报告摘要

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2009年4月21日-25人,中国数学会学术年会在厦门大学召开。这是与会者的报告摘要。

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报告摘要

  1. 1. I 1. John M. Ball: The Q-tensor theory of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Martin Grotschel: Mathematical aspects of infrastructure planning . . . . . . . . . . . . . . . . 1 ¨ 3. Manuel de Leon: Hamilton-Jacobi theory in nonholonomic dynamics . . . . . . . . . . . . . 1 ´ 4. L´ szlo Lov´ sz: Mathematical problems of very large networks . . . . . . . . . . . . . . . . . . . 2 a´ a 5. Zhiming Ma: Mathematics and internet information restirval . . . . . . . . . . . . . . . . . . . . . 2 6. Ragni Piene: Functors of infinitely near points on an algebraic surface . . . . . . . . . . . . . 3 7. Cheryl E. Praeger: The normal quotient philosophy for edge-transitive graphs . . . . . . 3 8. Claudio Procesi: From splines to the index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9. Rajat Tandon: Extending some results of Tunnell and Saito . . . . . . . . . . . . . . . . . . . . . . . 4 10. Victor A. Vassiliev: Solved and unsolved problems of integral geometry and monodromy theory arising from Newton’s ”principia” . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11. Marcelo Viana: TBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A 10 1. : Distribution of primes and dynamics of the w function . . . . . . . . . . . . . . . . . . . 6 2. : Applications of Lie theory to Finsler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. : Quivers, quasi-quantum groups and finite tensor categories . . . . . . . . . . . . . . . 7 4. : Anomalous primes for elliptic curves over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5. : Gorenstein projective injective and flat complexes . . . . . . . . . . . . . . . . . . . . . . . 7 6. : Composition factors of Kac-modules for general linear Lie superalgebras . . 8 7. : Linear groups over semilattices and automorphism groups for gradation shifting Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 8. : K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 9. : On generalized Schur’lemma and its application . . . . . . . . . . . . . . . . . . . . . . . . . 9 10. : Homological properties of Noetherian Hopf algebras . . . . . . . . . . . . . . . . . . . 9
  2. 2. II B 16 11. : - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 12. : Legendrian cable links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 13. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 14. : Balanced metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 15. : Complete classification of locally strongly convex affine hypersurfaces with parallel cubic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 16. : Amalgamations of Heegaard splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 17. : Variational problems in geometry of submanifolds . . . . . . . . . . . . . . . . . . . . 12 18. : Classification of equivariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 19. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 20. : The equivariant noncommutative Atiyah-Patodi-Singer index theorem . . . 13 21. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 22. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 23. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 24. : Local Gromov-Witten invariants in arbitrary genera . . . . . . . . . . . . . . . . . . . 14 25. : Geometric measures and geometric inequalities . . . . . . . . . . . . . . . . . . . . . . . 14 26. : Cannonical metrics on toric manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 C 27 27. : Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 28. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 29. : The Halpern open problem and viscosity approximation . . . . . . . . . . . . . . . 17 30. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 31. : Heat kernels on metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 32. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 33. : A generalization of the Lax-Oleinik formula in scalar conservation law . . 18 34. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 35. : A variational problem associated with the minimal speed of traveling waves for spatially periodic reaction-diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 36. : Heisenberg . . . . . . . . . . . . . . . . . . . . . . 19 37. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 38. : Global structure of positive solutions for nonlocal boundary value problems involving integral conditions . . . . . . . . . . . . . . . . . . . . . . . . . 20
  3. 3. III 39. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 40. : Asymptotic limits of compressible Euler-Maxwell system in Plasma Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 41. : Persistent homoclinic orbits for a spatial-dependently perturbed nonlinear Schr¨ dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 o 42. : Diffusion equations with degeneracy on the boundary . . . . . . . . . . . . . . . . . . 22 43. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 44. : New Besov-type spaces and Triebel-Lizorkin-type spaces . . . . . . . . . . . . . . 22 45. : Lorenz . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 46. : 23 47. : . . . . . . . . . . . . . . . . . . . . . 24 48. : Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 49. : Boltzmann BGK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 50. : Diffusive expansion for solutions of the Boltzmann equation in the whole space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 51. : p-Laplace . . . . . . . . . . . . . . . . . . . . 25 52. : . . . . . . . . . . . . . . . . . . . . . . . 25 53. : New progress on critical point theory with applications . . . . . . . . . . . . . . . . 25 D 20 54. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 55. : Convergence analysis of the Jacobi-collocation spectral methods for Volterra integral equations with a weakly singular kernel . . . . . . . . . . . . . . . 27 56. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 57. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 58. : Implicit/Explicit schemes for the Navier-Stokes equations . . . . . . . . . . . . . . 28 59. : Helmholtz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 60. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 61. : On the numerical simulation of chemical kinetic system . . . . . . . . . . . . . . . 29 62. : . . . . . . . . . . . . . . . . . . .30 63. : Exposing structure algorithms for nonsymmetric ARE arising in transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 64. : JASMIN . . . . . . . . . 31 65. : Adaptive RKDG method using different indicators . . . . . . . . . . . . . . . . . . . . 31
  4. 4. IV 66. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 67. : An introuduction to piecewise algebraic variety . . . . . . . . . . . . . . . . . . . . . . . 32 68. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 69. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 70. : An adaptive edge element method with perfectly matched absorbing layers for wave scattering by Bipe . . . . . . . . . . . . . . . . . . . . . . . . . . 33 71. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 72. : Relations between the multiscale methods for elliptic homogenization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 73. : Numerical investigation for a macroscopic model for hydrodynamic nematic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 E 17 74. : Design of computer experiments: construction and theory . . . . . . . . . . . . . . 37 75. : Feynman-Kac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 76. : Homeomorphic flows and large deviations : for stochastic differential equations driven by a G-Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 77. : Markov skeleton processes and their applications . . . . . . . . . . . . . . . . . . . . . . 38 78. : Fisher : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 79. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 80. : Limit theorems for super-diffusions and branching Hunt processes . . . . . . 39 81. : Wiener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 82. : Additive hazards regression with censoring indicators missing at random 40 83. : Model selection criteria for missing-data problems via the EM algorithm 40 84. : profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 85. : Stochastic maximum principles for partially observed optimal control problems and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 86. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 87. : Estimation for a partial-linear single-index model . . . . . . . . . . . . . . . . . . . . . 42 88. : Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 89. : Generalized profile LSE in varying-coefficient partially linear models with measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 90. : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
