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This book contains many substantial papers from distinguished speakers of a conference "Geometric Analysis: Present and Future" and an overview of the works of Professor Shing-Tung Yau. Contributors include E. Wit-ten, Y.T. Siu, R. Hamilton, H. Hitchin, B. Lawson, A. Strominger, C. Vafa, W. Schmid, V. Guillemin, N. Mok, D. Christodoulou. This is a valuable reference that gives an up-to-dated summary of geometric analysis and its applications in many different areas of mathematics.
目錄
part 1 summary of and commentaries on the work of shing-tung yau
　curriculum vitae of shing-tung yau
　a brief overview of the work of shing-tung yau
　lizhen ji
　1 introduction
　2 a summary of some major works of yau
　3 topics yau has worked on
　4 basics on kaihler-einstein metrics and calabi conjectures
　5 some applications of kaihler-einstein metrics and calabi-yau manifolds
　6 harmonic maps
　7 rigidity of kahler manifolds
　8 super-rigidity of spaces of nonpositive curvature
　9 survey papers by yau
　10 open problems by yau
　ll books written and co-written by yau
　12 books edited and co-edited by yau
　13 ph.d. students of yau
　14 partial list of papers and books of yau
　references
　yau's work on filtering problem
　wen-lin chiou, jie huang and lizhen ji
　1 filtering problem
　2 yau's two methods in solving nonlinear filtering problem
　2.1 direct method
　2.2 algorithm for real time solution without memory
　references
　from continues to discrete - yau's work on graph theory
　fan chung
　yau's work on moduli, periods, and mirror maps for calabi-yau manifolds
　charles f. doran
　1 construction of calabi-yau threefolds
　2 picard-fuchs equations and the mirror map
　3 arithmetic properties of mirror maps
　4 periods and moduli of complex tori and k3 surfaces
　references
　review on yau's work on the coupled einstein equations and the wave dynamics in the kerr geometry
　felix finster
　1 coupling the einstein equations to non-abelian gauge fields and dirac spinors
　2 the dynamics of linear waves in the kerr geometry
　references
　the work of witten and yau on the ads/cft correspondence
　gregory j. galloway
　1 introduction
　2 the witten-yau results on ads/cft
　3 further developments
　references
　yau's work on heat kernels
　alexander grigor'yan
　1 the notion of the heat kernel
　2 estimating heat kernels
　3 some applications of the heat kernel estimates
　references
　yau's contributions to engineering fields
　xianfeng david gu
　1 introduction
　2 computational conformal geometry
　2.1 conformal structure
　2.2 harmonic map
　2.3 surfacericci flow
　2.4 conformal mappings
　2.5 quasi-conformal mappings
　2.6 teichmiiller space
　3 geometric acquisition
　4 computer graphics
　5 geometric modeling
　6 medical imaging
　7 computer vision
　8 wireless sensor network
　9 summary
　references
　the syz proposal
　naichung conan leung
　1 pre-syz
　2 the birth of syz
　3 the growing up of syz
　3.1 special lagrangian geometry
　3.2 special lagrangian fibrations
　3.3 affine geometry
　3.4 syz transformation
　4 future of syz
　references
　yau- zaslow formula
　naichung conan leung
　yau's work on function theory: harmonic functions, eigenvalues and the heat equation
　peter li
　a vision of yau on mirror symmetry
　bong lian
　1 enumerative geometry
　2 geometry of calabi-yau manifolds and their moduli spaces
　references
　yau's work on group actions
　kefeng liu
　cheng and yau's work on the monge-ampere equation and affine geometry
　john loftin, xu-jia wang and deane yang
　1 introduction
　2 the monge-ampere equation
　3 cheng and yau's work on the dirichlet problem
　4 subsequent work on the monge-ampere equation
　5 affine spheres
　6 hyperbolic affine spheres and real monge-ampere equations
　7 affine manifolds
　8 maximal hypersurfaces in minkowski space
　9 the minkowski problem
　10 convex geometry without smoothness assumptions
　10.1 support function
　10.2 invariance properties of the support function
　10.3 minkowski sum
　10.4 mixed volume
　10.5 surface area measure
　10.6 invariance properties of the surface area measure
　10.7 the minkowski problem
　10.8 the brunn-minkowski inequality
　10.9 uniqueness in the minkowski problem
　10.10 variational approach to the minkowski problem
　11 convex geometry with smoothness assumptions
　11.1 the inverse gauss map
　11.2 the inverse second fundamental form
　11.3 the curvature function
　11.4 the surface area measure
　11.5 the minkowski problem
　11.6 the minkowski problem as a pde
　12 cheng and yau's regularity theorem for the minkowski problem.
