ChiunChuan Chen
Department of Mathematics
National Taiwan University
Taipei 106, Taiwan
chchchen@math.ntu.edu.tw
Abstract : Liouville type equations arise from problems in different fields. In this talk, we are going to discuss some recent development of these equations.
ChongQing Cheng^{*}
Department of Mathematics
Nanjing University
Nanjing 210093, China
chengcq@nju.edu.cn
Jun Yan
Institute of Mathematics
Fudan University
Shanghai 200422,
China
Abstract : We consider a priori hyperbolic and timeperiodic Hamiltonian systems with arbitrary n+1 degrees of freedom. The Hamiltonian has the form

H1, h_{1}+h_{2} is a convex function in v, i.e., Hessian matrix ¶^{2}_{vv}(h_{1}+h_{2}) is positive definite. It is finite everywhere and has superlinear growth in v, i.e., (h_{1}+h_{2})/v®¥ as v®¥.
H2, it is a priori hyperbolic in the sense that the Hamiltonian flow F^{t}_{h2} determined by h_{2} has a nondegenerate hyperbolic fixed point (x,y) = (0,0), the function h_{2}(x,0):\Bbb T^{n}®\Bbb R attains its strict maximum at x = 0 mod 2p. We set h_{2}(0,0) = 0.
Here, we do not assume the condition that the hyperbolic fixed point (x,y) = (0,0) is connected to itself by its stable manifold and unstable manifold, i.e., W^{s}(0,0) º W^{u}(0,0). Such condition appears not natural when n > 1.
Let B_{e,K} denote a ball in the function space C^{r}({(u,v,t) Î \Bbb T^{n+1}×\Bbb R^{n+1}×\Bbb T:v £ K}®\Bbb R), centered at the origin with radius of e. Our result is the following, which is a higher dimensional version of the theorem formulated by Arnold where it was assumed that n = 1.
Theorem 1.1 Let A < B be two arbitrarily given numbers and assume H satisfies the above two conditions. There exist a small number e > 0, a large number K > 0 and a residual set in S_{e,K} Ì B_{e,K} such that for each P Î S_{e,K} there exists an orbit of the Hamiltonian flow which connects the region with p < A to the region with p > B.
Jianqing Fan
Department of Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544, U.S.A
jqfan@Princeton.edu
Abstract : An overview is given on the recent development of nonparametric techniques that are useful for financial econometrics. The problems include estimation and inferences of instantaneous returns and volatility functions, timedependent stochastic models, estimation of transition densities and state price densities. We first briefly describe the problems and then outline main techniques and main results. In particular, we will discuss in detail the new developments on the generalized likelihood ratio tests for financial econometric models and on the dynamically integrating the time and statedomain methods for volatility estimation. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.
Tze Leung Lai
Department of Statistics
Stanford University
Stanford, CA
943054065, U.S.A
lait@stat.Stanford.edu
Abstract : A number of classical results on boundary crossing probabilities of Brownian motion and random walks are extended to certain classes of random fields, which include sums of independent random variables with multidimensional indices, multivariate empirical processes, and scan statistics in changepoint and signal detection as special cases. Key ingredients in these extensions are saddlepoint approximations for tail probabilities and geometric integration over tubular neighborhoods of extremal manifolds related to the large deviations theory of these fields.
JianShu Li
Department of Mathematics, HKUST
Clear Water Bay, Hong Kong
and
Zhejiang University
matom@ust.hk
Abstract : Let S_{H} Ì S be an inclusion of Shimura varieties. Suppose that f is a smooth square integrable function on S which is assumed to be an eigenfunction under the Laplace operator as well as all the Hecke operators. We are interested in period integrals of the form

The main objective is to express the period integral as a special value of the Lfunction attached to f, of the form

