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Elliptic operators, topology and asymptotic methods

Elliptic operators, topology and asymptotic methods BOOK REVIEWS 193 References 1. Guckenheimer, J. and Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Math. Sciences 42, Springer, Heidelberg, 1983. 2. Henry, D.: Geometric Theory of Semitinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer, New York, 1981. 3. Chow, S.-N. and Hale, J. (eds): Dynamics of lnfinite Dimensional Systems, NATO ASI Series, Series F: Computer and Systems Science 37, Springer, Heidelberg, 1987. Comenius University, PAVOL BRUNOVSKY Bratistava, Czechoslovakia John Roe: Elliptic Operators, Topoloyy and Asymptotic Methods, Pitman Research Notes in Mathematics No. 179, 1988, 184pp. The theory of elliptic operators on manifolds is enormously useful in topology, physics, and representation theory. This theory relates topological invariants of differential structures on manifolds on the one hand, to analytical invariants on the other. The oldest example goes back to C. F. Gauss and O. Bonnet for closed oriented Riemannian surfaces. The Gauss-Bonnet theorem states that z(M) = fM where ;~(M)= E(-l)idimHi(M,~) is the Euler characteristic of a surface M, Hi(M, I~) is the ith de Rham cohomology group of M, and E~ is the curvature. We may interpret this theorem as saying that the Euler characteristic of the de Rham complex is given by http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Elliptic operators, topology and asymptotic methods

Acta Applicandae Mathematicae , Volume 20 (2) – May 1, 2004
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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00046916
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Abstract

BOOK REVIEWS 193 References 1. Guckenheimer, J. and Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Math. Sciences 42, Springer, Heidelberg, 1983. 2. Henry, D.: Geometric Theory of Semitinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer, New York, 1981. 3. Chow, S.-N. and Hale, J. (eds): Dynamics of lnfinite Dimensional Systems, NATO ASI Series, Series F: Computer and Systems Science 37, Springer, Heidelberg, 1987. Comenius University, PAVOL BRUNOVSKY Bratistava, Czechoslovakia John Roe: Elliptic Operators, Topoloyy and Asymptotic Methods, Pitman Research Notes in Mathematics No. 179, 1988, 184pp. The theory of elliptic operators on manifolds is enormously useful in topology, physics, and representation theory. This theory relates topological invariants of differential structures on manifolds on the one hand, to analytical invariants on the other. The oldest example goes back to C. F. Gauss and O. Bonnet for closed oriented Riemannian surfaces. The Gauss-Bonnet theorem states that z(M) = fM where ;~(M)= E(-l)idimHi(M,~) is the Euler characteristic of a surface M, Hi(M, I~) is the ith de Rham cohomology group of M, and E~ is the curvature. We may interpret this theorem as saying that the Euler characteristic of the de Rham complex is given by

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Acta Applicandae Mathematicae – Springer Journals

Published: May 1, 2004

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