　12.1 statement
　12.2 sketch of proof
　13 generalizations of the minkowski problem
　references
　yau's work on minimal surfaces and 3-manifolds
　feng luo
　the work of schoen and yau on manifolds with positive scalar curvature
　william, minicozzi ii
　0 introduction
　1 topological restrictions on manifolds with positive scalar curvature
　1.1 stable minimal surfaces and scalar curvature
　1.2 inductively extending this to higher dimensions
　1.3 preserving positive scalar curvature under surgery
　2 locally conformally flat manifolds
　2.1 the new invariants
　2.2 a positive mass theorem
　references
　yau's contributions to algebraic geometry ndrey todorov
　1 introduction
　1.1 yau's program-plenary talk at icm 1982
　2 monge-ampere equation and applications to algebraic geometry
　2.1 solution of the calabi conjecture
　2.2 existence of canonical metrics on zariski open sets
　3 stable vector bundles over kahler manifolds
　3.1 donaldson-uhlenbeek-yau theorem
　3.2 applications to kodaira's classification of surfaces
　4 moduli spaces
　4.1 existence of kiihler-einstein metrics on domain of holomorphy and teichmfiller spaces
　4.2 moduli spaces of k3 surfaces
　4.3 moduli spaces of cy manifolds
　4.4 generalization of shwarz lemma by yau and baily-borel compactification
　5 contributions of yau to string theory
　5.1 mirror symmetry and syz conjecture
　5.2 large radius limit
　5.3 string theory and number theory
　5.4 rational curves on algebraic k3 surfaces
　6 rigidity
　6.1 yau's conjecture about rigidity of some complex manifolds
　6.2 geometric proof of margulis' superrigidity
　6.3 geometric proof of kazhdan theorem about galois
　conjugation of shimura varieties
　references
　yau's work on positive mass theorems
　mu-tao wang
　yan's conjecture on kaihler-einstein metric and stability
　xiaowei wang
　on yau's pioneer contribution on the frankel conjecture and
　related questions
　fangyang zheng
　yau's work on inequalities between chern numbers and
　uniformization of complex manifolds
　kang zuo
　part 2 differential geometry and differential equations
　geometry of complete gradient shrinking ricci solitons
　huai-dong cao
　1 gradient shrinking ricci solitons
　2 classification of 3-dimensional gradient shrinking solitons
　3 geometry of complete gradient solitons
　references
　the formation of black holes in general relativity
　demetrios christodoulou
　pagerank as a discrete green's function
　fan chung
　1 introduction
　2 preliminaries
　3 dirichlet eigenvalues
　4 connections between pagerank and discrete green's function
　5 relating the cheeger constant to the pagerank
　6 relating the pagerank of a graph to that of its subgraphs
　7 the pagerank and the hitting time
　references
　a geodesic equation in the space of sasakian metrics
　pengfei guan and xi zhang
　some inverse spectral results for the two-dimensional schrodinger operator
　v. cuillemin and a. uribe
　1 introduction
　2 the weyl calculus
　3 some bracket identities
　4 the quantum birkhoff canonical form
　references
　li-yau estimates and their harnack inequalities
　richard s. hamilton
　1 the heat equation
　2 the dirichlet problem for the heat equation
　3 the heat equation in the plane
　4 the castaway
　5 endangered species equation
　6 the migration equation
　7 motion of a curve by its curvature
　8 motion of a surface by its mean curvature
　9 motion of a surface by its gauss curvature
　references
　plurisubharmonicity in a general geometric context
　f. reese harvey and h. blaine lawson, jr
　1 introduction
　2 geometrically defined plurisubharmonic functions
　3 more general plurisubharmonic functions defined by an elliptic cone p+
　4 p+-plurisubharmonic distributions
　5 upper-semi-continuous p+-plurisubharmonic functions
　6 some classical facts that extend to p+-plurisubharmonie functions
　7 the dirichlet problem uniqueness
　8 the dirichlet problem existence
　9 p+-convex domains
　10 topological restrictions on p+-convex domains
　11 p+-free submanifolds
　12 p+-convex boundaries
　references
　poisson modules and generalized geometry
　nigel hitchin
　1 introduction
　2 poisson modules
　2.1 definitions
　2.2 a construction
　3 the serre construction
　3.1 the algebraic approach
　3.2 the analytical approach
　3.3 the second section
　4 generalized geometry
　4.1 basic features
　4.2 generalized dolbeault operators
　4.3 the canonical bundle
　5 a generalized construction
　5.1 the problem
　5.2 generalized complex submanifolds
　5.3 the construction
　6 an application
　references
　uniqueness of solutions to mean field equations of liouville type in two-dimension
　chang-shou lin
　1 introduction
　2 uniqueness in r2
　3 uniqueness in bounded domains of r2
　4 onofri inequality and its generalization
　5 mean field equation and green functions on torus
　6 generalized liouville system
　references
　monotonicity and holomorphic functions
　lei ni
　decay of solutions to the cauchy problem in the kerr geometry for various physical systems: stability of black holes
　j. a. smoller
　1 introduction
　2 main
　references
　the calabi-yau equation, symplectic forms and almost complex structures
　valentino tosatti and ben weinkove
　1 background- yau's theorem
　2 donaldson's conjecture and applications
　3 estimates for the catabi-yau equation
　4 methods
　5 a monotonicity formula
　references
　understanding weil-petersson curvature
　scott a. wolpert
　1 introduction
　2 basics of teichmiiller theory
　3 wp intrinsic geometry
　4 methods
　5 applications of curvature
　5.1 the work of liu, sun and yau
　5.2 the model metric 4dr2 + rs do2
　5.3 projection and distance to a stratum
　references
　examples of positively curved complete kahler manifolds
　hung-hsi wu and fangyang zheng
　1 introduction
　2 the abcd functions
　3 characterization by the function
　4 some examples
　5 characterization by surface of revolution
　6 correlation between volume growth and curvature decay
references

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