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

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.

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.

^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.

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.

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.

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.

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.

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.

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.

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.

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^* ×GL_n-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.

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]<sup>BS</sup> 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.

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.

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.

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) = u^p±u^q 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.

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.

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.

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

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.)

Abstract : Let X = {X(t),t Î 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.

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.

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

Abstract : Let C be a smooth projective curve of genus g ³ 2 and L a line bundle of degree d on C. Let U_C(r,d) (resp. S U_C(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 U_C(r,L) and applications.
2. Factorizations of generalized theta functions on U_C(r,d).
3. Degeneration of the moduli spaces when C degenerates.
4. Differential operators on moduli spaces and Hitchin's connection.

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.

The first result is the compact case.

Theorem 1. A generic hypersurface of degree d ³ 5 in P³ is Kobayashi hyperbolic.

The second result is the non-compact case.

Theorem 2. For a generic curve C of degree d ³ 5 in P² the complement P² \C is Kobayashi hyperbolic.

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.

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 ò|D^ks(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 ò|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

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.

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.

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 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

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.