45-Minut Invited Talks The Ricci Flow on Kähler Manifolds

Huai-Dong Cao
Department of Mathematics
Lehigh University
Bethlehem, PA 18015, U.S.A
huc2@lehigh.edu

Abstract : I will survey the Ricci flow on compact Kähler manifolds and describe recent developments and some open problems.




Combining PDE and Wavelet Techniques for Image Processing

Tony F.C. Chan
Mathematics Department
University of California, Los Angeles
Box 951555 Los Angeles
CA 90095-1555, U.S.A.
TonyC@college.ucla.edu

Joint work with Haomin Zhou, Math Dept, Georgia Tech

Abstract : Standard wavelet linear approximations (truncating high frequency coefficients) generate oscillations (Gibbs' phenomenon) near singularities in piecewise smooth functions. Nonlinear and data dependent methods are often used to overcome this problem. Recently, partial differential equation (PDE) and variational techniques have been introduced into wavelet transforms for the same purpose. In this talk, I will present our work on two different approaches that we have been working on in this direction. One is to use PDE ideas to directly change the standard wavelet transform algorithms so as to generate wavelet coefficients which can avoid oscillations in reconstructions when the high frequency coefficients are truncated. We have designed an adaptive ENO wavelet transform by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet transforms retains the essential properties and advantages of standard wavelet transforms without any edge artifacts. We have shown the stability and a rigorous error bound which depends only on the size of the derivative of the function away from the discontinuities. The second one is to stay with standard wavelet transforms and use variational PDE techniques to modify the coefficients in the truncation process so that the oscillations are reduced in the reconstruction processes. In particular, we use minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have fewer oscillations near edges. Examples in applications including image compression, denoising, inpainting will be presented.




The Geometry Induced by a Class of Subelliptic Operators

Der-Chen Chang
Department of Mathematics
Georgetown University
Washington DC, 20057 U.S.A.
chang@georgetown.edu

Abstract : In this article we study the geometry induced by a class of second-order subelliptic operators. This class contains degenerate elliptic and hypoelliptic operators (such as the Grusin operator, the Baouendi-Goulaouic operator and the sub-Laplacian on the Heisenberg group). Given any two points in the space, the number of geodesics and the lengths of those geodesics are calculated. We also find the modified complex action function and show that the critical points of this function will recover the lengths of the corresponding geodesics. Using this action function, we may obtain the fundamental solution and the heat kernel of the corresponding operator.




Some Recent Results on Combinatorial Number Theory

Mei-Chu Chang
Department of Mathematics
University of California, Riverside
Riverside, CA 92521-0135, U.S.A
meichu.chang@ucr.edu

Abstract : Let A be a finite subset of a ring R. The sum set and the product set of A are
A+A
=
{a1 + a2   |   ai A },
A·A
=
{a1 a2   |   ai A }.
In a 1983 paper Erdös and Szemerédi conjectured that for a set of integers, either the sum set or the product set is large. More precisely, either the sum set or the product set should have nearly n2 elements.

This problem is still unsolved, despite a certain amount of recent results.

We will describe the present status of it and some related topics. For example, we give results for rings that are different from \Bbb Z or \Bbb C, for some noncommutative generalizations and we give some applications to the theory of exponential sums and cryptography.




Global Existence and Convergence for a Fourth-Order Flow in Conformal Geometryf

Shu-Cheng Chang
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan
scchang@math.nthu.edu.tw

Abstract : Let (M,[g0]) be a closed smooth Riemannian 3-manifold with the fixed conformal class [g0]. In this talk, we first review the so-called Bondi-mass type estimates for the Calabi flow on (M,[g0]). With its applications, we show that the solution of the Q-curvature flow exists on M×[0,. Moreover, if we assume that the Q0-curvature with respect to g0 is positive, then the solution converges smoothly to a metric of constant positive Q-curvature. As a consequence, we show that there exists a positive constant Q-curvature metric in such a fixed conformal class [g0] with the Paneitz operator of the negative base eigenvalue. Finally we will deal with the Q-curvature flow on a CR 3-manifold.

f This program is proposed by S.-T. Yau during his visiting at the National Tsing Hua University at Hsinchu in 1991-92. The author would like to thank him for the constant encouragement and inspirations during the work.




Normal Dilations

Man-Duen Choi
Department of Mathematics
University of Toronto
Toronto, M5S 3G3, Canada
choi@math.toronto.edu

Abstract : Normal dilations usually serve as a sort of standard models for the intrinsic structure of Hilbert space operators. Here, we will look into some old and new results of normal dilations related to numerical ranges and spectral sets.




A Semi-discrete, Linear Curve Shortening Flow

Bennett Chow
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093, U.S.A
benchow@math.ucsd.edu

Abstract : We consider a curve shortening type flow of polygons in the form of a system of linear ODEs. This flow has similarities with the curve shortening flow as considered by Gage and Hamilton and is even more similar to the work of Bruckstein, Sapiro, and Shaked on fully discrete evolutions of polygons. We are interested in studying nonlinear versions of this flow. This is joint work with David Glickenstein of the University of Arizona.




Convolution Operators and Harmonic Functions

Cho-Ho Chu
School of Mathematical Sciences
Queen Mary, University of London
Mile End Road, London E1 4NS
United Kingdom
c.chu@qmul.ac.uk

Abstract : We discuss some recent results concerning convolution operators and harmonic functions on locally compact groups. A special feature is that the functions and measures involved are allowed to take matrix values.




Non-stationary Tight Frames of Wavelets

Charles Chui
UMSL and Stanford
ckchui@stanford.edu

Abstract : I will present some highlights of my two recent joint papers with Wenjie He (UMSL) and Joachim Stoeckler (Dortmund, Germany) that will appear (or just appeared) in the second and third issues of the ACHA Special Issues on Recent Development of Frames. The goal of these two papers is to develop a general theory of non-stationary wavelets with small (local) supports and arbitrarily desirable orders of vanishing moments under the tight frame structure, with the first paper being focused on bounded intervals and the second on unbounded intervals. Our theory is based on a very weak notion of multiresolution approximation (MRA), in which dilation and translation operations are no longer needed (but only follow as special cases), and the main ingredients are just approximation properties of nested subspaces with locally supported basis functions satisfying certain uniform "matrix bandwidth" upper bound condition. The order of vanishing moments is a focal point of our study. Hence, for polynomial splines of order m on arbitrary nested knot sequences, locally supported tight-frame spline wavelets can be constructed to possess vanishing moments of order up to m. It is important to remark that while the study of orthonormal wavelets on a bounded interval is a follow-up investigation of that of orthonormal wavelets on the real-line (by introducing certain wavelet functions near the end-points of the bounded interval that preserve the orthogonality property), this approach cannot be (easily) extended to tight frames, where the key property of vanishing moments is no longer a by-product (due to the lack of orthogonality in tight frames). In this regard, our two papers do not directly depend on each other, though the results in our first paper on bounded intervals definitely facilitate our discussion in the second paper on tight frames on unbounded intervals.




The Asymptotic Expansion of the Bergman Kernel

Xianzhe Dai
Department of Mathematics
University of California, Santa
Barbara Santa Barbara, CA 93106
dai@math.ucsb.edu

Abstract : The Bergman kernel in the context of several complex variables (i.e. for pseudoconvex domains) has long been an important subject. Its analogue for complex projective manifolds is studied by G. Tian, W. Ruan, S. Zelditch, Catlin, Z. Lu, establishing the asymptotic expansion for high powers of an ample line bundle. Moreover, the coefficients in the asymptotic expansion encode geometric information of the underlying complex projective manifolds. This asymptotic expansion plays a crucial role in the recent work of Donaldson where the existence of Kähler metrics with constant scalar curvature is shown to be closely related to Chow-Mumford stability.

We study the asymptotic expansion of Bergman kernel for high powers of an ample line bundle in the more general context of symplectic manifolds and orbifolds. One of our motivations is to extend Donaldson's work to orbifolds. This is joint work with K. Liu and X. Ma.




Vertex Operator Superalgebras

Chongying Dong
Department of Mathematics
University of California Santa Cruz
1156 High Street Santa Cruz, CA 95064, U.S.A
dong@math.ucsc.edu

Abstract : The modular invariance property of trace functions in the orbifold theory for a vertex operator superalgebra will be discussed. This result generalizes many known ones.




Phase Field Models and Simulations of Vesicle Membranes

Qiang Du
Department of Mathematics
Penn State University
University Park, PA 16802, USA
qdu@math.psu.edu

Abstract : In this talk, we report some joint works with colleagues at PSU on a variational phase field approach developed for modeling the vesicle membranes under elastic bending energy and its possible impact on the study of complex bio-membrane systems. The effectiveness of such a phase field approach is substantiated via careful analysis and extensive computation. Various membrane configurations have been captured in the numerical simulations and the numerical procedure has been shown to be insensitive to topological events. We also discuss the problem of numerically retrieving useful topological information of the membrane from the phase field model for both tracking and control purposes which may be of even broader interests.