  5. 5. V F 10 91. : Primal-dual IPMs for conic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 92. : . . . . . . . . . . . . . . . . . . . . . . . . . 45 93. : Applications of Steiner tree problem in network design . . . . . . . . . . . . . . . . 46 94. : Error estimates for an optimal control problem of the heat equation with state and control constrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 95. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 96. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 97. : Lowner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 98. : Inverse and generalized inverse optimization problems . . . . . . . . . . . . . . . . . 49 99. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 100. : . . . . . . . . . . . . . . . . . . . . 50 G 8 101. Christian Reidys: Canonical RNA pseudoknot structure: combinatorics and folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 102. : On the coverings of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 103. : On the spectral radius of unicyclic graphs and H-shape trees graph with applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 104. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 105. : Acyclic coloring of planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 106. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 107. : Jones polynomials of links and limits of their zeros . . . . . . . . . . . . . . . . . . . 53 108. : Lattice structure on the perfect matchings of a plane bipartite graph with applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
  6. 6. 1 The Q-tensor Theory of Liquid Crystals John M. Ball Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, UK E-mail: ball@maths.ox.ac.uk The lecture will survey what is known about the mathematics of the de Gennes Q- tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe. Mathematical Aspects of Infrastructure Planning Martin Grotschel ¨ Konrad-Zuse-Zentrum f¨ r Informationstechnik Berlin, Takustraße, Berlin, Germany u E-mail: groetschel@zib.de The costs of establishing, maintaining, and extending a country’s infrastructure such as highway, railroad, public transport, pipeline, power, water, or telecommunication net- works are immense. The utilization of mathematics in these processes ranges consider- ably from one infratsructure area to another and from country to country. In this talk I will address some of the planning and operational issues arising in infrastructure planning in Germany, and I will demonstrate how mathematics can be employed to solve some of the problems coming up. Hamilton-Jacobi Theory in Nonholonomic Dynamics Manuel de Leon ´ Instituto de Ciencias Matematicas, Consejo Superior de Investigaciones Cient´ ificas, Madrid, Spain E-mail: mdeleon@imaff.cfmac.csic.es The Hamilton-Jacoby theory is a useful and classical tool to integrate the equations of motion in Mechanics. There were many attempts in the last decades to extend this theory
  7. 7. 2 to nonholonomic mechanics. In this lecture we will discuss the geometric framework of the Hamilton-Jacobi theory and show how it permits to extend it for the nonhonomic case. Mathematical Problems of Very Large Networks L´ szlo Lov´ sz a´ a Department of Computer Science of the Eotvos Lor´ nd University , Budapest, Hungary ¨¨ a E-mail:lovasz@cs.elte.hu Suppose that we have a huge graph (we don’t even know its size), and the only way to obtain information about it is to draw a sample of the node set of bounded size. What properties of the graph can be deduced from this sample? What should we mean by an answer to an algorithmic question like finding the conencted components, if we cannot even list all nodes? What does it mean if two graphs are “close” in the sense that they cannot be distinguished by such tests? How to model such huge graphs, and how to approximate them by smaller ones? The graph property testing model was first introduced by Goldreich, Goldwasser and Ron (but related questions were considered before). In the context of dense graphs, a very general result is due to Alon and Shapira, who proved that every hereditary graph property is testable. In this language, the Regularity Lemma of Szemeredi states that every graph can be approximated by a weighted graph with k nodes so that the error tends uniformly to 0 as k tends to infinity. In the other direction, the theory of graph limits yields an approximation by a 2-variable measurable function. which allows us to use tools from analysis. This analytic version allows for simpler formulation of many graph theoretic problems, and leads to various characterizations of testable properties. We survey these results, along with analogous results for graphs with bounded degree. Mathematics and Internet Information Restirval Zhiming Ma Institute of Applied Mathematics, Academy of Math and Systems Science, Beijing, China E-mail: mazm@amt.ac.cn In this public talk I shall briefly review some of our recent joint work (in collaboration with Microsoft Research Asia) concerning Internet Information Retrival. I shall tell you how can we construct a Markov process describing Web users’ real browsing behaviors,
  8. 8. 3 and thus yield an algorithm for page importance computation (referred to as BrowseR- ank). The algorithm is likely to be a competitor to the conventional ParRank used in Google search. Our work reveals that mathematics is becoming more and more important in information retrival. Functors of Infinitely Near Points on an Algebraic Surface Ragni Piene Centre of Mathematics for Applications, Dept. of Maths., University of Oslo, Norway E-mail: ragnip@math.uio.no In joint work with Steven Kleiman we study sequences of infinitely near points of a family of algebraic surfaces that are associated to given Enriques diagrams and show that they form a functor. The functor is representable by a smooth scheme, which maps naturally to the Hilbert scheme of the family. I will focus on applications of this theory to the ”Gromov-Witten” problem of enumerating singular curves on a surface. The Normal Quotient Philosophy for Edge-transitive Graphs Cheryl E. Praeger School of Mathematics and Statistics, University of Western Australia, Australia E-mail: praeger@maths.uwa.edu.au Studying normal quotients has proved an effective way to describe the structure of many families of finite edge-transitive graphs. The normal quotient approach was initiated in my investigation of s-arc transitive graphs, and then refined in collaboration with Giudici and Li to develop our theory of locally s-arc transitive graphs. I will attempt to present the essence of this philosophy with reference to a new analysis of an infinite family of edge-transitive graphs. From Splines to the Index Theorem Claudio Procesi Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Universit` di Roma, Italia a E-mail: procesi@mat.uniroma1.it
