Victor Nistor - US grants

Nistor will continue his work in that part of the theory of operator algebras relating to noncommutative geometry, and particularly index theory and cyclic cohomology. His immediate goals include a formula for the bivariant Chern-Connes character and the identification of the Atiyah-Singer integrand, computation of the cyclic cohomology of foliation algebras using Bott's simplicial methods, and a combination of these two into an index theorem for foliations. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.

9457859 Nistor This research, supported by a National Science Foundation Young Investigator award, will involve research in "non- commutative" geometry. This very active field is a program to extend the subject of the usual differential geometry of space to "non-commutative" settings that are typical of quantum mechanics and other related phenomena. The work centers around new character and index invariants previously pioneered by Nistor. The National Science Foundation Young Investigator award recognizes outstanding young faculty. The award recognizes the recipient's strong potential for continued professional growth as a research mathematician and for significant development as a teacher and academic leader. ***

Abstract
Nistor

The project is devoted to application of certain areas of Analysis, especially Index theory, K-theory, and Hochschild and cyclic homology, to a broad range of problems that have implications in several other areas of mathematics and physics. A common issue in these problems is to identify the relevant algebras that model specific situation of interest. Sometimes these algebras already exist, sometimes they have to be constructed. These algebras will then be studied with the indicated homological tools, and the results will be interpreted as providing information on the specific situations that are studied. Some of the applications include Index theory on singular spaces, the spectral and Index theory of Dirac operators coupled with vector potentials, and determinants of elliptic operators. A different but closely related type of application is a cohomological study of the representation space of a p-adic group.

The index of an operator in its simplest form is a number. Many quantities from mathematics and even from physics and chemistry can be identified with the index of a suitable operator - for example, the number of electrons occupying a certain energy level in an atom is the index of a suitable operator. The Dirac equation (or operator) is one of the fundamental equations in physics; the number of solution of the Dirac equation is very closely related and can often be identified with the index of that operator.

9981251
Nistor

This three-year award for U.S.-France collaboration in mathematical sciences involves Victor Nistor of Pennsylvania State University and Thierry Fack and Moulan Benameur of the University of Lyon. The project, supported under the joint program of the National Science Foundation and the French National Center for Scientific Research (CNRS), is a study of algebras relevant to foliations. The investigators will use homological methods (K-theory, Hochschild, and cyclic homology) in their study. They are interested in obtaining index theorems and in studying the spectral invariants of operators acting along the leaves of foliations. Their approach to index theorem for foliations will involve the bivariant Chern-Connes character of the pseudodifferential extension of operators along the leaves of foliations.

This award represents the U.S. side of a joint proposal to the NSF and the French National Center for Scientific Research (CNRS). NSF will cover travel funds and living expenses of the U.S. investigator and the CNRS will support the visits of the French mathematicians to the United States. The project takes advantage of complementary expertise, in particular, the French team's expertise in the proposed approach. It will advance understanding of index theorems and other related topics.

Abstract
Nistor

This project is devoted to the analysis and spectral theory of differential operators on non-compact manifolds. The investigator is especially interested in generalizing the classical results of elliptic theory for compact manifolds, including index theory. Little can be said about all non-compact manifolds in general, but results of Bismut, Beunning, Cordes, Mazzeo, Melrose, Meuller, Shubin, and others have singled
out a class of manifolds that is more amenable to study: the class of manifolds with a uniform structure at infinity. The local theory (regularity, local existence) for these manifolds is the same as for compact manifolds, so our methods will necessarily be global. Thus, in addition to the methods of Partial differential equations and Differential geometry used by the above mentioned authors, methods from Operator algebras have come to play an increasingly important role in the study of non-compact manifolds with
a uniform structure at infinity, as is seen from the work of Connes, V.F.R. Jones, Lauter, Monthubert, Skandalis, Taylor, and the investigator.

Manifolds with a uniform structure at infinity appear naturally in Scattering theory, Differential geometry, Representation theory, Mathematical physics, and certain problems of Applied mathematics. The results of the proposed research will have, in the long run, applications to all these domains. The main methods that are proposed belong to Analysis, especially Partial differential equations, Operator algebras, Spectral theory, and K-theory. A main technical tool will be provided by algebras generated by differential
operators on non-compact manifolds with a uniform structure at infinity. By using Sobolev spaces, one can reduce many of our basic questions to questions about algebras of bounded operators. A novel feature of this proposal is the study of boundary value problems for manifolds with a uniform structure at infinity that have Lipschitz boundaries, as in the work of Mitrea and Taylor on such compact domains.