Some simulation results can be found at

http://www.math.psu.edu/qdu/Res/Pic/gallery5.html.

Related references can be downloaded from

http://www.math.psu.edu/qdu/Res/year.html.




Birkhoff Ergodic Averages

Ai-Hua Fan
Faculté de Mathématiques et d'Informatique
Université de Picardie Jules Verne
33, rue Saint Leu
80039 Amiens CEDEX 1, France
ai-hua.fan@u-picardie.fr

Abstract : We will present some recent works on the Birkhoff ergodic averages. It concerns the Banach valued Birkhoff ergodic averages relative to a compact dynamical system, their convergence speeds, their relation to the topological entropy, their applications in the number theory and in the multifractal analysis.




Kaehler Manifolds with Numerically Effective Ricci Class

Fuquan Fang
Department of Mathematics
NanKai University
No.94, Weijin Road, Tianjin, P.R.of China
ffang@nankai.edu.cn

Abstract : In this talk I will first brief some background and known results on Kaehler manifolds with nef. Ricci class, and then I will discuss and prove a theorem concerning the rigidity of Kaehler manifolds with nef. Ricci class.




Number-theoretic Methods in Experimental Designs

Kai-Tai Fang*
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong
ktfang@hkbu.edu.hk

Yuan Wang
Academy of Mathematics and System Sciences
Chinese Academy of Sciences
Beijing 100080, P.R. China
ywang@math.ac.cn

Abstract : Computer experiments have been widely used in various fields of industry, system engineering, and others because many physical processes are difficult or even impossible to study by conventional experimental methods. Design and modeling of computer experiments have become a hot topic since late Seventies of the Twentieth Century. Almost in the same time two different approaches are proposed for design of computer experiments: Latin hypercube sampling (LHS) and uniform design (UD). The former is a stochastic approach and the latter proposed us in 1978 is a deterministic one. In this talk we will review the developments in the past 25 years of these two approaches and discuss their advantages and shortcomings. A uniform design is a Low-discrepancy set in the sense of number-theoretic methods (or quasi-Monte Carlo methods).

In the past four years the Ford Motor Company has employed the uniform design for their computer experiments. Dr. Agus Sudjianto, Engineering Manager in his letter invitation to Kai-Tai Fang said: ``In the past few years, we have tremendous in using Uniform Design for computer experiments. The technique has become a critical enabler for us to execute ``design for Six Sigma" to supper the new product development, in particular, automotive engine design. Today, computer experiments using uniform design have become standard practices at Ford Motor Company to support early stage of product design before hardware is available. We would like to share with you our successful real world industrial experiences in applying the methodology that you developed. Additionally, your visit will be very valuable for us to gain more insight about the methodology as well as to learn the latest development in the area." Hundreds of such successful applications of uniform designs in practice can be found in literature.

Besides computer experiments, the number-theoretic methods can be applied to factorial and supersaturated designs. In the past years there is an essential development on the uniformity in the above designs. The talk will review the development in this direction and address related discussions.




Local Monodromy of the Kloosterman Sheaf

Lei Fu*
Institute of Mathematics
Nankai University
Tianjin, P.R. China
leifu@nankai.edu.cn

Daqing Wan
Institute of Mathematics
Chinese Academy of Sciences
Beijing, P. R. China
and
Department of Mathematics
University of California
Irvine, CA 92697, U.S.A.
dwan@math.uci.edu

Abstract : The classical Kloosterman sum gives rise to a Kloosterman sheaf, which defines a galois representation of the function field unramified outside 0 and . Based on the work of N. Katz, we study the local monodromy of this representation at . We then apply our result to study the bad factors of the L-functions of symmetric products of the Kloosterman sheaf.




Groups and Factors

Liming Ge
Department of Mathematics and Statistics
University of New Hampshire
Durham, NH 03824, U.S.A

liming@spicerack.sr.unh.edu

Abstract : Connections between groups and their group von Neumann algebras will be discussed.




Experimental Design, Coding Theory and Reproducing Kernel Hilbert Spaces

Fred J. Hickernell
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong
fred@hkbu.edu.hk

Abstract : Experimental design is an important area of statistical research that investigates how to determine the values (levels) of independent variables (factors) so that the data obtained from an experiment provide the maximum information about the underlying function (response). The original motivation for this subject came from laboratory or the field experiments, but today it is also important to design computer experiments (numerical simulations) well.

Constructing good designs relies on finite fields, Hamming distances, parity check matrices, and other ideas from coding theory. Measures of design quality may be formulated as discrepancy or dispersion measures that arise from the theory of reproducing kernel Hilbert spaces. Thus, experimental design is a practical subject that requires input from both algebraists and analysts.

This talk highlights the interplay of experimental design, coding theory and reproducing kernel Hilbert spaces. Examples will be used to illustrate how algebraic and analytic methods can be used to solve problems in experimental design. Some open problems will be described.




Collineation Groups of Translation Planes

Chat-Yin Ho
Department of Mathematics
University of Florida
358 Little Hall, PO Box 118105
Gainesville, FL 32611-8105, U.S.A
cyh@math.ufl.edu

Abstract : Fundamental Theorem of Projective Geometry distinguishes the role of projective planes. In the last four decades the emphasis in this area has on finite planes. Among projective planes, translation planes have rich algebraic structures. A translation plane can be considered as an analogue of the Euclid plane. A long outstanding but still open question in the theory of translation plane is the following Main Problem: Determine quasi-simple collineation groups of translation planes.

For a translation plane of odd order, not one simple group has been shown or eliminated to be a collineation group. Define a simple translation plane to be a translation plane that admits a simple collineation group. This suggests the following Odd Order Translation Planes Conjecture: A simple translation plane has order divisible by 4.

Not much has been known for the Main Problem if the group in question does not contain non central perspectivities. On the other hand, the Hering-Ostrom Theorem (Hering 1972, Ostrom 1970, 1974) provides a complete answer to the Main Problem when the group in question is generated by elations. We present the progress and results for the odd order translation planes conjecture and the Main Problem. In the case in which the linear collineation group is generated by perspectivities, we will show that if a minimal normal subgroup is non abelian, the the characteristic of the plane must be even.




Geometric Properties and Non-Blowup of 3D Incompressible Euler Flow

Thomas Yizhao Hou
Applied & Computational Mathematics
California Institute of Technology
Pasadena, CA 91125, U.S.A.
hou@acm.caltech.edu

Abstract : Whether the 3D incompressible Euler equation can develop a finite time singualrity from smooth initial data has been an outstanding open problem. It has been believed that a finite singularity of the 3D Euler equation could be the onset of turbulence. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equation. Further, we show that there is a sharp relationship between the geometric properties of the vortex filament and the maximum vortex stretching. By exploring this geometric property of the vorticity field, we have obtained a global existence of the 3D incompressible Euler equation under the some mild and localized regularity assumption on vortex filaments. Our assumption on the local geometric regularity of vortex filaments seems consistent with numerical computations. This is a joint work with Dr. Jian Deng and Mr. Xinwei Yu.




Stochastic Analysis on Path and Loop Spaces over a Riemannian Manifolds

Elton P. Hsu
Department of Mathematics
Northwestern University
Evanston, IL 60208, U.S.A
elton@math.northwestern.edu

Abstract : The path or the loop spaces over a Riemannian manifold are typical examples of (non-flat) infinite dimensional spaces. They are equipped with a distinguished probability measure (Wiener measure) determined by the Laplace-Beltrami operator. This measure is the law of Brownian motion on the manifold. At the same time the concept of parallel translation (with respect to the Levi-Civita connection) gives rise to a gradient operator (generalized Malliavin derivative) on the path and loop spaces. An analytic theory of path and loop spaces based on these two fundamental objects can be developed with the help from probability theory and stochastic analysis. We will give a brief survey of this theory and discuss some open problems.




Factorization Theorem and Geometric Invariant Theory

Yi Hu
Department of Mathematics
The University of Arizona
617 N. Santa Rita Ave., P.O. Box 210089
Tucson, AZ 85721V0089, USA
yhu@math.arizona.edu

Abstract : We will survey some new developments in the last decade or so in Geometric Invariant Theory and their applications to Birational Geometry of Projective Varieties.