  9. 9. 4 The celebrated index theorem of Atiyah–Singer led to a theory of transversally elliptic operators, with respect to some compact group of symmetries. In that case an explicit computation of the equivariant K−theory and the associated index for linear representa- tions of tori exists only in an implicit form. Some rather surprising connection arose with the work of Dahmen– Micchelli on partition functions and the box–spline which allows us to make the constructions extremely explicit (joint work with Corrado De Concini and Michele Vergne). Extending Some Results of Tunnell and Saito Rajat Tandon Department of Mathematics and Statistics, University of Hyderabad, Hyderabad, India E-mail:rtsm@uohyd.ernet.in One of the problems representation theorists often look at is the following: Given a topological group G, an irreducible “representation” π of G and a closed subgroup H how does π|H decompose into irreducibles, if at all. Tunnel looked at this problem when G =GL(2,F), where F is a nonarchimedian local field with residue characteristic not two, √ π is an irreducible irreducible representation of G and H = K ∗ = K − {0}, K = F( d) being a quadratic extension of F, embedded in G under the embedding    a bd   √     a+b d → .       b a    His answer was that the multiplicity of a character χ of K ∗ whose restriction to the cen- tre of G is equal to the central character ωπ of π in π|K∗ is given by 1 + (Π ⊗ χ−1 , ψ0 ) /2 where Π is the base change lift of π to GL(2,K), ψ0 is a nontrivial additive character of K trivial on F and (Π ⊗ χ−1 , ψ0 ) is the corresponding epsilon factor (local root number). It can easily be shown that in the given circumstance (Π ⊗ χ−1 , ψ0 ) = ±1 so either χ does not occur in πK∗ or it occurs with multiplicity one. Saito gave a residue characteristic free proof of the same result. It was apparent from the work of Tunnel that the problem posed a challenge only in the case that π is supercuspidal ,i.e., its coefficients have compact support modulo the centre of G. Dipendra Prasad looked for a criterion (for χ occurring in π|K∗ ) in terms of the local root number of the character itself. Suppose π is an irreducible supercuspidal representation of G associated to a character θ of a separable quadratic L of F. Let G+ be
  10. 10. 5 the subgroup of index 2 in G consisting of those 2 × 2 matrices whose determinant is in NK/F (K ∗ ). If LK then π|G+ remains irreducible but if L = K then π|G+ breaks up into two ˆ irreducible representations π+ and π− . If χ ∈ K ∗ occurs in π|K∗ then χ must occur in either π+ or π− . Prasad showed that if the residue characteristic is odd then χ will occur in π+ if and only if (χθ−1 , ψ0 ) = (χσ θ−1 , ψ0 ) = 1 ( σ is the nontrivial element in the Galois group G(K/F)) and in π− if and only if (χθ−1 , ψ0 ) = (χσ θ−1 , ψ0 ) = −1. In a joint work with N.K. Vishnu we give a residue characteristic free proof of this result. Saito has also given a completely different proof of this. We also count the number of χ’s of a given conductor that occur in π|K∗ . Solved and unsolved problems of integral geometry and monodromy theory arising from Newton’s ”Principia” V.A.Vassiliev Steklov Institute of Mathematics Russian Academy of Science Mathematics College Independent University of Moscow E-mail: vassil@vassil.mccme.rssi.ru Two statements from the Newtons ”Principia Mathematicae...” will be discussed: Lemma XXVIII on the non-squarability of plane ovals, and Theorem XXXI on the attraction by spheric layer outside the sphere. Both these results can be extended to arbitrary dimen- sions and much more general surfaces. However, unsolved problems still exist in both arising theories. Here are two of them: 1. For any n, do there exist compact algebraically squarable hypersurfaces in Rn other than the ellipsoids for odd n (which are squarable by an Archimedes’ theorem)? 2. Do there exist new examples of hyperbolic layers in Rn (besides the ones described in the talk), whose attraction fields are single-valued (or at least algebraic) in some com- ponents of their complements, except for the hyperbolicity domain? TBA Marcelo Viana IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil E-mail: viana@impa.br
  11. 11. 6 A Distribution of Primes and Dynamics of the w Function , , 210097 E-mail: ygchen@njnu.edu.cn Let P be the set of all primes, and A3 = {pqr|p, q, r ∈ P}{p3 |p ∈ P}. For n = pqr ∈ A3 , let w(n) = P(p + q)P(q + r)P(r + p), where P(m) denotes the largest prime factor of m, and let wi+1 (n) = w wi (n) , (i = 1, 2, . . .). It is known that for any n ∈ A3 , there exists an i with wi (n) ∈ {20, 98, 63, 75}. We will talk about the properties of the shifted set a + P of P and dynamics of the w function. In particular we confirm a conjecture of Wushi Goldring on dynamics of the w function by employing the Green-Tao Theorem on arithmetic progressions in primes. Applications of Lie Theory to Finsler Geometry , , 300071 E-mail: dengsq@nankai.edu.cn In this talk, we will introduce some recent progress on the applications of Lie theory to the study of Finsler geometry. In Section 1, we introduce the motivations to study Finsler geometry using Lie theory. In section 2, we recall some basic definitions and results on Finsler geometry. In section 3, we consider the group of isometries of a Finsler space. In Section 4, we present some algebraic method to construct homogeneous Finsler spaces. In section 5, we study symmetric Finsler spaces. Finally, we present some result concerning the relationship between Finsler metrics and the degree of symmetry of a smooth manifold.
  12. 12. 7 Quivers, Quasi-Quantum Groups and Finite Tensor Categories , , 250100 E-mail: hualin@sdu.edu.cn In this talk I report the project of quiver approaches to quasi-quantum groups. Firstly I introduce the general quiver framework, then I give some classification results of finite quasi-quantum groups, finally I apply these results to study finite tensor categories. Anomalous Primes for Elliptic Curves over Q with Complex Multiplication , , 210093 E-mail: qingzhji@nju.edu.cn In this paper, we prove that there exist elliptic curves defined over Q which possess an infinite number of anomalous primes if and only if the corresponding quadratic poly- nomials represent infinitely many primes. In particular, we show that a prime p is an anomalous prime for the elliptic curve E : y2 + y = x3 − 21870x + 2489899/2 if and only if p = 432n2 ± 108n + 7 for some integer n. Gorenstein Projective Injective and Flat Complexes , , 730070 E-mail: liuzk@nwnu.edu.cn A complex C is called Gorenstein projective if there exists an exact resolution of com- plexes . . . → P−1 → P0 → P1 → · · · such that each Pi is projective, C = Ker(P0 → P1 ) and the resolution remains exact when Hom(−, Q) is applied to it for any projective com- plex Q. Gorenstein injective complexes are defined similarly. A complex G is called Gorenstein flat if there exists an exact sequence of complexes · · · → F−1 → F0 → F1 → · · · such that each Fi is flat, G = Ker(F0 → F1 ) and the sequence remains exact when E − is applied to it for any injective complex E. We will talk some characterizations
  13. 13. 8 of Gorenstein projective, injective and flat complexes of left R-modules. Also Gorenstein homological dimensions and Gorenstein flat covers of complexes are considered. Composition Factors of Kac-modules for General Linear Lie Superalgebras , , 230026 E-mail: ycsu@ustc.edu.cn To study finite-dimensional modules of Lie superalgebras, Kac introduced certain in- decomposable modules, now known as Kac-modules. It is known that the Kac-module is simple if and only if the corresponding highest weight is typical. When the highest weight is atypical, the structure of the Kac-module has been a subject of intensive study. In this talk, we present results on composition factors of Kac-modules for general linear Lie superalgebras. Linear Groups Over Semilattices and Automorphism Groups for Gradation Shifting Lie Algebras , ,361005 E-mail: tans@xmu.edu.cn Under similarity a semilattice S in the Euclidean space Rn can be expressed as a union of distinct cosets of 2Zn in Zn including the trivial coset. It is known that there is a one to one correspondence from the set of all similarity classes of semilattices onto the set of all isomorphic classes of extended affine root systems of type A1 . The automorphism group of a semilattice S is the group of permutations preserving all cosets in S . In this paper, we study the structure of the automorphism groups of semilattices in Rn . The auto- morphism groups of all non-similar semilattices with dimension 2 and 3 are determined. As an application, we use the automorphism group of a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra L(t1 , · · · , tn , 1), which extends our earlier work.
  14. 14. 9 K2 , ,100049 E-mail: tanggp@gucas.ac.cn F p Cp p G K2 (F[G × C p ]) K2 (FG) p- 2 G p- 1 F FG K2 K2 (FG) K2 (FG) ΩFH → K2 (FH[t]/(tk ), (t)) k ≡ 1modp k>p H p- 1 p- H HDR (FH) H p- On Generalized Schur’Lemma and its Application , , 100871 E-mail: lwang@math.pku.edu.cn In this talk, the Schur’s lemma on the basis of centralized ring is generalized to en- domorphsim ring of any induced module. This generalized lemma can unify some old results and get some new results. Homological Properties of Noetherian Hopf Algebras , , 200433 E-mail: qswu@fudan.edu.cn I will survey the progress on the homological aspects of noetherian Hopf algebras/quantum groups in the talk. Some classification results of noetherian Hopf algebras with low di- mensional GK dimension will also be discussed.