DMS Award Abstract
Award #: 0209497
PI: Xu, Jinchao
Institution: Pennsylvania State University
Program: Computational Mathematics
Program Manager: Catherine Mavriplis

Title: Multiscale Methods for Partial Differential Equations

The focus of this work is on the development and applications of a two-scale discretization technique, namely the finite element method based on partition of unity. One main application is on the design of efficient discretization for nonmatching (either overlapping or nonoverlapping) grids. The main idea of nonmatching grids is to divide a physical domain into a set of overlapping or nonoverlapping subregions which can accommodate smooth, simple, easily generated grids. In this approach, a grid generation for complex geometries can be made simple, refinement grids can be added or removed without changing other grids, different equations/numerical methods may be used on different grids, efficient structured grid solvers may be used. Furthermore, overlapping grids are well suited for parallelization and vectorization. The proposed generalized finite element method based on partition of unity provides a general and powerful discretization framework for this type of grids. Another major task is the development of a multigrid iterative method for solving the resulting algebraic systems for these new discretization schemes. As divide and conquer techniques, the proposed multiscale algorithms are suitable for parallel and high-performance computers.

A class of new multiscale techniques are proposed to study for efficient numerical solution of partial differential equations. Multiscale methods in general are proven to be among the most powerful mathematical tools for the investigation of a broad range of models that are described by partial differential equations. Their pivotal role in the design of fast, reliable, and robust numerical methods for the solution of various problems places them among the most important research areas in the applied mathematics in the recent years. Since these methods are in some sense problem-independent, they are expected to have many important applications in science and engineering such as composite materials and subsurface flows in environmental applications.


Date: May 28, 2002

Abstract
Nistor

The proposed research extends and strengthens
the field of applications of Operator Algebras. It leads to new
results on the structure and representations of groupoid algebras associated
to singular spaces and non-compact manifolds. It leads also
to a new characterization of the Fredholm operators on non-compact
and singular spaces based on representations of groupoid C_-algebras.
Using also methods from Noncommutative Geometry (cyclic homology,
Chern character, smooth subalgebras) the indices and relative indices
of operators on singular and non-compact spaces will be computed.
These index theorems are non-local, so they will lead to spectral invariants,
generalizing the eta-invariant, that will be investigated. The
homology, K-theory, and other invariants of the relevant operator algebras
will also be determined. The spectrum and the structure of the
distribution kernels of operators on singular spaces will be investigated.

The proposed research will have applications to numerical
methods for polyhedral domains, which are important in Engineering
(an example is the method of layer potentials), and to the analysis on
non-compact manifolds with nice ends, which arise in String Theory
and General Relativity. This proposal will contribute to the development
of the general techniques necessary to approach mathematical
and computational problems from Biology, Chemistry, Engineering,
and Physics. It will also contribute to applying mathematical results
in practice by interactions with researchers from other fields, by organizing
conferences, and by advising students, which will lead to the
creation of specialists able to use theoretical tools to handle practical
problems.

Proposal Title: Collaborative Research: Multigrid QCD at the Petascale
Institution: Trustees of Boston University
Abstract Date: 10/10/07
0749300
Brower
0749202
Brannick
0749317
McCormick
Numerical solutions to Quantum Chromodynamcs on a lattice are critical to high
precision experimental tests of the standard model and an ab-initio understanding of
nuclear matter. The core of these calculations involves inverting a Dirac matrix which
becomes increasingly ill conditioned as the lattice is refined. Consequently while
Terascale computing hardware has exposed this new physics, it is incapable of fully
accommodating it. On the other hand, if lattice QCD algorithms are reformulated to
exploit and reveal the physics at this finer microsale, Petascale hardware does have the
potential for opening up a new era of physics discovery. This award brings together a
close collaboration of leading experts in applied mathematics and theoretical physics to
meet this challenge by the application of new multi-level algorithms for QCD
simulations. The central mission of the proposed Multigrid Quantum
Chromodynamics at the Petascale project (MGQCD) is: to develop new and significantly
more robust multigrid methods for enabling more complex and higher fidelity physics for
lattice QCD calculations; to support their migration into Petascale simulations; and to
engage the broader scientific community through collaborative research and
educational activities that highlight the multigrid methodology.
NATIONAL SCIENCE FOUNDATION
Proposal Abstract