In searching for a birational model of a projective variety, it is desirable to know how we can obtain one projective variety from another one within the same birational class. This was known long time ago for surfaces due to the classification theory. But arbitrarily higher dimensional cases were only discovered recently. As it turns out, given any two birational projective orbifolds X and Y, we can always realize them as two geometric quotients by C* ×GLn-action on a nonsingular projective variety, then the Variation of Geometric Invariant Theory (VGIT) implies that X and Y are related by a sequence of weighted blowups and blowdowns. This is the so-called Factorization Theorem for projective varieties with (at worst) finite quotient singularities, which was proved by the speaker in early 2004. When both X and Y are nonsingular, the Factorization Theorem was due to Wlodarczyk and Abramovich et. al. In this smooth case and in terms of our approach, X and Y can be realized as two geometric quotients by C* -action, which was proved in late 90's jointly with Keel. (There is a strong version of Factorization Theorem which is still open.)

We will start with basic ideas of Geometric Invariant Theory and explain its relation with symplectic reductions. And then, we will explain the Variational Geometric Invariant Theory and its application to the Factorization Theorem.




Dirac Operators and Group Representations

Jing-Song Huang
Department of Mathematics
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
mahuang@ust.hk

Abstract : Dirac invented his operator for describing elementary particles. He made astonishing discoveries using this operator. In this talk we will demonstrate the important role various Dirac operators play in representation theory, in particular, the recent developments of Dirac cohomology of admissible and unitray representations of reductive Lie groups. Most of the new results that will be presented are joint work with Pavle Pandzic and David Renard.




The Integral Novikov Conjectures and Compactifications for S-arithmetic Groups

Lizhen Ji
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
lji@umich.edu

Abstract : An important conjecture in topology is the Novikov conjecture on (oriented) homotopy invariance of the higher signatures, which can be formulated equivalently as the rational injectivity of the assembly map in algebraic surgery theory. The injectivity of the assembly map is called the integral Novikov conjecture in L-theory. There are also assembly maps in algebraic K-theory and C*-algebras, and the rational injectivity of each assembly map is called the Novikov conjecture, and the injectivity of the assembly map the integral Novikov conjecture in the corresponding theory.

Various results on the integral Novikov conjectures have been obtained for discrete subgroups of Lie groups. In this talk, I will present some results for S-arithmetic groups of semisimple linear algebraic groups.

The Novikov conjectures are closely related to the large scale geometry, in particular compactifications, of the universal covering of finite classifying spaces. For any torsion free arithmetic group G of a semisimple linear algebraic group G, the associated locally symmetric space G\X is a classifying space of G, where X = G/K is the symmetric space. If G\X is noncompact, the Borel-Serre compactification G\[`X]BS of G\G/K is a finite classifying space. In the approach to the Novikov conjecture by Carlsson and Pederson, a compactification of the partial Borel-Serre compactification [`X]BS is needed. For an S-arithmetic subgroup G of a semisimple linear algebraic group G, the product of X together with some Bruhat-Tits buildings plays the role of X for arithmetic groups, and certain compactifications of such products are needed for the integral Novikov conjectures.




Bessel Sequences and Riesz Bases in Sobolev Spaces

Rong-Qing Jia
Department of Mathematical & Statistical Sciences
University of Alberta
Edmonton, Alberta T6G 2G1 Canada
rjia@ualberta.ca

Abstract : In this talk we will present some recent results on Bessel sequences, affine frames, and Riesz bases induced by dilation and translation of one or several basic functions. We will discuss their properties with respect to the scale of Sobolev spaces. On the basis of theory of function spaces, we are able to obtain very general results on Bessel sequences and affine frames without requiring the structure of multiresolution analysis. We will also give a general principle for the construction of wavelet bases for Sobolev spaces.




Geometry in Low-Dimensional Dynamical Systems

Yunping Jiang
Department of Mathematics
CUNY Graduate Center and Queens College
65-30 Kissena Blvd
Flushing, NY 11367, U.S.A
yunqc@forbin.qc.edu

Abstract : In this talk, I will summarize my research in geometric aspects of low-dimensional dynamical systems. I will define the space of geometrically finite maps. I will mention classification of this space up to conjugacy by quasisymmetric homeomorphisms and up to conjugacy by diffeomorphisms. I will show that every topological conjugacy class is also a quasisymmetric conjugacy class in this space. The scaling function is a useful tool in my further classification. I will define it and show that it exists and is Hölder continuous for a non-critical geometrically finite map, while it exists and is discontinuous for a critical geometrically finite map. I will show the smooth classification by using scaling functions. I will show that the conjugacy between two geometrically finite maps is a diffeomorphism if and only if their scaling functions and the asymmetries and exponents at corresponding singular points are the same. The scaling functions of Ulam-von Neumann transformations, which are geometrically finite maps conjugating to the map f(x) = -x2+2 of [-2, 2], will also mentioned. We will show that the conjugacy between two Ulam-von Neumann transformations is a diffeomorphism if and only if their eigenvalues at corresponding periodic points and their exponents at a unique critical point are the same. A more complete picture will be given in a rigidity theorem for a wider class of certain one-dimensional maps, which contains all geometrically finite maps. The theorem says that the topological conjugacy between two such maps is piece-wise C1 if and only if it is differentiable at one general point with bound, in addition, it is piece-wise diffeomorphic if the exponents at corresponding power law singularities are also the same.

We used to call the space of Ulam-von Neumann transformations as the boundary of hyperbolicity. I will explain the reason. That is, I will show the deformation from hyperbolic systems to a non-hyperbolic system, which is defined as a family of Cantor systems. I will show that the bridge geometry of a Cantor system in the family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap. Asymptotical behaviour of the family of scaling functions corresponding to a family of Cantor systems is also investigated. I will also mention a recent work with Fan and Wu. The result in this work says that the Hausdorff dimension of the maximal invariant set of a map in a family of Cantor systems is also regulated by the size of the leading gap.




Computations of Multivalued Solutions of Nonlinear PDEs

Shi Jin
Department of Mathematics
University of Wisconsin
Madison, WI 53706, USA
jin@math.wisc.edu

Abstract : Many physical problems arising from high frequency waves, dispersive waves or Hamiltonian systems require the computations of multivalued solutions which cannot be described by the viscosity methods. In this talk I will review several recent numerical methods for such problems, including the moment methods, kinetic equations and level set methods. Applications to the semiclassical Schroedinger equation and Euler-Piosson equations with applications to modulated electron beams in Klystrons, and general symmetric hyperbolic systems will be discussed.




Results and Open Problems on Semi-Linear Elliptic Equations

Man Kam Kwong
Lucent Technologies
kwong@nwsgpa.ih.lucent.com

Abstract : We survey some conjectures and open problems in the study of semi-linear elliptic equations of the form Du+f(u) = 0 either in a bounded domain or the entire Rn . A classical example is f(u) = upuq where 0 < q < p , but f(u) can be more general.

Topics covered include: symmetry of solutions, symmetry of the discretized equations, uniqueness and multiplicity of the ground state solutions, numerical algorithm based on the mountain pass lemma, the De Giorgi conjecture, and the Lazer and McKenna equations of suspension bridges.




On an Invariant Complementation Property of the Group Von Neumann Algebra of a Locally Compact

Anthony To-Ming Lau
Department of Mathematics
University of Alberta
Edmonton, Alberta T6G 2G1 Canada
tlau@math.ualberta.ca

Abstract : In this talk, I discuss some recent results on an invariant complementation property of the group von Neumann algebra V N(G) of a locally compact group G and its relationship with the separation property of closed subgroups of G by continuous positive definite functions.




G2 Geometry

Conan Nai-Chung Leung
Institute of Mathematical Sciences
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong
leung@ims.cuhk.edu.hk

Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong
ncleung@math.cuhk.edu.hk

Abstract

We explain the geometry of G2-manifolds using vector cross product and octonion structure. We will also discuss its relationships with mirror symmetry of Calabi-Yau threefolds, M-theory, Seiberg-Written theory, triality and so on.




The Implications of the Third International Mathematics and Science Study for Mathematics Curriculum Reform in Chinese Communities

Frederick Koon-Shing Leung
Faculty of Education
The University of Hong Kong
Pokfulam Road, Hong Kong
FrederickLeung@hku.hk

Abstract : In this paper, results of the Third International Mathematics and Science Study (TIMSS) and the TIMSS 1999 Video Study will be presented, and implications for mathematics curriculum reform in Chinese communities will be discussed in light of the results.

From the TIMSS studies, it was found that students from Chinese communities (including Hong Kong, Taiwan, and Singapore) consistently did extremely well in mathematics. However, their superior performance was not accompanied by correspondingly positive attitudes towards mathematics. Also, variables due to societal resources which usually explain within-country differences in achievement failed to explain across-country differences in achievement as far as the Chinese communities are concerned. In fact, from the findings of the TIMSS studies, the societal resources for education in these communities were found to be relatively unfavourable.