  15. 15. 10 B - , , 100190 E-mail: qunchen@whu.edu.cn Spin M L M Legendrian cable links , , 100871 E-mail: dingfan@math.pku.edu.cn Hansjorg Geiges Open and Closed string theories in Geometry , , 100871 E-mail: fanhj@math.pku.edu.cn The geometrical realizations of the 2D topological field theory are the quantization of the classical geometrical theories. The Geometrical realizations of the N=2 supersymmet- ric nonlinear sigma model and the Landau-Ginzburg model have brought new quantum invariants to the fields of symplectic geometry and the classical singularity theory and various duality phenomena. This big theory can be divided into two type theories: the closed string theory and the open string theory. The closed string theories in geometry correspond to the Gromov-Witten theory and the quantum singularity theory; The open string theories in geometry correspond to various derived categories of branes. There are many problems arising in understanding each geometrical theory and their mutual con- nections. So far as I understand, those known geometrical theories are only some isolated
  16. 16. 11 points in a big 2D topological field theory. I will talk about the progress in this field and propose some problems in which I am interested in this field. Balanced metrics , , 200433 E-mail: majxfu@fudan.edu.cn A balanced metric is a hermitian metric on a complex n-dimensional manifold such that its hermitian form ω satisfies d(ωn−1 ) = 0. The balanced metric plays an important role in the study of non-K¨ hler complex geometry and superstring theories. In this talk I a will review the existence result of balanced metrics on complex manifolds k (S 3 × S 3 ) for any k ≥ 2 in my joint work with Jun Li and Shing-Tung Yau. Complete classification of locally strongly convex affine hypersurfaces with parallel cubic form , , 450052 E-mail: huzj@zzu.edu.cn The fundamental theorem of equiaffine hypersurfaces in affine differential geometry states that the affine Blaschke-Berwald metric h and the cubic form C uniquely determine a non-degenerate hypersurface M in the real unimodular-affine space Rn+1 up to equiaffine equivalence. Moreover, the classical Blaschke-Pick-Berwald theorem shows that C ≡ 0 on M if and only if M is an open part of a non-degenerate quadric. The classification of locally strongly convex hypersurfaces with ˆ ≡ 0 is a fundamental and unsolved problem, where ˆ ≡ 0 is the Levi-Civita connection of h. The known results are only for lower dimensional cases (n ≤ 7). In this talk, we will give a survey about the history of the study of this problem and give an outline of our complete classifications of locally strongly convex hypersurfaces with ˆ ≡ 0 for all dimensions (recent joint works with H. Li and L. Vrancken), which gives a complete answer of the above unsolved problem.
  17. 17. 12 Amalgamations of Heegaard Splittings , ,116024 E-mail: ffcclei@yahoo.com.cn Let Mi be a compact 3-manifold with boundary, Fi ⊂ ∂Mi a boundary component, i = 1, 2, and h : F1 → F2 a homeomorphism. The 3-manifold M = M1 ∪h M2 is called an amalgamation of M1 and M2 along F = F1 = F2 , and is denoted by M = M1 ∪F M2 . A Heegaard splitting for M1 and M2 can naturally induce a Heegaard splitting for the amalgamated 3-manifold M, which is called the amalgamated Heegaard splitting of the two Heegaard splittings. The study of the questions if and when the genus of an amalgamated Heegaard splitting is degenerate has addressed much attention in recent years. In the present talk, we will survey the results on the aspect, including some quite new progress. Variational problems in Geometry of Submanifolds , , 100084 E-mail: hli@math.tsinghua.edu.cn In this talk, we give a survey of some variational problems in geometry of subman- ifolds, which includes recent results about r-minimal submanifolds, Wulff shape and stability of hypersurfaces with constant (r + 1)-th anisotropic mean curvature and the anisotropic version of the Alexandrov Theorem. Classification of Equivariant Manifolds , , 200433 E-mail: zlu@fudan.edu.cn In the recent years, a new research field called “ Toric Topology” is emerging. Toric Topology is not only related to Toric Geometry but also related to many other fields, such as Symplectic Geometry, Combinatorics, Algebra and so on. This new field is getting active, and can gather many different ideas and theories together. Based upon this, we
  18. 18. 13 shall consider the classification of equivariant manifolds; especially for the classification up to (equivariant) homeomorphism and equivariant cobordism. We shall give a necessary and sufficient condition for equivariant homeomorphism, and calculate the equivariant cobordism groups etc.. , , 100190 E-mail: xsun@math.ac.cn The Equivariant Noncommutative Atiyah-Patodi-Singer Index Theorem , , 130024 E-mail: wangy581@nenu.edu.cn In this paper, we prove an equivariant noncommutative Atiyah-Patodi-Singer index theorem. , , 200092 E-mail: yhyang@tongji.edu.cn , , 215006 E-mail: zhangyiing@gmail.com
  19. 19. 14 SL(2,C) B.H.Bowditch S.P.Tan Y.L.Wong Bowditch McShane Bowditch , ,510275 E-mail: zhenghao@mail.sysu.edu.cn Abel Abel Local Gromov-Witten Invariants in Arbitrary Genera , ,100084 E-mail: jzhou@math.tsinghua.edu.cn We will report some results on computations of Gromov-Witten invariants of some local Calabi-Yau 3-folds in arbitrary genera. Geometric Measures and Geometric Inequalities , ,400715 E-mail: zhoujz@swu.edu.cn The isoperimetric inequality is one of the well-known geometric inequalities. One may list many of known geometric inequalities involving global differential geometry, inte- gral geometry, convex geometry, and functional analysis. For example, there are Fenchel inequality, Alexandrov Fenchel inequality, Bonnesen inequality, Ros inequality and the
  20. 20. 15 Sobolev inequality in analysis. Integral geometry, originated from Buffon s needle prob- lem, closely related to probability. Buffon, Crafton, Poincare, Santalo, Blashke and Chern made great contribution to integral geometry. Minkowski investigates convex bodies. Hadwiger s work on geometric measures is the milestone for convex theory. Via geo- metric measures, integral and convex geometric analysis, we will present the recent results and the future possible research on this beautiful and fascinating branch of mathematics. Cannonical Metrics on Toric Manifolds , , 100871 E-mail: xhzhu@math.pku.edu.cn In this talk, I will discuss some geometry on toric manifolds. We will talk about exis- tence problems of Kahler-Einstein metrics as well as Calabi’s extremal metrics on such manifolds. We can also extend our method to toric orbifolds.
