvmanifold is a solvable Lie group with a left-invariant metric; it is Einstein if it has constant Ricci curvature.)
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The second project is joint with Chuu-Lian Terng at Northeastern University and Andreas Kollross at Augsburg University, in Germany. This work involves investigating a system of soliton equations determined by a particular symmetric space. A family of soliton equations is a natural non-linear PDE system. With some work, one can find a one-to-one correspondence between solutions of the PDE system and a class of submanifolds in space forms. The goal is to study the class of submanifolds -- the parametrizing space of a class of submanifolds can be unwieldy, whereas soliton theory provides a nice description of the space of solutions. The deep connection between soliton systems and geometry has not been fully explored.