From the quantitative analysis of the TIMSS 1999 Video Study data, classrooms in the Chinese community of Hong Kong were found to be dominated by whole-class interaction. Teachers talked most of the time, while students were found solving procedural problems unrelated to real-life following prescribed methods. On the other hand, a qualitative analysis of the same data set revealed that relatively advanced content was taught in the Hong Kong classrooms, and the lessons were more coherent. The mathematics presentation was more fully developed and students were more likely to be engaged in the lesson, and the overall quality of the lessons was found to be high.

The seemingly contradicting results highlight that fact that outcomes of international comparative studies should not be interpreted rashly. Finally implications of the results discussed in the paper will be drawn for mathematics curriculum reform in Chinese communities.




Moduli of Sheaves on K3 Surfaces

Jun Li
Department of Mathematics
Stanford University
Stanford, CA 94305-2125, U.S.A
jli@math.stanford.edu

Abstract : We will discuss the possible generalization of the work of Nakajima on moduli of sheaves on K3 surfaces and representation of infinite Lie algebras.




Strictly Pseudo-convex Pseudo-hermitian CR manifolds

Song-Ying Li
Department of Mathematics
University of California, Irvine
Irvine, CA 92697-3875, U.S.A.
sli@math.uci.edu

Abstract : In this talk, I present some recent results on strictly pseudoconvex pseudo-hermitian CR manifolds of hypersurface type. It includes some joint works with D.C. Chang, and with H.-K. Luk. It contains estimating eigenvalues for sub-Laplacian, expression for Riemann zeta function of sub-Laplacian, formula for the Webster Ricci curvatures and characterizations of balls in Cn.




Symplectic Topology in Dimensions 4 and 6

Tian-Jun Li
Department of Mathematics
University of Minnesota
127 Vincent Hall, 206 Church St. S.E.
Minneapolis, MN 55455 USA
tjli@math.umn.edu

Abstract : In this talk we will survey the recent results of symplectic structures on 4 dimensional and 6 dimensional manifolds.




On Bubbling behaviors of the Toda System in Two Dimension

Chang-Shou Lin
Department of Mathematics
National Chung Cheng University
160 San-Hsing, Min-Hsiung, Chia-Yi 621 Taiwan
cslin@math.ccu.edu.tw

Abstract : In this talk, we want to consider the following Toda system: On a Rie- mann Surface S with the area |S| = 1

Dui + N

j = 1 
aij rj
hjeuj

hj euj
- 1
= 0 in S
for i = 1, 2, , N, where
(aij) =







2
-1
0
-1
2
0
0
-1
:
:
···
:
0
2








is the Cartan Matrix, rj R and hj (x) is a positive C1 function on S. The Toda System is a natural generalization of the well-know Liouvill equation. Here, we want to talk about the existence of minimizers at critical values of rj and also the bubbling behavior of solutions. This is a joint work with J. Jost and G.-F. Wang.




Separation of Bound State Solutions of Coupled Nonlinear Schrodinger Equations

Tai-Chia Lin
Department of Mathematics
National Taiwan University
Taipei, Taiwan
tclin@ntu.edu.tw

Abstract : Here we study bound state solutions of a system of nonlinear Schrodinger equations with a large parameter. As the parameter goes to infinity, we prove rigorously that the bound state solution converges uniformly (up to a subsequence) to its limit on any compact subset of the whole domain. Furthermore, each component of the limit function is locally Lipschitz continuous, and the associated nodal domains may cover the whole domain. Such a mathematical theorem may support a physical phenomenon called phase separation in multispecies Bose-Einstein condensates.




Patterns Generation and Spatial Entropy in Multi-Dimensional Lattice Models

Song-Sun Lin*
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
sslin@math.nctu.edu.tw

Jung-Chao Ban
National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu 300, Taiwan
jcban@math.cts.nthu.edu.tw

Yin-Heng Lin
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
yhlin.am91g@nctu.edu.tw

Abstract : In this lecture, we study the patterns generation problem in multi-dimensional models. Patterns may be realized as stationary solutions of infinite dimensional lattice dynamical system on integer lattices Zd. We first review the problems and associated transition matrices Hn on Z2×n which have been obtained earlier. To study spatial entropy h = h(H2), we need to identify all admissible patterns on Z(m+1)×n. By introducing a Reduction operator Rm, Hn+1m can be expressed in terms of Hnm. Rm can also be used to given a lower bound of h. We then introduce trace operator Tm, and show h = limsupm[(logr(Tm))/ m], where r(Tm) is the maximum eigenvalue of Tm. When H2 is symmetric, [(logr(T2m))/ 2m] are upper bounds of h.




Knot Adjacency and Classification of Knots

Xiao-Song Lin
Department of Mathematics
University of California, Riverside
Riverside, CA 92521, U.S.A.
xl@math.ucr.edu

Abstract : Two knots in the 3-space may differ by a single operation of cutting and pasting of strands. Such a relationship between knots occupies a central position in knot theory. The notion of knot adjacency generalizes this relationship to a multiplex relationship between two knots. We apply techniques and results from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer the question of the extent to which non-isotopic knots can be adjacent to each other. Our main theorem provides a partial verification of the conjecture of V. Vassiliev that finite type knot invariants distinguish all knots. (This is a joint work with Effie Kalfagianni of Michigan State University.)




Some Hausdorff Properties for a Multi-parameter Stable Process

Zhengyan Lin
Department of Mathematics
Zhejian University
Hangzhou 310028, China
zlin@zju.edu.cn

Abstract : Let X = {X(t),t R+N} be a d-dimension stable process with N-parameter and index alpha. We investigate Hausdorff measures and Hausdorff dimensions of the level set, range, graph, k multiple time set and multiple point set of this process. We solve the problem of Ehm (1981) and answer the conjecture of Khoshnevisan, Xiao and Zhong (2003) about the Hausdorff measure of the level set; extend the conclusion of Rosen (1984) and sharpen the result of Zhong and Xiao (1995) about Hausdorff measure and Hausdorff dimension of k multiple times, moreover, extend the result of Talagrand (1998) about the Hausdorff measure of multiple points of the fractional Brownian motion to the case of a stable process.




The Enumeration of Nodal Curves and the Harvey-Moore Problem

Ai-Ko Liu
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840 USA
akliu@math.berkeley.edu

Abstract : In this talk, we will survey the algebraic technique about nodal curve enumeration on algebraic surfaces and the universality theorem using the algebraic Seiberg-Witten Invariants. Based on the algebraic proof of the universality theorem, we will discuss the enumeration problem of immersed nodal curves on K3 fibered Calabi-Yau three-folds and the resolution of Harvey-Moore conjecture on the Gopakumar-Vafa numbers.




Marino-Vafa Formula of One-partition Hodge Integrals

Chiu-Chu Melissa Liu
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138, USA
ccliu@math.harvard.edu

Abstract : Marcos Marino and Cumrun Vafa conjectured a remarkable formula of one-partition Hodge integrals based on duality between open Gromov-Witten theory and Chern-Simons theory. I will describe a proof of this formula based on joint works with Kefeng Liu and Jian Zhou.




Divorcing Pressure from Viscosity in Incompressible Navier-Stokes Dynamics

Jian-Guo Liu
Department of Mathematics
University of Maryland
College Park, MD 20742-4015, U.S.A.
jliu@math.umd.edu

Abstract : The pressure term has always created difficulties in treating the Navier-Stokes equations of incompressible flow, reflected in the lack of a useful evolution equation or boundary conditions to determine it. In joint work with Bob Pego and Jie Liu, we show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain with no-slip boundary conditions, we can treat the Navier-Stokes equations as a perturbed vector diffusion equation instead of as a perturbed Stokes system. We illustrate the advantages of this view in a number of ways. In particular, we provide simple proofs of (i) the unconditional stability of a time-differencing scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solutions of stationary Stokes systems, and (ii) existence and uniqueness of strong solutions based on the difference scheme.




Analysis of Singular Set of the Landau-Lifshitz System

Xiangao Liu
Institute of Mathematics
Fudan University
Shanghai 200433, China
xgliu@fudan.edu.cn

Abstract : The aim of this work is to analyze the singular sets of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from a m-dimensional manifold M into the unit sphere S2 of R3. The main barrier to Landau Lifshtiz system is that there is not monotonicity inequality which plays an important role in getting regularity. First we study the partial regularity to the weak stationary solution of the system. Furthermore, suppose that uk u weakly in W1,2(M×R+, S2) and that St is the blow up set for fixed t. In the present paper we first prove that St is a Hm-2-rectifiable set for almost all t R+. And then we verify that St moves by the quasi-mean curvature under some assumptions, which is a new 2-codimension curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas.

After the seminal work contributed to the study of the construction of singular sets of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.