  21. 21. 16 C Hamilton , ,010021 E-mail: alatanca@imu.edu.cn Fourier Hilbert-Schmidt Sturm-Liouville 20 60 Hamilton 90 Hamilton Fourier Sturm-Liouville Hamilton Hamilton Hamilton Hamilton Hamilton Hamilton , ,215008 E-mail: ylcao@suda.edu.cn Henon-like Henon-like
  22. 22. 17 The Halpern Open Problem and Viscosity Approximation , ,300160 E-mail: chenrd@tjpu.edu.cn In this talk, I shall introduce Halpern open problem and viscosity iterative and some results on viscosity approximation on fixed points, solutions of a generalized equilibrium problem recently by our group. , ,100871 gansb@math.pku.edu.cn Poincare Heat Kernels on Metric Spaces , , 100084 E-mail: hujiaxin@mail.tsinghua.edu.cn This talk gives some recent developments on the study of heat kernels on metric mea- sure spaces, including some interesting fractals. , , 050016 E-mail: cljiang@mail.hebtu.edu.cn 1978 M.Cowen R.Douglas Cowen-Douglas
  23. 23. 18 A Generalization of the Lax-Oleinik Formula in Scalar Conservation Law , , 100190 E-mail: thli@math.ac.cn For the scalar conservation law ut + f (u) x = 0, when f is uniformly convex; i.e. f (u) ≥ α > 0, the Lax-Oleinik formula provides the weak solution. We generalize this formula to the case when f is convex and f (u) is strictly increasing. For example, f (u) = exp(u) or u4 .(Joint with Banghe Li). , ,730000 E-mail: wtli@lzu.edu.cn G- Profile Nicholson Lyapunov
  24. 24. 19 Lotka-Volterra A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations , , 230026 E-mail: xliang@ustc.edu.cn We consider the equation ut = u xx + b(x)u(1 − u), x ∈ R, where b(x) is a nonnegative measure on R that is periodic in x. In the case where b(x) is a smooth periodic function, it is known that there exists a travelling wave – more precisely a “pulsating travelling wave” – with average speed c if and only if c ≥ c∗ (b), where c∗ (b) is a certain positive number depending on b. This constant c∗ (b) is called the “minimal speed”. In this paper, we first extend this theory by showing the existence of the minimal speed c∗ (b) for any nonnegative measure b with period L. Next we study the question of maximizing c∗ (b) under the constraint [0,L) b(x)dx = αL, where α is an arbitrarily given positive constant. This question is closely related to the problem studied by mathematical ecologists in late 1980’s but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac’s delta functions αL δ(x + kL). k∈Z Heisenberg , , 100871 E-mail: hpliu@pku.edu.cn Heisenberg 2 Heisenberg Heisenberg Heisenberg Siegel Heisenberg Heisenberg Heisenberg Laplace ¯ ∂-Neuman H- Heisenberg Laplace
  25. 25. 20 H- H- L1 - KK 1 Iwasawa H- H- 1 H- AN 50 Lichnerowicz Riemanm Heisenberg (1)H- Wiener Feynman-Kac Schrodinger ¨ (2)H- Schrodinger ¨ Hardy Hardy Littlewood- Paley Hardy Schrodinger ¨ Riesz Hardy (3) Heisenberg Laplace Heisenberg Cauchy-Szego ¨ , , 610064 E-mail: matian56@sina.com Global Structure of Positive Solutions for Nonlocal Boundary Value Problems Involving Integral Conditions , ,730070 E-mail: mary@nwnu.edu.cn We consider the nonlinear eigenvalue problems u + λh(t) f (u) = 0, 0 < t < 1 with 1 1 u(0) = 0, u(1) = u(s)dA(s), where u(s)dA(s) is a Stieltjes integral with A is non- 0 0 decreasing and A(t) is not a constant on (0, 1); h ∈ C((0, 1), [0, ∞)) and h(t) 0 on
  26. 26. 21 any subinterval of (0, 1); f ∈ C([0, 1), [0, ∞)) and f (s) > 0 for s > 0, and f0 = f∞ = 0, f0 = lim s→0+ f (s)/s, f∞ = lim s→∞ f (s)/s. We investigate the global structure of positive solutions by using global bifurcation techniques. , ,230026 E-mail: wxshen@ustc.edu.cn Julia Rivera-Letelier Asymptotic Limits of Compressible Euler-Maxwell System in Plasma Physics , ,100124 E-mail: wangshu@bjut.edu.cn In this talk we will discuss asymptotic limit problems of compressible Euler-Maxwell system in plasma physics. Some recent results about the convergence of compressible Euler-Maxwell system to the incompressible Euler or e-MHD equations are given, and some new methods or ideas are reviewed. Persistent Homoclinic Orbits for a Spatial-dependently Perturbed Nonlinear Schr¨ dinger Equation o , ,230026 E-mail: wangyi@ustc.edu.cn The existence of homoclinic orbits for nearly integrable Hamiltonian PDEs is closely related to Chaos. In this talk, we consider a certain diffusive perturbations of the inte- grable nonlinear Schr?dinger equation under periodic boundary conditions. The spatial- dependence of the damped-driven term and unboundedness of the diffusion destroy the
  27. 27. 22 invariance of the plane of constants and some geometric structures. We overcome these difficulties and prove the existence of homoclinic orbits for the perturbed NLS. This is a joint work with Shui-Nee Chow and Chongchun Zeng. Diffusion Equations with Degeneracy on the Boundary , , 130012 E-mail: wangcp@jlu.edu.cn This talk contains two parts. In the first part, we formulate the boundary problems of quasilinear diffusion equations with boundary degeneracy and gradient nonlinearity and establish their well-posedness. Different from the classical theory by G. Fichera and O. A. Oleinik for the linear equations, it is shown that on the degenerate part of the boundary, the equations may exhibit not only hyperbolic characteristics but also parabolic characteristics. In the second part, we study the approximate controllability of a class of semilinear systems with boundary degeneracy. The equations may be weakly degenerate and strongly degenerate on a portion of the boundary. We prove that the control systems are approximate controllable and the controls can be taken to be of quasi bang-bang form. , ,400715 E-mail: wendi@swu.edu.cn New Besov-type Spaces and Triebel-Lizorkin-type Spaces , ,100875 E-mail: dcyang@bnu.edu.cn
  28. 28. 23 ˙ p,q Let s, τ ∈ R and q ∈ (0, ∞]. We introduce Besov-type spaces Bs,τ (Rn ) for p ∈ (0, ∞] ˙ s,τ and Triebel-Lizorkin-type spaces F p,q (Rn ) for p ∈ (0, ∞), which unify and generalize the Besov spaces, Triebel-Lizorkin spaces and Q spaces. We then establish the ϕ-transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the ϕ- ˙ p,q ˙ s,τ transform characterization of Bs,τ (Rn ) and F p,q (Rn ), we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molec- ˙ p,q ˙ s,τ ular decomposition characterizations of Bs,τ (Rn ) and F p,q (Rn ). For s ∈ R, p ∈ (1, ∞) and 1 q ∈ [1, ∞) and τ ∈ [0, (max{p,q}) ], via the Hausdorff capacity, we introduce certain Besov- ˙ s,τ ˙ s,τ Hausdor spaces BH p,q (Rn ) and Triebel- Lizorkin-Hausdorff F H p,q (Rn ) and prove that the ˙ s,τ ˙ s,τ ˙ ˙ dual spaces of BH p,q (Rn ) and F H p,q (Rn ) are just, respectively, B−s,τ (Rn ) F −s,τ (Rn ) ,where p ,q p ,q t denotes the conjugate index of t ∈ (1, ∞). Applications to trace theorems and bound- edness of the pseudo-differential operators with homogeneous symbols are obtained. The inhomogeneous versions of these results are also given. Lorenz , ,510640 E-mail: qgyang@scut.edu.cn Lorenz Lorenz Hopf Silinkov 2 Marotto , , 430074 E-mail: yangxs@mail.hust.edu.cn
  29. 29. 24 , ,210093 E-mail: huicheng@nju.edu.cn Newton , ,610064 E-mail: zhangshiqing@msn.com Newton Boltzmann BGK , ,430074 E-mail: xwzhang@mail.hust.edu.cn The BGK model of the Boltzmann equation plays an important role in the kinetic theory of rarefied gases. Some existence and uniqueness results of global solutions to its Cauchy problem and boundary value problem were established both for large and small initial data under various circumstances. In this talk, we first review its L1 theory established by B. Perthame and its L∞ existence and uniqueness results due to B. Perthame, M. Pulvirenti and S. Mischler, and some related topics as well. Then, we establish certain weighted L p estimates of the hydrodynamical quantities and local Maxwelians for rarefied gases. By those estimates and the L∞ results, we construct approximate solutions and prove their uniform L p bounds. Finally, we prove the existence theorem of the L p solutions to the Cauchy problem and establish some propagation properties of the L p moments for this kind of solutions.