Computational methods are used in almost any area in which computers are used. Improved numerical and computational methods are one of the main venues to improve the speed of modern computer applications. The "Finite Element Method" is a very popular numerical method that is used by engineers, physicists, biologists, meteorologists, in financial mathematics, and in many other areas. More concretely, the finite element method is used in the study of heat propagation, radar detection, medical imaging, structural analysis of buildings, design of aircraft wings, and in many other practical problems.

The first step in the use of the Finite Element Method is to fictitiously divide the domain to be studied (the domain occupied by a building, aircraft, body to be imaged) into many small triangles, parallelepipeds, or tetrahedra. This is to a large extent an elementary task, but very time-consuming. Moreover, additional care has to be paid close to the vertices and the edges of the domain. Without this additional extra care in the way we divide our domain, obtaining the desired precision in calculations would take much longer. For example, some recent results in which the Principal Investigator is also involved lead to the estimate that, for a precision of three exact digits, a careful division procedure can decrease the time of calculation by one million times or more. For a precision of five exact digits, the estimated improvement is one trillion times or more.

While the mathematical techniques to decide the shape of the improved divisions of the computational domain are quite sophisticated (anisotropic mesh refinement, elliptic partial differential equations, non-compact manifolds, functional analysis), once such a division algorithm has been formulated, it can be taught to a good beginning undergraduate student. The main purpose of this proposal is to train undergraduate and graduate students in state-of-the-art Finite Element Method techniques (including, but not limited to, anisotropic mesh refinement, a priori and adaptive mesh refinement, meshless methods, and multigrid methods for solving the resulting linear systems). The more advanced the students, the more opportunities they will be given to learn about the inner workings of these methods. Each of the students involved will be expected to produce original research at their corresponding level of mathematical training. The resulting training of the students will provide the necessary numerical experience needed by the students, by the Principal Investigator, and by others to conduct cutting-edge research in the future. Another important quality of the Finite Element Method is that it can be taught to students with a very wide scale of mathematical backgrounds and hence sophistication. As such, by training more people with various backgrounds to use such methods and to do research on them, it is expected, based on previous experience, that a wider range of students, including underrepresented groups, will be attracted to mathematical research.

Numerical Methods for Partial Differential Equations (PDEs) are an
essential ingredient in many areas of Sciences and Engineering. Often
in the applications of PDEs, one has to deal with additional
difficulties caused by the singularities in the geometry, the
coefficients, or the boundary conditions. The main goals of this
proposal are to obtain more efficient numerical methods for equations
with singular solutions and to develop user friendly implementations.
The PI and his collaborators will design improved meshes that provide
optimal rates of convergence for the Finite Element Method in three
dimensions for elliptic equations on polyhedral domains with
non-smooth interfaces. These equations form one of the basic
ingredients in many other applications. The results will be extended
to quasi-linear elliptic equations and to evolution equations. For
some of these equations, spaces with higher smoothness or other additional
properties are sometimes needed, and the Generalized Finite Element
Method will be used for these purposes.



The proposal will contribute to the formation of graduate and
undergraduate students by advising and mentoring them and by
integrating them in research projects. The PI also plans to continue
to run an Interdisciplinary Seminar featuring outside speakers,
including non-mathematicians. The proposed research will lead to the
design and testing of new and improved numerical methods for Partial
Differential Equations (PDEs) of interest in Sciences and
Engineering. The resulting numerical methods will be presented and
implemented in a way that makes them accessible to
non-mathematicians. It will also contribute to applying mathematical
results in practice by interaction with researchers from other fields,
including the private sector. It will also contribute to the creation
of specialists able to use advanced theoretical tools to handle
practical problems. He will also introduce numerical methods and the
use of computers in more traditional undergraduate courses. The PI
also plans to popularize mathematical questions that arise from
practice among mathematicians. The theoretical results and the
resulting numerical methods will have applications in Mathematical
Physics, Quantum Chemistry, Biology, Finance (Risk Management and
pricing of Financial Derivatives, or Options), and other areas.