On a Class of Double Cosets in Reductive Algebraic Groups

Jiang-Hua Lu
Department of Mathematics
The University of Hong Kong
Pokfulam Rd., Hong Kong
jhlu@maths.hku.hk

Abstract : We study a class of double coset spaces in the product of two reductive algebraic groups. We will explain applications of the classification to some problems in Poisson geometry.




A Variational Method to Find Geometric Structures on 3-manifolds

Feng Luo
Department of Mathematics
Rutgers University
New Brunswick, NJ 08854, U.S.A.
fluo@math.rutgers.edu

Abstract : The concept of linear geometric structure on the triangulated 3-manifold is introduced and the volume of the structure is defined. The space of all linear geometric structures is an open convex set. We prove that the critical points of the volume functional are either totally geodesic triangulations in constant sectional curvature metrics or the manifold admits a linear Euclidean structure of non-negative curvature. The later means that one can realize each 3-simplex in the triangulation by a Euclidean 3-simplex so that the sum of the dihedral angles at each edge is at most 2 . We conjecture that the later class of manifolds have zero Gromov norm.




Parabolic Methods in Kahler Geometry

Lei Ni
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093 U.S.A.
lni@math.ucsd.edu

Abstract : In this talk we discuss certain parabolic methods in the study of Kahler geometry.




Middle School Algebra in California

Yat-Sun Poon
Department of Mathematics
University of California at Riverside
Riverside, CA 92521, U.S.A
ypoon@ucr.edu

Abstract : Middle school algebra is known to be a bottleneck subject for many students in their academic lives. From the perspective of teacher training providers, we examine the standard required by the State of California.




Model Assessment, Selection and Averaging

Xiaotong Shen
School of Statistics
University of Minnesota
224 Church Street, S.E. Minneapolis, MN 55455, U.S.A.
xshen@stat.umn.edu

Abstract : In this talk, I will discuss a number issues involved in model assessment, selection and averaging from a prediction viewpoint. A general technique of model assessment will be presented based on data perturbation, yielding optimal selection, in particular-model selection and combination. From a frequentist perspective, model combination over a selected subset of modeling procedures is attractive, as it controls bias while reducing variability, and hence yielding better performance in terms of the accuracy of estimation and prediction. To realize the potential of model combination, I will present methodologies for estimating the optimal tuning parameter such as weights as well as subsets for combining via data perturbation. Simulated and real data examples are presented to illustrate main aspects.




Recent Progress on Boundary Value Problems in Nonsmooth Domains

Zhongwei Shen
Mathematics Science
University of Kentucky
Lexington, Kentucky 40506-0027, U.S.A.
shenz@ms.uky.edu

Abstract : In this talk we will discuss a new approach to the boundary value problems with Lp data, via L2 estimates. The approach may be used for elliptic systems as well as higher-order elliptic equations on Lipschitz domains. Related results on the Riesz transforms associated with second order elliptic equations of divergent form will also be discussed.




Unitary Extension Principle and Applications

Zuowei Shen
Department of Mathematics
University of Singapore
2, Science Drive 2, Singapore 117543
matzuows@nus.edu.sg

Abstract : Since the unitary extension principle was published in 1997, it has led to much theoretic development and has been used in various applications. The unitary extension principle provides a great flexibility of designing tight frame wavelet filters and makes constructions of tight frame wavelets painless. In this talk, I will first briefly review the recent development based on or motivated by the unitary extension principle. Then I will focus on a few applications that use the unitary extension principle to design tight frame based algorithms. In particular, the power of this principle is illustrated by showing how it is used in solving various problems in the area of the high resolution image reconstruction.




Non-slip Vs Slip: The Hydrodynamic Boundary Condition and the Moving Contact Line*

Ping Sheng
Department of Physics
Hong Kong University of Science & Technology
Clear Water Bay, Hong Kong
sheng@ust.hk

Abstract : Immiscible two-phase flow in the vicinity of the contact line (CL), where the fluid-fluid interface intersects the solid wall, is a classical problem that falls beyond the framework of conventional hydrodynamics. In particular, molecular dynamics (MD) studies have shown clear violation of the no-slip boundary condition. Numerous ad hoc models were proposed to resolve this incompatibility, but none can give realistic predictions in agreement with MD simulations. Consequently, a breakdown in the hydrodynamic description for the molecular-scale CL region has been suggested. We have uncovered the boundary condition governing the moving contact line, denoted the generalized Navier boundary condition (GNBC), and used this discovery to formulate a continuum hydrodynamics whose predictions are in remarkable quantitative agreement with the MD simulation results at the molecular level.

f Molecular Scale Contact Line Hydrodynamics of Immiscible Flows, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. E68, 016306 (2003).




Multifractal Spectra of Branching Measure on a Galton-Watson Tree

Narn-Rueih Shieh
Department of Mathematics
National Taiwan University
Taipei 106, Taiwan
shiehnr@math.ntu.edu.tw

Abstract : If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < , then there is a branching measure m defined on the boundary of the tree. We establish three dimension spectra for the thin points and the thick points of the measure m. Two different thin spectra are shown respectively in the case P[Z = 1] > 0 and the case P[Z = 1] = 0. The thick spectrum is shown is the case either 0 < sup{t:E[exp(tZ)] < } < or esssup Z < . These are joints works with S.J. Taylor and P. Mörters, J. Appl. Probab.(2002, 2004) and Stat. Probab. Lett.(2002).




Mathematics, Mathematics Education, and the Mouse

Man-Keung Siu
Department of Mathematics
University of Hong Kong
Pokfulam Road, Hong Kong
mathsiu@hkucc.hku.hk

Abstract : The speaker will offer some reflections on the learning and teaching of mathematics in the age of information technology, mainly from the perspective of a mathematics teacher.




Some Results of Moduli Spaces of Vector Bundles on Curves

Xiaotao Sun
Institute of Mathematics
Chinese Academy of Sciences
Beijing 100080, P. R. of China
xsun@mail.math.ac.cn

Abstract : Let C be a smooth projective curve of genus g 2 and L a line bundle of degree d on C. Let UC(r,d) (resp. S UC(r,L)) be the moduli space of semistable vector bundles of rank r and degree d (resp. with fixed determinant L). In this talk, I will survey some results about the moduli spaces that we have achieved in recent years. I will address the following aspects

  1. Rational curves in S UC(r,L) and applications.
  2. Factorizations of generalized theta functions on UC(r,d).
  3. Degeneration of the moduli spaces when C degenerates.
  4. Differential operators on moduli spaces and Hitchin's connection.




Spectral Methods for Partial Differential Equations in Unbounded Domains

Tao Tang
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong
ttang@hkbu.edu.hk

Abstract : In this talk, we will review some recent progress on spectral methods for partial differential equations in unbounded domains. In particular, we will consider Hermite and Laguerre sepctral methods which are natural choice for a class of problems in unbounded domains. The importance of using appropriate scaling factors will be demonstrated. Applications of the Hermite and Laguerre sepctral methods to some practical problems, including the Fokker-Planck equations and Bose-Einstein condensates, will be discussed.




Generalized Poincare Inequality and Topology of Manifolds

Jiaping Wang
School of Mathematics
University of Minnesota
127 Vincent Hall, 206 Church St.
S.E. Minneapolis, MN55455, U.S.A
jiaping@math.umn.edu

Abstract : In this joint work with Peter Li, we consider a version of generalized Poicare inequality on complete manifolds and derive some sharp estimates for a class of harmonic functions. As applications, we prove splitting theorems under suitable Ricci curvature assumptions. Our results bring the Cheeger-Gromoll splitting theorem for manifolds with nonnegative Ricci curvature and our recent splitting result concerning manifolds with maximum bottom spectrum into a unified framework.




The Mean Curvature Flow of Lagrangian Submanifolds

Mu-Tao Wang
Department of Mathematics
Columbia University
4406 New York, NY 10027, U.S.A
mtwang@math.columbia.edu

Abstract : The mean curvature flow is the heat equation of submanifolds. A submanifold evolves in order to decrease its area as fast as possible along this process and the stationary phase corresponds to minimal submanifolds. A distinguished class of minimal submanifolds of Calabi-Yau manifolds are called special Lagrangians. Several important conjectures on Calabi-Yau manifolds demand deep understanding of the structure of special Lagrangians. However, so far there is no general procedure of constructing special Lagrangians. We propose to deform a Lagrangian submanifold to a special one by the mean curvature flow. The flow may develop singularities along this process. We shall discuss regularity and global existence results in this talk.