  30. 30. 25 Diffusive Expansion for Solutions of the Boltzmann Equation in the Whole Space , , 430072 E-mail: hhjjzhao@whu.edu.cn This talk is concerned with the diffusive expansion for solutions of the Boltzmann equation in the whole space. It is based on a recent work joint with Dr. Shuangqian Liu. p-Laplace , , 361005 E-mail: jnzhao@xmu.edu.cn 1 p-Laplace 2 3×3 Navier-Stokes , ,730000 E-mail: ckzhong@lzu.edu.cn Sobolev New Progress on Critical Point Theory with Applications , ,100084 E-mail: wzou@math.tsinghua.edu.cn
  31. 31. 26 The following topics will be involved: 1. Multi-Bump solutions and Critical Groups; 2. Bahri-Lions perturbation problem; 3. Brezis-Nirenberg problem; 4. Perturbed Brezis- Nirenberg problem; 5. A pure critical exponent problem.
  32. 32. 27 D , ,100190 E-mail: clq@lsec.cc.ac.cn 1. 2. / 3. 4. Maxwell Convergence Analysis of the Jacobi-collocation Spectralmethods for Volterra Integral Equations with a Weakly Singular Kernel , , 510631 E-mail: yanpingchen@scnu.edu.cn In this talk, a Jacobi-collocation spectralmethod is developed for Volterra integral equa- tions of the second kind with a weakly singular kernel. We use some function transfor- mation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corre- sponding to the Lagrange interpolation polynomials, polynomials approximation theory for orthogonal polynomials and the operator theory. The spectral rate of convergence for the proposed method is established in the L∞ -norm and weighted L2 -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.
  33. 33. 28 , ,510275 E-mail: lnsczy@mail.sysu.edu.cn Fast multilevel augmentation methods for solving operator equations are developed. Each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace. A subspace at a level is obtained from the subspace at the previous level by adding a dif- ference subspace. Accordingly, the projection equation at the present level is obtained by augmenting the projection equation at the previous level. Using this idea recursively, we solve the operator equation only at an initial lower resolution level while obtain its solution at a higher resolution level. We prove that the proposed methods require only linear computational complexity and have the optimal convergence order. A relationship between the proposed method and the multigrid method is discussed. Two specific fast methods based on the Galerkin projection and the collocation projection are developed. These algorithms are applied to solve Fredholm integral equations of the second kind, ill-posed integral equations of the first kind, Hammerstein nonlinear equations, and two- point boundary value problems of differential equations. Numerical results are presented to confirm the theoretical estimates and the efficiency of the methods. , ,410083 E-mail: xlhan@mail.csu.edu.cn B Implicit/Explicit Schemes for the Navier-Stokes Equations , ,710049
  34. 34. 29 E-mail: heyn@mail.xjtu.edu.cn In this paper, we study the stability and convergence of the implicit/explicit schemes for the two-dimensional Navier-Stokes equations. The finite element method and spec- tral method are applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit schemes for the linear terms and the explicit schemes for the nonlinear term. Moreover, we prove the stabilities and optimal error estimates under the corresponding stability conditions, where the schemes are almost un- conditionally stable and convergent for the smooth initial data u0 ∈ H 2 , i.e., the time step size τ satisfies τ ≤ C0 ; and the schemes are almost weak unconditionally stable and convergent for the non-smooth initial data u0 ∈ H 1 , i.e., the time step size τ satisfies τ|logh| ≤ C0 for the mesh size 0 < h < 1; and the schemes are conditionally stable for the non-smooth initial data u0 ∈ L2 , i.e., the time step size τ satisfies τh−2 ≤ C0 . Helmholtz , ,100190 E-mail: hqy@lsec.cc.ac.cn Helmholtz Maxwell Helmholtz Helmholtz Maxwell , ,510631 E-mail: liwen@scnu.edu.cn On the Numerical Simulation of Chemical Kinetic System
  35. 35. 30 , , 100871 E-mail: tieli@pku.edu.cn The numerical simulation of chemical kinetic system opens a new way for computa- tional mathematics community. Mathematically it deals with the stochastic simulation for the Q-process in the theory of probability. SSA and tau-leaping methods are two typ- ical methods, which correspond to the exact and Euler approximations to the stochastic differential equations. In this lecture, I will talk about some recent mathematical results about these algorithms done by our group, which include the mathematical analysis, the construction of new algorithms, etc. I will try to show the interesting interplay of the traditional numerical analysis and the stochastic analysis. , ,300401 E-mail: mathlxw@hebut.edu.cn - Structure Exploited Algorithms for a Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory , ,361005 E-mail: lzlu@xmu.edu.cn The nonsymmetric algebraic Riccati equation arising in transport theory is a special algebraic Riccati equation whose coefficient matrices having some special structures. By reformulating the Riccati equation, they were found that its arbitrary solution matrix is of
  36. 36. 31 a Cauchy-like form and the solution matrix can be computed from a vector form Riccati equation instead of the matrix form that. In this talk, we review some classical-type and Newton-type iterative methods for solving the vector form Riccati equation and present some work under investigation. JASMIN , ,100088 E-mail: zeyao mo@iapcm.ac.cn JASMIN http://www.iapcm.ac/ jasmin JASMIN PIC JASMIN Adaptive RKDG Method Using Different Indicators , ,210093 E-mail: jxqiu@nju.edu.cn In this presentation, we systematically investigate adaptive Runge-Kutta discontinu- ous Galerkin (RKDG) methods for hyperbolic conservation laws with different indicators which were based on the troubled cell indicators studied by Qiu and Shu [SIAM J. Sci. Comput., 27 (2005), 995-1013], with an objective of obtaining efficient and reliable indi- cators to obtain better performance for adaptive computation to save computational cost. Both h-version and r-version adaptive methods are considered in the paper. The idea is to first identify “troubled cells” by different troubled-cell indicators, namely those cells
  37. 37. 32 where limiting might be needed and discontinuities might appear, then adopt an adaptive approach in these cells. A detailed numerical study in one dimensional case is performed, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. , ,100871 E-mail: hztang@pku.edu.cn An Introduction to Piecewise Algebraic Variety , ,116024 E-mail: renhong@dlut.edu.cn It is well known that the face of any object in the nature is combined by some sur- faces, and their intersection. Therefore to study the geometric properties on intersection of surfaces will be very important in both theory and practice. The algebraic variety is one of most important subjects in the classical algebraic ge- ometry which deals with the geometric properties of intersection of surfaces defined by multivariate polynomials. Because of most surfaces have been recently represented by multivariate piecewise polynomials, the study of so-called “piecewise algebraic variety” defined as an intersec- tion of surfaces represented by multivariate piecewise polynomials should be also most important in both theory and practice. In fact, the piecewise algebraic variety is also very useful in Computer Aided Design, Computer Aided Geometric Design, Computer Aided Manufacture, Computer Aided Engineering, Computer Graphics, as well as Image Processing. The main problem is how we can study the piecewise algebraic variety. However, there are some essential difficulties in studying the piecewise algebraic variety, because
  38. 38. 33 of some reasons as follows: • The multivariate piecewise polynomial has very strong local property; • Zeros of a multivariate polynomial determined by multivariate piecewise polynomial on a given cell may be outside this cell; • The multivariate piecewise polynomial defined on a given partition of a domain not only depends on topological property of the partition, but also sometimes depends on geometric property of the partition. In fact, most methods for studying the classical algebraic variety can not be used to study the piecewise algebraic variety. Therefore, we have to find some special methods to study the piecewise algebraic variety, much more to study “real piece-wise algebraic variety”. In this talk, we will introduce some results on the piecewise algebraic variety. Moreover some open problems will also be introduced too. , , 100088 E-mail: wang shuanghu@iapcm.ac.cn jet , ,116023 E-mail: wuweiw@dlut.edu.cn An Adaptive Edge Element Method with Perfectly Matched Absorbing Layers for Wave Scattering by Bipe
  39. 39. 34 , ,210093 E-mail: hjw@nju.edu.cn An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a poste- riori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method. , ,361005 E-mail: cjxu@xmu.edu.cn In this talk, we present some new developments in spectral element methods (SEM) for the unsteady Navier-Stokes equations. The main ingredients include: 1) a fast PN × PN spectral element solver using preconditioned Schur complement al- gorithm for the Navier-Stokes equations, and detailed comparison and discussion of some new approaches. The link of different methods will be clarified. The key feature of our method is that only one grid is needed for the velocity and pressure approximations. Al- though not yet proven by rigorous theoretical analysis, the stability and accuracy of this simple method are demonstrated by a series of the numerical experiment. 2) an efficient stabilization method, which consists in employing SVV technique in the standard SEM for accurate computations of high-Reynolds number flows. Our new formulation yields an algorithm which can be easily implemented and does not require additional computational time. 3) applications of the Legendre-SVV-SEM to the LES of turbulent flows. The SVV- SEM is used as a no-model approach, i.e. no modeling of the sub-grid scale tensor which results from the spatial filtering of the Navier-Stokes equations. Some simulation in the 3D driven cavity flow and the wake flow behind a circular cylinder give very satisfactory results. 4) development and implementation of a triangular spectral element method for the Navier-Stokes equations in complex geometries.
