By P Li, Gang Tian, Shiu-Yuen Cheng

This quantity is a set of study papers on nonlinear partial differential equations and similar parts, representing many facets of latest advancements in those components. specifically, the next are integrated: nonlinear conservation legislation; semilinear elliptic equations, nonlinear hyperbolic equations; nonlinear parabolic equations; singular restrict difficulties; and research of actual and numerical strategies. very important components equivalent to numerical research, leisure idea, multiphase concept, kinetic concept, combustion concept, dynamical structures and quantum box conception also are lined The lifestyles and arithmetic of Shiing-Shen Chern / R.S. Palais and C.-L. Terng -- My Mathematical schooling / S.S. Chern -- A precis of My medical lifestyles and Works / S.S. Chern -- S.S. Chern as Geometer and pal / A. Weil -- a few Reflections at the Mathematical Contributions of S.S. Chern / P.A. Griffiths -- Shiing-Shen Chern as good friend and Mathematician / W.-L. Chow -- Abzahlungen fur Gewebe -- On critical Geometry in Klein areas -- an easy Intrinsic evidence of the Gauss-Bonnet formulation for Closed Riemannian Manifolds -- at the Curvatura Integra in a Riemannian Manifold -- attribute periods of Hermitian Manifolds -- Sur une Classe Remarquable de Varietes dans l'espace Projectif a N Dimensions -- A Theorem on Orientable Surfaces in 4-dimensional house / S.S. Chern and E. Spanier -- at the Kinematic formulation within the Euclidean house of N Dimensions -- On a Generalization of Kahler Geometry -- at the overall Curvature of Immersed Manifolds / S.S. Chern and R.K. Lashof

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**Extra info for A mathematician and his mathematical work : selected papers of S.S. Chern**

**Example text**

In [62] Chern and Lashof generalized Fenchel's theorem by showing that reM, f) 2:: 2, with equality if and only if M is a convex hypersurface of an (m + 1)-dimensional affine subspace V . In [66] they obtained 27 44 The Life and Mathematics of Shiing-Shen Chern the sharper result that where f3i(M) is the i-th Betti number of M. An inunersion f :M -t ~n is called tight if T(M, f) is equal to the infimum, T(M), of the absolute total curvature among all the inunersions of M into Euclidean spaces of arbitrary dimensions.

Q(O) is exact. They next write down an explicit formula in terms of Q, w, and 0 for a U-1 form TQ(w) on P, and show that dTQ(w) = Q(O). TQ(w) is natural under pull-back of a bundle and its connection. Now suppose U > n. , in this case TQ(w) is closed, and so defines an element [TQ(w)J of HU-1(P). If 2f > n+ 1 Chern and Simons show this cohomology class is independent of the choice of connection w, and so defines a "secondary characteristic class" However if U = n+ 1 then they show that [TQ(w)J does depend on the choice of connection w.

The Generalized Gauss-Bonnet Theorem Geometers tend to make a sharp distinction between "local" and "global" questions, and it is common not only to regard global problems as somehow more important, but even to consider local theory "oldfashioned" and unw0rthy of serious effort. , the equivalence problem), and moreover, once one had discovered the local invariants of a theory, one was well on the way towards finding its global invariants as well! We shall next explain how Chern came to this contrary attitude, for it is an interesting and revealing story, involving the most exciting and important events of his research career: his discovery of an "intrinsic proof" of the Generalized Gauss-Bonnet Theorem and, flowing out of that, his solution of the characteristic class problem for complex vector bundles by his striking and elegant construction of what are now called "Chern classes" from 29 46 The Life and Mathematics of Shiing-Shen Chern his favorite raw material, the curvature forms of a connection.