Superpotentials and the Cohomogeneity One Einstein Equations

McKenzie Yuen-Kong Wang
Department of Mathematics and Statistics
McMaster University
Hamilton, Ontario L8S 4K1, Canada
wang@univmail.cis.mcmaster.ca

Abstract : A Riemannian manifold (M, g) is of cohomogeneity one if a compact Lie group acts on it by isometries such that the generic orbits have codimension one. In this situation, the Einstein condition can be formulated as a Hamiltonian system with constraint \sf H = 0, where \sf H is the Hamiltonian. A superpotential is a globally defined function on configuration space satisfying the first order PDE
\sf H(duq, q) = 0.
Such a superpotential naturally gives rise to a first order subsystem of the Einstein system. Many physicists have used superpotentials to study cohomogeneity one metrics with exceptional holonomy. In this talk I will report on joint work with Andrew Dancer (Oxford University) trying to classify which principal orbits G/K admit a superpotential for the associated cohomogeneity one Ricci-flat system.




On Piece Algebraic Variety

Ren-Hong Wang
Institute of Mathematical Sciences
Dalian University of Technology
Dalian 116024, P.R. China
renhong@dlut.edu.cn

Abstract : It is well known that a most important topic in the algebraic geometry is the algebraic variety. The so-called ``piecewise algebraic variety" is defined as a set of common zeros of the multivariate splines. Because of the multivariate spline is a piece- wise polynomial, moreover its properties depend on both topological and geometric properties of the partition for the given domain. Therefore the piecewise algebraic variety has a very complicated construction. In fact, a lot of results and methods on the classical algebraic geometry can not be used to study the piecewise algebraic variety. The purpose of this paper is to introduce some researches concerning the piecewise algebraic variety.




Numerical Study of Self-focusing
of Super-Gaussian Laser Beam

Xiaoping Wang
Department of Mathematics
Hong Kong University of Science and Technology
Clear Water Bay, Kowloon Hong Kong
mawang@ust.hk

Abstract : In this talk, we study the super-Gaussian laser beam propagation based on the nonlinear Schrodinger equation. A new type of singular solutions are discovered numerically, which has ring structure. This ring structure is unstable with respect to 2d perturbations, leading to filamentation of laser beam.




Algebraic Properties in Wavelet Analysis

Yang Wang
Department of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160, U.S.A
wang@math.gatech.edu

Abstract : Compactly supported wavelets are constructed using refinable functions. Since the seminal work of Ingrid Daubechies on wavelets, refinable functions have been studied extensively. Most of these studies focus on the anlytical properties of refinable functions with integer dilation via their Fourier transforms. An interesting but virtually neglected area is the role of algebraic properties of the dilations on the analytical properties of refinable functions. A classical result in this direction was established by P. Erdös in 1939, who proved that if the dilation of a refinable function is a Pisot number then the function is not in L1. Kahane in 1971 proved a analogous result for dilations that are Salem numbers. In this talk we will survey results in this area, and state some new results and problems.




Geometry of Polynomials

Yuefei Wang
Institute of Mathematics, AMSS
Chinese Academy of Sciences
Beijing 100080, China
wangyf@math.ac.cn

Abstract : We will talk about recent results related to the geometry of complex polynomials, mainly on Smale's mean value conjecture. A weak form of Smale's conjecture is proved. We also compare the problem with Sendov's conjecture.




Bubbling Solutions for Nonlinear Elliptic Equations with Critical Nonlinearity

Juncheng Wei
Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong
wei@math.cuhk.edu.hk

Abstract : We consider the following nonlinear elliptic equations
Du-mu + u[(N+2)/( N-2)] = 0,  u > 0  in  W, u
n
= 0  on  W
where W is a bounded smooth domain in RN. Two results will be presented: first we consider the case of N = 4,5,6 and m << 1. (Joint work with O. Rey.) We show that for any positive integer K, there exists a solution with K interior bubbles. No condition on the domain topology, nor domain geometry, nor symmetry is needed. Next we consider the case of N 7 and m >> 1. (Joint work C.S. Lin.) Under some nondegeneracy condition, we prove that for any positive integer K, there exists a bubbling solution with K bubbles concentrating at the same point which has positive mean curvature and is a local minimum point of the mean curvature function.




Geometry of Whitney Set

Zhi-Ying Wen
Department of Mathematics
Tsinghua University
Beijing, P. R. China
wenzy@mail.tsinghua.edu.cn

Abstract : (A joint work with XILi-Feng) In the paper, the complete geometric characteristics, including decomposition theorem and compression theorem, are obtained for a connected and compact set to be a critical set in the sense of Whitney's, which is a set such that there exists a differentiable function critical and not constant on it. The problem, how to characterize these stes geometrically, was posed by H. Whitney [W1] in 1935. We also provide a complete geometrical characteristics for monotone Whitney arc, i.e., the above differentiable function is also increasing along the arc. This monotone class includes all the examples appearing in the literature, for example, the examples of Whitney's [W1] and Becovitch's [BS], Norton's t-quasi arc with Hausdorff dimension < t [N2], and self-similar arc researched by Wen and Xi [WX]. Furthermore, by the homogeneous Moran arc defined in this paper, we can completely characterize all the monotone Whitney arc of criticality >1, including t-quasi arc and self-conformal arc. Some applications to arcs which are attractors of Iterated function System are discussed, including the self-conformal arc, self-similar arc and self-affine arc. Finally, we obtain an example of critical arc such that any subarc of it fails to be a t-quasi-arc for any t, providing a positive answer to an open question of Norton's.




Higgs Rigidity of Holomorphic Bundles over Kahler Surfaces

Bun Wong
Department of Mathematics
University of California at Riverside
Riverside, CA 92521, U.S.A
wong@math.ucr.edt

Abstract : In this talk we will discuss some local rigidity phenomena of certain holomorphic bundles associated with the tangent bundle on a compact complex twofold with constant negative holomorphic sectional curvature. This is a joint work with Dr. Wing Sum Cheung at the University of Hong Kong partially supported by the HK RGC grant HKU 7130/00P.




Disjointness Preserving Operators of C*-algebras

Ngai-Ching Wong
Department of Applied Mathematics
National Sun Yat-sen University
Kaohsiung 80424, Taiwan
wong@math.nsysu.edu.tw

Abstract : In this talk, we will discuss the structure of disjointness preserving operators between C*-algebras. In particular, they preserve Jordan triple products {a,b,c} = (ab*c + cb*a)/2.




On Complex Hyperbolic Manifolds

Pit-Mann Wong
Department of Mathematics
University of Notre Dame
Notre Dame, Indiana 46556, U.S.A
pmwong@nd.edu

Abstract : The theory of complex hyperbolic manifolds in (complex) dimension 2 or higher was first investigated systematically by S. Kobayashi in 1967 when he introduced the famous intrinsic pseudo-distance, now known as the Kobayashi pseudo-distance. The infinitesimal version of this, known as the Kobayashi/Royden pseudo-metric, was first formulated by H. Royden in 1971. A complex manifold M is said to be Kobayashi hyperbolic if the Kobayashi pseudo-distance is indeed a distance. Equivalently, M is Kobayashi hyperbolic if the Kobayashi/Royden pseudo-metric is a metric. The Kobayashi/Royden metric is in general not a Hermitian metric but a Finsler metric. A geometrically simple concept of hyperbolicity was introduced by R. Brody in 1978: a complex manifold M is Brody hyperbolic if there is no non-constant holomorphic map f : C M. It is known that the two concepts of hyperbolicity are equivalent if M is compact. For non-compact manifolds it is known that Kobayashi hyperbolicity implies Brody hyperbolicity but the converse is not true in general. There is now a vast literature in the theory of complex hyperbolic manifolds, the readers are encouraged to consult the monograph by Kobayashi on this subject. In this talk I shall present two results in the case of complex surfaces (complex dimension 2). These results are joint works with K. Chandler.

The first result is the compact case.

Theorem 1. A generic hypersurface of degree d 5 in P3 is Kobayashi hyperbolic.

The second result is the non-compact case.

Theorem 2. For a generic curve C of degree d 5 in P2 the complement P2 \C is Kobayashi hyperbolic.




On Developing a Nonlinear Recursive Predictor

Yau-Shu Wong
Department of Math & Stat Sciences
University of Alberta
Edmonton, Alberta, Canada, T6G 2G1
yaushu.wong@ualberta.ca

Abstract : Developing techniques for long-term predictions has been an interesting and active topic for centuries. The ability to perform a reliable prediction is important in a wide range of disciplines ranging from science, engineering, medical research and econometric studies. Consider in the design and control of a vibration system, if one could predict that the amplitude of the vibrations is going to increase without bound, certain control laws could be applied to suppress the undesirable oscillations in order to ensure a stable system. Clearly, a poor prediction can lead to disastrous results.

Traditionally, statistical models have frequently been applied for time series forecasting.Several time series techniques, such as autoregressive(AR), moving average (MA), combined AR and MA (ARMA) and autoregressive integrated moving average (ARIMA) models, have been developed. However, the major shortcoming of these models is the assumption that the time series are generated from linear process. Nonlinear statistical approach such as threshold model, exponential model, polynomial and bilinear models have been proposed in the literature to provide accurate predictions over a wider range of applications. For a practical problem and without a priori knowledge about the time series under consideration, it is a very difficult task to select an appropriate nonlinear model. Moreover, the capability of time series models has been showed to be limited to short-term predictions.