  40. 40. 35 Relations Between the Multiscale Methods for Elliptic Homogenization Problems , ,230026 E-mail: xyyue@ustc.edu.cn We will give a short review on multiscale methods for elliptic homogenization prob- lems. We will focus on the intrinsic links between some popular methods such as gener- alized finite element methods (GFEM), residual- free bubble methods (RFB), variational multiscale methods (VMS), multiscale finite element method (MSFEM) and heteroge- neous multiscale methods (HMM). Numerical Investigation for a Macroscopic Model for Hydrodynamic Nematic Liquid Crystals , , 100875 E-mail: hzhang@bnu.edu.cn We use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain. We reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields. This enables us to use convenient conformal C 0 finite elements in solving the problem. Moreover, a discrete energy law is derived for a modified midpoint time discretization scheme. A fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately. A number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method. At last we will present an example of this macroscopic model for complex fluids in “1+2” dimension case. It be found that the direction of the molecules will tumble from the boundary layer and later on the inner layer with a much longer time period. This is consistent with the theoretical predict of special case. Moreover, we find some complex phenomena, where the tumbling rises from boundary layer then is deep into the middle
  41. 41. 36 area more clearly when the viscosity coefficient of the macro flow has a larger value. The norm of the molecular director would endure greater change as well. This implies that the viscosity of flow plays the role of an accelerator in the whole complex fluids. Comparing these results with the theoretical analysis, we can find that the gradient of the velocity has direct impact on the tumbling phenomena. These results roughly show that such a scheme is capable of giving rich phenomena embedded in the macro-micro model.
  42. 42. 37 E Design of Computer Experiments: Construction and Theory , ,100871 E-mail: myai@math.pku.edu.cn Computer models are widely used in business, engineering, and sciences to study com- plex real world systems. The corresponding physical experimentation might otherwise be time-consuming, costly, or even infeasible to conduct. Space-filling designs have been widely used for conducting computer experiments. A large computer code, like a finite element analysis model, is often run at variable degrees of sophistication, resulting in mul- tiple computer experiments with different levels of accuracy and varying computational times. In this talk we consider the situation in which two such experiments are available and one source is generally more accurate than the other but also more expensive to run. A new type of designs called nested space-filling designs has been proposed for conduct- ing multiple computer experiments with different levels of accuracy. Several approaches to constructing such designs have also been developed. The development of these meth- ods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices. Feynman-Kac , ,571158 E-mail: czchen@hainnu.edu.cn Feynman-Kac (E, D(E)) Xt (E, D(E)) u ∈ D(E) u(Xt ) Fukushima u(Xt ) − u(X0 ) = Mtu + Ntu Mtu Ntu Fukushima Ntu
  43. 43. 38 u Pu f (x) = E x [e−Nt f (Xt )] t Feynman-Kac Feynman-Kac Pu Ntu Feynman-Kac t Girsanov h− Pu t Homeomorphic Flows and Large Deviations for Stochastic Differential Equations Driven by a G-Brownian Motion , ,430072 E-mail: fqgao@whu.edu.cn Some moment estimates and H¨ lder continuity of the G-stochastic integral and the o solutions of stochastic differential equations driven by a G-Brownian motion are obtained. Homeomorphic property with respect to the initial values and large deviation principle for the stochastic differential equations are established. Markov Skeleton Processes and Their Applications , ,410075 E-mail: zthou@csu.edu.cn A stochastic process X(t) is called a Markov skeleton process(MSP) if it has the Markov property on a sequence of stopping times τn ↑ ∞. The usual Markov process, semi- Markov process, deterministic Markov process and semi-regenerative process can be re- garded as special cases of MSP. In this paper, first, backward and forward equations with which we can compute one-dimensional distribution is derived, and then formulas to com- pute finite-dimensional distribution and the existence and computation of limit distribu- tion are also obtained. Based ourselves upon the above the results, we give a tentative study of queueing system, reliability system, and storage system. Transient distribu- tion and formulas to compute limit distribution of the stochastic processes introduced for studying these system are presented in the latter half the paper.
  44. 44. 39 Fisher : , ,100190 E-mail: luosl@amt.ac.cn Fisher Fisher , ,510275 E-mail: renjg@mail.sysu.edu.cn Limit Theorems for Super-diffusions and Branching Hunt Processes , ,100871 E-mail: yxren@math.pku.edu.cn First, we establish a scaling limit theorem for a large class of Dawson-Watanabe su- perprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with Cb -coefficients or a symmetric Levy process on Rd whose Levy 1 exponent Ψ(η) is bounded from below by c|η|α for some c > 0 and α ∈ (0, 2) when |η| is large, a stronger almost sure limit theorem is established for the superprocess. Second, we establish Kesten-Stigum L log L type theorems for super-diffusions and branching Hunt processes using spine decompositions. Wiener
  45. 45. 40 , ,100875 E-mail: shaojh@bnu.edu.cn Wiener Ambrosio-Gigli-Savare Ornstein-Uhlenbeck Additive Hazards Regression with Censoring Indicators missing at random , ,100190 E-mail: slq@amt.ac.cn In this article, we consider a semiparametric additive hazards regression model for right-censored data that allows some censoring indicators to be missing at random. We develop a class of estimating equations and use an inverse probability weighted approach to estimate the regression parameters. Nonparametric smoothing techniques are employed to estimate the probability of non-missingness and the conditional probability of an un- censored observation. The asymptotic properties of the resulting estimators are derived. Simulation studies show that the proposed estimators perform well. We motivate and illustrate our methods with data from a brain cancer clinical trial. Model Selection Criteria for Missing Data Problems via the EM Algorithm , ,650091 E-mail: nstang@ynu.edu.cn We propose a class of novel and computationally attractive model selection criteria for missing data problems based on the output of the expectation-maximization (EM) algorithm. The proposed criteria do not require numerical integration or Laplace approx- imations to integrals. The methodology is very general and can be applied to numerous
  46. 46. 41 situations involving incomplete data within an EM framework, from covariates missing at random (MAR) in arbitrary regression models to nonignorably missing longitudinal responses and/or covariates. Towards this goal, we develop a class of information criteria for missing data problems, called ICH, Q, which yields the Akaike information criterion (AIC) and the Bayesian Information Criterion (BIC) as special cases. The computation of ICH, Q requires an analytic approximation to a complicated function, called the H- function, along with output from the EM algorithm used in obtaining maximum likelihood estimates. To eliminate the analytic approximation to the H-function, we also propose a computationally simpler approximation to ICH, Q, called ICQ, whose computation de- pends solely on the Q-function of the EM algorithm. Advantages and disadvantages of ICH, Q and ICQ are discussed and examined in detail in the context of missing data prob- lems. Theoretical properties of ICH, Q and ICQ are also investigated in detail, including consistency. Extensive simulations are given to demonstrate the methodology and exam- ine the small and large sample performance of ICH, Q and ICQ in missing data problems. An AIDS data set is also presented to illustrate the proposed methodology. profile , ,300071 E-mail: zjwang@nankai.edu.cn linear profile, general profile, nonparametric profile Stochastic Maximum Principles for Partially Observed Optimal Control Problems and Applications , ,250100 E-mail: wuzhen@sdu.edu.cn In this talk, we first give some stochastic maximum principles-the necessary condition of the optimal control–for partially observed risk-sensitive optimal control problems. Ap- plying these theoretical results we study a partially observed linear-quadratic non-zero sum stochastic differential game problem and give an explicit observable Nash equilib- rium point. As a natural deduction, a general maximum principle is also given for a fully
  47. 47. 42 observed risk-sensitive case. This result is applied to study a risk-sensitive optimal portfo- lio problem. An explicit optimal investment strategy and a cost functional are obtained. A numerical simulation result shows an influence of a risk-sensitive parameter on an optimal investment proportion, this coincides with its economic meaning and theoretical results. At last, we study a partially observed stochastic recursive optimal control problems. This kind of problems have wide applications in finance and economic such as the celebrated principal-agent problems. We also give a maximum principle for this kind of optimization problems in this talk. , ,100081 E-mail: xuxz@bit.edu.cnn p p Fiducial p p Fiducial Estimation for a Partial-linear Single-index Model , ,100124 E-mail: lgxue@bjut.edu.cn In this paper, we study the estimation for a partial-linear single-index model. A two- stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components. For the index, a constrained estimating equation leads to an asymptotically more efficient
  48. 48. 43 estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates; and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched. Markov , ,200433 E-mail: jgying@fudan.edu.cn In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that thisirreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition. Generalized Profile LSE in Varying-Coefficient Partially Linear Models with Measurement Errors / , / ,100190/200433 E-mail: yzhou@amss.ac.cn This paper is concerned with the estimating problem of a semi-parametric varying- coefficient partially linear errors-in-variables model. Due to the measurement errors the usual profile least squares estimator of the parametric component, local polynomial es- timator of the nonparametric component and profile least squares based estimator of the error variance are biased and inconsistent. By taking the measurement errors into account we propose a generalized profile least squares estimator for the parametric component and show it is consistent and asymptotically normal. Correspondingly, the consistent esti- mations of the nonparametric component and error variance are proposed as well. These results can be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performances of these proposed esti- mations.
  49. 49. 44 : , ,100190 E-mail: ghzou@amss.ac.cn , , , :
  50. 50. 45 F Primal-Dual IPMs for Conic Optimization , ,200444 E-mail: yqbai@shu.edu.cn After the path-breaking paper of Karmarkar, the Interior-point Methods (IPMs) has been among the most effective methods for solving linear optimization problems. With the development of IPMs, it has been extended to the conic optimization, a class of special convex optimization such as second-order cone optimization and semidefinite optimiza- tion. Because the scheme of primal-dual IPMs includes three blocks: the selection of bar- rier functions, the method of unconstrained optimization and the policy of updating the penalty parameter. We introduce a new class of univariate function called kernel function which is defied by some simple conditions. The barrier function is determined by kernel function with nice properties. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on the kernel functions. Both in the algorithm and in its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of primal- dual IPMs for conic optimization, including linear optimization, second-order cone opti- mization and semidefinite optimization. Iteration bounds both for large- and small-update methods are derived. It is shown that small-update methods based on the new kernel func- √ tions all have the same complexity as the classical primal-dual IPM, namely O( nlog n ). √ For large-update methods the best obtained bound is O( n(logn)log n ), which is up till now the best known bound for such methods. , ,100190 E-mail: htfang@iss.ac.cn
  51. 51. 46 , , Wiener Applications of Steiner Tree Problem in Network Design , ,100190 E-mail: xdhu@amss.ac.cn The study of the Steiner tree problem has received lots of attentions since many impor- tant open problems were solved in 1990s. Those achievements have greatly influenced not only the general theory of designs and analysis of approximation algorithms for combina- torial optimizations but also the discovery and research on many new important applica- tions. Those applications usually require some modifications on the classical Steiner tree problem, and hence they demand new techniques for solving them. As a result, studying various Steiner tree problems forms a very hot topic in the past decade.In this talk we will study some of those applications to the design of computer communication networks. Error estimates for an optimal control problem of the heat equation with state and control constrain , ,430072 E-mail: wanggs62@yeah.net We study the priori error analysis for numerical approximations of the optimal con- trol governed by the heat equation with certain control constraint and ending point state constraint. By making use of the classical space-time discretization scheme, namely, finite element method with the space variable and implicit Euler discretization for the time variable, We first project the original optimal control problem into a semi-discrete
  52. 52. 47 control and state constrained optimal control problem governed by an ordinary differ- ential equation, and then project the aforementioned semi-discrete problem into a fully discrete optimization problem with constraints. With the help of Pontryagin maximum principle, we obtain, under a certain reasonable condition of Slater style, not only a con- vergence order for L2 −error of the optimal controls between the original problem and the semi-discrete problems, but also the solutions between the semi-discrete problem and fully discrete problem. Thus, we provide a base to compute the optimal control of the heat equation with certain control constraint and ending point state constraint. The main difficulty arising in this work is the lack of the quantitative analysis for the Lagrange mul- tipliers appeared in Pontryagin maximum principles in the original problem and discrete problems. These multipliers are produced by the state constraint. , ,100081 E-mail: jmwang@bit.edu.cn Riesz Riesz 1 2 3 Riesz Riesz , ,100190 E-mail: zrxi@iss.ac.cn
  53. 53. 48 I A B , A C II Markov 10-6 Markov Markov Markov Markov Markov III
  54. 54. 49 A. 0 B. 0 Lowner , ,100044 E-mail: nhxiu@bjtu.edu.cn 1934 Karl Lowner Sun-Sun Lowner Lowner Lowner-Heinz Lowner-type Taylor 1 Lowner Euclidean Jordan Algebra 2 Lowner 3 Lowner 4 Inverse and Generalized Inverse Optimization Problems , ,100190 E-mail: xgyang@iss.ac.cn The inverse optimization problems are to find a minimal modification cost of the param- eters of the original problem such that some given solutions become optimum under the new parameters, while the generalized inverse optimization problems are to find a mini- mal modification cost of the parameters of the original problem such that a given objective is achieved. In this talk, we will give a survey about the applications, major developments, major techniques of the inverse and generalized inverse optimization problems. Then we present a unified model which can combine the inverse and generalized inverse problems.

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