In recent years, considerable progress has been reported in the field of artificial neural networks (ANNs). The development of ANNs is originally inspired by the study of biological neural systems, in particularly, research in the human brain. An ANN can be regarded as an information processing system which is capable to model highly nonlinear and complex systems. In this presentation, we present a novel approach to develop a nonlinear recursive predictor based on ANNs. Using limited time series data as input, our goal is to develop a predictor which is capable to provide a reliable long-term forecasting. A combination of network architecture, training algorithm, special procedures for scaling the weight coefficients and initialization is proposed. For time series arising from nonlinear dynamical systems, the performance and effectiveness of the proposed method is discussed, and results on testing to data sets obtained from numerical simulation and actual experiments arising from practical engineering problems will also be reported.




The Joint Distribution of Three Important Actuarial Diagnostics

Rong Wu
School of Mathematics, Nankai University
Tianjin 300071, China
rongwu@nankai.edu.cn

Abstract : Starting with the classical risk model with constant interest rate, we are to introduce a method of deriving the explicit expression for the joint distribution of the following three important actuarial diagnostics: the time of ruin,the surplus immediately before ruin and the deficit at ruin. Some recursive formulae are obtained, which can be used to calculate the joint distribution and the Gerber-Shiu's expected discounted penalty function at ruin for the risk model. The above method has been used in some risk model with stochastic interest and renewal models.




Generalized B- Spline f

ZongMin Wu*

Shanghai Key Lab. of Contemporary Applied Mathematics
Department of Mathematics
Fudan University
Shanghai 200433, P.R. China
zmwu@fudan.edu.cn

Abstract : A well systematically theory of the generalized spline will be established in this talk.

The approach is try to find a suitable function space simply for curve representation, recognition, reproduction and approximation.

The most common function, which we prefer using is the polynomial. We can use interpolation or the least square approximation to simulate a real exist curve (parametric or non parametric type). However we know that the polynomial interpolation possesses Lunge's phenomena and require to solve a large scaled linear system of equation on the other hand side the least square approximation or the Bernstein's approximation possesses only very lower convergence order. Therefore the function space of polynomial is not very efficient for the curve simulation.

The spline, which is constructed by piecewise polynomial (more generally the the piecewise rational polynomial- the NURBS) is now the most favorite basis both for Mathematician and Engineers, which can be found by most of standard computer softwares.

Simplest case of the spline is the piecewise linear function, which is already used to construct the quadrature form for the numerical integration and the the finite elements for numerical solution of partial differential equation.

To the spline theory, the B- spline the basic (best) spline basis should be mentioned. There are a lot of advantages of using the B- spline: such as the approximations capacity, the local control property (compactly supported), the shape preserving property, easy to evaluation (recursive evaluations scheme), multi resolutions property (refinable, subdivisions algorithm, wavelets). The spline possesses a disadvantage that it does not cover the arc of circle, which is the most common curve in engineering (draw by rule and compass). One can use rational spline to represent conic function, however difficult to construct an algorithm to keep the conic reproducing property.

Back to the history, the spline is at first only defined for odd degree that minimizes the bend energy |Dks(x)|2dx (D denote the differential operator) subjects to some interpolatory condition. We find that the solutions are piecewise polynomials of degree 2k-1 (order 2k). Later the spline is generalized to the piecewise polynomials of any order, or satisfying Dns(x) = 0 piecewisely.

Now we define the generalized spline to be the function minimizes the generalized energy |Q(D)s(x)|2dx subjects to some interpolatory condition, moreover we generalize the approach that the generalized spline of order n is defined to be the function satisfying P(D)s(x) piecewisely, where P(l) is a polynomial of degree n. We assume P(D) = Q(D)*Q(-D).

The existence of the B- spline, the uniqueness of the B- spline, the recursive computations formula, the Bezier form, the shape preserving properties, the Schoeberg Whitnay's theory, the refinable as well as the wavelets constructed by such generalized spline function are discussed.

fSupported by NSFC Project No. 19971017 and 10125102




Quiver Representations, Hall Algebras and Quantum Groups

Jie Xiao
Department of Mathematical Sciences
Tsingua University
100084 Beijing, P.R.China
jxiao@math.tsinghua.edu.cn

Abstract : In my talk, I will outline a progress on the interaction of representation theory of algebras and infinite dimensional Lie theory. This new development will be demonstrated in the following examples: (1) Representations of the Kronecker quivers, (2) Coherent sheaves over the projective lines, (3) Affine Kac-Moody algebras of type A1(1), (4) Quantum groups of type A1(1), and (5) Affine canonical bases of type A1(1). Finally I will introduce an algebraic method to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras, by using the Ringel-Hall algebras of the representations of tame quivers.




Ear Modeling and Auditory Transforms

Jack Xin
Department of Mathematics
University of Texas at Austin
University Station C1200 Austin, TX 78712-0257, U.S.A.
jxin@math.utexas.edu

Abstract : A class of nonlinear nonlocal dispersive PDEs are developed to model the nonlinear vibration patterns in the inner ear as reponse to multi-frequency sound input. Nonlinear interaction at different frequencies reveal sound masking phenomenon essential in sound compression and processing. Structure of PDE solutions in the linear regime motivates an invertible transform with built-in auditory filter characteristics. Properties and examples of discrete versions of such auditory transforms will be shown.




Algebraic Curves over Finite Fields and Applications

Chaoping Xing
Department of Mathematics
National University of Singapore
10 Kent Ridge Crescent, Singapore 119260, Singapore
matxcp@nus.edu.sg

Abstract : Algebraic curves over finite fields with many points have found some interesting applications in the past two decades. In the first part of the talk, we introduce the problem and report the recent development on this topic. The applications to coding theory and quasi-Monte Carlo methods are presented.




Continuous-Time Mean-Risk Portfolio Selection

Jia-An Yan
Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing, China
jayan@mail.amt.ac.cn

Abstract : This paper is concerned with continuous-time portfolio selection models in a complete market where the objective is to minimize the risk subject to a prescribed expected payoff at the terminal time. The risk is measured by the expectation of a certain function of the deviation of the terminal payoff from its mean. First of all, a model where the risk has different weights on the upside and downside variance is solved explicitly. The limit of this weighted mean-variance problem, as the weight on the upside variance goes to zero, is the mean-semivariance model which is shown to admit no optimal solution. This negative result is further generalized to a mean-downside-risk portfolio selection problem where the risk has non-zero value only when the terminal payoff is lower than its mean. Finally, a general model is investigated where the risk function is convex. Sufficient and necessary conditions for the existence of optimal portfolios are given. Moreover, optimal portfolios are obtained when they do exist. The solution is based on completely solving certain static, constrained optimization problems of random variables. This a joint work with Hanqing Jin and Xun Yu Zhou, The Chinese University of Hong Kong.




Contact Discontinuity with General Perturbation for Gas Motion

Tong Yang
Department of Mathematics
City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
matyang@cityu.edu.hk

Abstract

In this talk, we will present our recent results on the stability of contact dicontinuity for gas motion with general perturbation. Our results apply to the Navier-Stokes equations and the Boltzmann equation and thus give a satisfatory approach to this problem. This is a joint work with Feimin Huang and Zhouping Xin.




Dynamics of Bose-Einstein Condensate

Horng-Tzer Yau
Department of Mathematics
Stanford University
Stanford, CA 94305-2125, U.S.A
yau@math.stanford.edu

Abstract : Consider a system of N bosons on the three dimensional unit torus interacting via a pair potential N2 V(N(xi-xj)), where x = (x1, ,xN) denotes the positions of the particles. Suppose that the initial data yN,0 satisfies the condition (yN,0,HN2 yN,0) C N2, where HN is the Hamiltonian of the Bose system. Let yN,t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k 1 } solves the GP hierarchy. We prove that as N the limit points of the k-particle density matrices of yN,t are solutions of the GP hierarchy. The uniqueness of the solutions to this hierarchy remains an open question.




Deformation of CR Manifolds and Deformation of Isolated Singularities

Stephen S.T. Yau
Department of Mathematics, Statistics & Computer Science
University of Illinois at Chicago
851S. Morgan Street, Chicago, IL 60607-7045, U.S.A.
yau@uic.edu

Abstract : Recently we have introduced a new nonnegative Bergman function for any strongly pseudoconvex complex manifold. This Bergman function is invariant under biholomorphic maps and vanishes precisely on the exceptional set of the strongly pseudoconvex complex manifold. It can be used to study the deformation of strongly pseudoconvex CR manifolds within an variety. In the joint work with H.S. Luk and X.J. Huang, we study the simultaneous embedding and filling problems for a CR family of CR manifolds. As a corollary, we showed that Buchweitz-Milson-Miyajima theorem is true even for singularities with dimension 3. Thus if (V,0) is a normal isolated singularity with dimension and depth at least 3, then the Kuranishi family of the link of (V,0) is realized as a real hypersurface of the versal family of deformation of (V,0). Thus the deformation of strongly pseudoconvex CR manifolds inside V will produce an analytic trivial family inside the versal deformation of (V,0).




Steady Flows and Transonic Shocks

Huicheng Yin
Department of Mathematics
Nanjing University
Nanjing 210093, P.R. China
huicheng@nju.edu.cn

Abstract : In this talk, I will discuss some recent progresses in the theory of multi-dimensional systems of nonlinear conservation laws. In particular, I will present the recent results on the existence and stability of transonic shocks for the steady flow through a general multi-dimensional nozzle with variable sections or past a curved wedge under the appropriate pressures. These are free boundary value problems for a quasilinear mixed type equation. Our results answer some questions of Courant-Friedrichs on the transonic phenomena in a nozzle or past a wedge. Other related problems including the subsonic steady Euler equations will be discussed too. These works are joint with Prof. Xin Zhouping.




Statistical Issues in Educational and Psychological Measurements

Zhiliang Ying
Department of Statistics
Columbia University
1255 Amsterdam Avenue
New York, NY 10027, U.S.A.
zying@stat.columbia.edu

Abstract : This talk covers some statistical models and methods for educational and psychological tests. Emphasis will be given to adaptive tests in which sequential analysis and computer technology play important and interactive roles. Related challenging issues in statistical design, modelling, as well as incorporation of computer technology will be illustrated through examples.




Backward Stochastic Integral Equations and Some Related Problems

Jiongmin Yong
Department of Mathematics
Fudan University/University of Central Florida
Orlando, Florida 32816, U.S.A
jyong@mail.ucf.edu

Abstract : Forward stochastic differential equations have an equivalent integral form. This leads to general stochastic integral equations naturally. Likewsie, backward stochastic differential equations also have an equivalent integral form. What is the general form of backward stochastic integral equations? In this talk, we will explore that and establish a general theory for backward stochastic integral equations. Moreover, it turns out that backward stochastic integral equations have some interesting relations to the Pontryagin's maximum principle for optimal control of stochastic integral equations, time-inconsistent preferences, and (forward-backward) stochastic differential utility theory.




Recent Mathematical Development on Boltzmann Equation

Shih-Hsien Yu
Department of Mathematics
City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
mashyu@cityu.edu.hk

Abstract : In this talk we will review recent developments of mathematical theories for Boltzmann equation. In particular, we are interested in linear and nonlinear wave propagation of Boltzmann equation related to those in fluid mechanics. A particle-like and wave-like dual structure in solution of Boltzmann equation is an important key component for the recent development. Its application to construct the Green function will be mentioned also.




Stepsizes for the Steepest Descent Direction Method

Ya-xiang Yuan
Institute of Computational Mathematical
and
Scientific/Engineering Computing
AMSS, Chinese Academy of Sciences
Beijing 100080, China
yyx@lsec.cc.ac.cn

Abstract : The steepest descent method is the simplest method for minimization that use gradients. It is well know that the steepest descent method with the ``best'' stepsize in the sense of reducing the objective function, namely the exact line search converges only linearly and would lead to zig-zag, giving very slow convergence, particularly when the function is ill-conditioned. However, a superising result given by Barzilai and Borwein indicates that a specific stepsize would ensure the steepest descent method converging superlinearly for two dimensional problems. This talk will present recent advances on different choices for the stepsize of the steepest descent method, in order to improve the Barilai and Borwein method.




The Mathematical Problem of Waves and Oscillations in Rotating Planets and Stars

Keke Zhang
Department of Mathematical Sciences
University of Exeter, EX4 4QE
Devon, UK
kzhang@ex.ac.uk

Abstract : The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal planets and stars was first formulated and studied by Poincaré (1885). We report the first explicit general analytical solution of this classical problem in a rotating oblate spheroid of arbitrary eccentricity. The explicit general solution of the Poincaré equation, given by a new polynomial in spheroidal polar coordinates, represents a possibly complete set of the inertial modes.

The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity.

On the basis of the general explicit solution, some unusual and intriguing properties of the the Poincaré equation are discovered. In particular, we are able to prove that




V 
( u ·2 u )   dV 0,
where u is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and V denotes three-dimensional integration over the volume of the spheroidal cavity.




The Ca Regularity of an Ultraparabolic Equations

Liqun Zhang
Institute of Mathematics
Chinese Academy of Sciences
Beijing 100080, China
lqzhang@mail.math.ac.cn

Abstract : In this talk, I will discuss the regularity of ultraparabolic equations which from the study of boundary layer theory. We have proved the existence of weak solutions which is global in time. Then we try to prove that the solution is actually a classical solution. The key point is to obtain the Holder estimates. This is a jointed work with Xin Zhouping and Zhao Junning.




High Frequency Limit of the Helmholtz Equation with Variable Refraction Index

Xue Ping Wang
Département de Mathématiques
Laboratoire Jean Leray, UMR 6629 du CNRS
Université de Nantes F-44322 Nantes Cedex 3, France
wang@math.univ-nantes.fr

Ping Zhang*
Academy of Mathematics & Systems Science, CA
Beijing 100080, China
zp@mail.amss.ac.cn

Abstract

In the propagative regime, we study the high frequency limit of the Helmhlotz equation with variable refraction index and a source term concentrated near a p-dimensional affine subspace. Under some conditions, we first derive unform estimates in Besov spaces for the solutions. Then, we prove that the Wigner measure associated with the solutions satisfies the stationary Liouville equation with an explicit source term and has certain radiation property at infinity.




The Effects of Information Accuracy in a Financial Market

Qiang Zhang
Department of Mathematics
City University of Hong Kong
Tat Chee Avenue, Kowloon Tong, Hong Kong
mazq@cpmaspc07.cityu.edu.hk

Abstract : Information is an important driving force of financial markets. Usually information has uncertainty. We consider a financial market that consists of insiders, random traders and market makers (Kyle's model of insider trading). Each insider possesses private information of different accuracy and tries to choose a trading strategy that maximizes his profit from his private information. We examine how the information accuracy will affect the trading strategy and profit allocation. In particular, we study the interaction between the first-mover advantage and the information asymmetry in Stackelberg duopoly and show that in most cases the second-mover will gain more profit than the first-mover. Only when the first-mover's information is quite accurate and the second-mover's is much more accurate, the first-mover will outperform the second-mover. We also compare the outcomes from Stackelberg duopoly with those from Cournot duopoly and show that the first-mover's (second-mover's) profit in a Stackelberg duopoly can be smaller (larger) than his profit in Cournot duopoly. All these results are in a sharp contrast to those from classical models in which all quantities are deterministic.




The Positive Mass Theorem Near Null Infinity

Xiao Zhang
Institute of Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100080 P.R. China
xzhang@mail.amss.ac.cn

Abstract : We prove a positive mass theorem near null infinity in asymptotically flat spacetimes. We use it to study the complete and rigorous proof on the positivity of the Bondi mass in gravitational radiation.




Statistics of Diffusion

Weian Zheng
Department of Mathematics
University of California, Irvine
Irvine, CA 92697-3875, U.S.A.
wzheng@uci.edu

Abstract : Given a realized path of a diffusion process with unknown parameter in its coefficients, we define the maximum likelihood estimator of the unknown and show the rate of convergence. Our condition is much more general than the previous known results.




Complexifications of Invariant Domains

Xiangyu Zhou
Department of Mathematics
Chinese Academy of Sciences
Beijing 100080, China
xyzhou@math.ac.cn

Abstract : In this talk, we'll consider the Steiness of the complexifications of Stein domains with real group actions. We'll outline two methods to approach the problem. One is based on the minimum principle due to Kiselman and Loeb, another is based on the invariant version of Cartan's lemma, here L2 method could play an important role.




Inexact Preconditioned Uzawa Algorithms and Domain Decomposition Methods for Three-dimensional Maxwell's Equations

Jun Zou
Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong
zou@math.cuhk.edu.hk

Abstract : In this talk, we shall present a few inexact preconditioned Uzawa algorithms for solving the saddle-point systems arising from the discretization of Maxwell system and Navier-Stokes system. The motivation and convergence analysis will be discussed. Then we shall address how to combine these inexact Uzawa methods with our new nearly optimal non-overlapping domain decomposition methods for solving the three-dimensional Maxwell equations in non-homogeneous media. The work was supported by Hong Kong RGC grants (project CUHK4048/02P and project 403403).