In this talk we will try to describe formulas of the form (*), obtained in joint work with S.W. Zhang, for Shimura varieties attached to classical groups, and for subvarieties of Hodge type arising from fairly general subgroups. We shall also try to describe applications to some equidistribution problems.
^{f} Supported in part by NNSFC Grant No. 19928103, RGCCERG grants HKUST6126/00P, HKUST6115/02P, and the Cheung Kong Scholars Programme
Ming Li
School of Computer Science
University of Waterloo
Waterloo, Ontario N2L 3G1, U.S.A.
mli@uwaterloo.ca
Abstract : Kolmogorov complexity has influenced the development of many fields during the last 4 decades. We will talk about its colorful history, its mathematical theory and its applications in computer science. In particular, we will demonstrate how to use Kolmogorov complexity to prove lower bounds and analyze averagecase complexity of algorithms. We will also use Kolgomorov complexity to develop a theory of shared information between two objects. This theory is then applied to compare documents, detect plagiarism, classify languages and music scores, construct phylogenetic histories from genomes, and control junk emails.
Some popular writings related to this talk can be found in ``An introduction to Kolmogorov complexity and its applications" by Ming Li and Paul Vitanyi, SpringerVerlag, 1997, 2n Edition; and Charles B. Bennett, Ming Li, and Bin Ma, Chain letters and evolutionary history, Scientific American, June 2003, 7681; Dana MacKenzie, On a roll, New Scientist, Nov. 6, 1999, 4447.
Peter Li
Department of Mathematics
University of California, Irvine
CA 926973875 U.S.A
pli@math.uci.edu
Abstract : In this talk, I will discuss some rigidity and structural results (joint work with Jiaping Wang) for a class of complete manifolds on which a weighted Poincar?inequality is valid. More specifically, if the Ricci curvature of such a manifold is bounded from below by a certain negative multiple of the weigh function given by the weighted Poincar?inequality, then the manifold must either be a specific warped product or that it has only one nonparabolic end. As a special case, this recovers our previous result on manifolds with positive spectrum. This generalizes a theorem of Witten and Yau concerning an outstanding issue in the AdS/CFT correspondence.
Fanghua Lin
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012, U.S.A.
linf@cims.nyu.edu
Abstract : The classical incompressible fluids are described by the (Incompressible) Euler and navierStokes equations.The lecture will start with a very brief look at some recent mathematical prograsses. The main goal of the lecture would to describe some most recent works concerning incompressible complex fluids(fluids with multiscales ettecfs).These fluids are regarded as complex because the marcoscopic (continum)fluid flows are nonlinearly coupled with evolutions of microscopic quantities(such as molecule dynamics and molecular structures,kinetic behavior and microlocal potential effects and theromal fluctuations).There are many challenging issues regarding such fluids from modellings,computations and analysis.
Ngaiming Mok
Department of Mathematics
The University of Hong Kong
Pokfulam Road, Hong Kong
nmok@hkucc.hku.hk
Abstract : Let X be a uniruled projective manifold, i.e., a projective manifold that can be filled up by rational curves. Fixing a polarization and minimizing degrees among free rational curves we have the notion of minimal rational curves. Let K be an irreducible component of the Chow space of minimal rational curves. A general minimal Krational curve is immersed, and the normal bundle has only direct summands of degrees 1 and 0. We call these standard minimal rational curves. Collecting the set of all tangents to minimal rational curves passing through a given general point we obtain the variety of minimal rational tangents. Let x Î X be a general point and U_{x} be the normalization of the Chow space of minimal rational curves marked at x. Then, there is a tangent map t_{x}: U_{x} ® \Bbb PT_{x}(X) which assigns to each element [C] Î U_{x} the tangent direction [T_{x}(C)] at the given marking at x, and the variety of minimal rational tangents C_{x} Ì \Bbb PT_{x}(X) is the strict transform of U_{x} under the rational map t_{x}. The dimension dimC_{x} for a general point x Î X agrees with the number p of direct summands of degree 1 in the Grothendieck decomposition of the tangent bundle T_{X} over a standard minimal rational curve. We call p = p(K) the positivity index. Together with JunMuk Hwang we have been studying uniruled projective manifolds in terms of the geometry of varieties of minimal rational tangents. We put the emphasis on the case of Fano manifolds X of Picard number 1, the hard nuts according to Miyaoka.
Our general philosophy is to recover complex structures and algebrogeometric properties of uniruled projective manifolds from their varieties of minimal rational tangents. When p > 0 we study the double fibration r: U ® K, m: U® X given by the universal family of Kcurves. Here we have interesting distributions in general defined on U and on K, and sometimes on X itself. We study these distributions in conjunction with the deformation theory of rational curves. Various integrability theorems and uniqueness results on tautological foliations lead to interesting geometric consequences. This study involves the local geometry of families of varieties of minimal rational tangents. By contrast, when p = 0 the local geometry is more difficult, and we have in its place a geometry dictated by global objects, notably the extended discriminantal divisor, defined by the singularities of the total space of varieties of minimal rational tangents p: C ®X.
In this lecture we will explain the basic geometric setup and illustrate its use by high points of various applications. These include
ChiWang Shu
Division of Applied Mathematics
Brown University
Providence, Rhode Island 02912, USA
shu@dam.brown.edu
Abstract : Convection dominated partial differential equations include hyperbolic equations as well as convection dominated convectiondiffusion equations, convection dominated convectiondiffusiondispersion equations, etc. Such partial differential equations appear often in physical and engineering applications. The solutions of such equations contain discontinuities or sharp gradient regions, making it difficult to design stable and accurate numerical methods to approximate them. The discontinuous Galerkin methods form a class of finite element methods for solving such problems, which have been gaining popularity both in computational mathematics and in physical and engineering applications over the past decade. The advantages of this method includes its strong nonlinear stability, high order accuracy, flexibility in adaptivity, and high parallel computing efficiency. In this talk we will first give an overview of the discontinuous Galerkin methods, and then present recent results on their development, analysis and applications.
YumTong Siu
Department of Mathematics
Harvard University
One Oxford Street Cambridge, MA 02138, U.S.A.
siu@math.harvard.edu
Abstract : We will discuss the regularity problem of the [`(¶)] equation for pseudoconvex domains and its relation with problems in algebraic and complex geometry. We will explore a number of open problems in algebraic and complex geometry which can be approached with the techniques of [`(¶)] estimates, explain the progress up to this point, and point out the remaining obstacles. We will also discuss the application of algebraic geometric methods to the regularity problem of the [`(¶)] equation.
LuenFai Tam
Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong
lftam@math.cuhk.edu.hk
Abstract : We shall discuss some results on the structure of complete Kähler manifolds with nonnegative sectional or holomorphic bisectional curvature. The results are obtained by making use of two evolution equations: the heat equation which is linear and the KählerRicci flow which is nonlinear. We shall discuss some structure results by considering solutions of the heat equation with plurisubharmonic initial data. We shall also discuss attempts to study a uniformization conjecture of Yau with the use of KählerRicci flow.
ChinLung Wang
Department of Mathematics
National Central University
ChungLi, Taiwan 32054
dragon@math.nthu.edu.tw
Abstract : A birational class of projective manifolds is naturally partialordered by the first Chern class. It is expected that this partial order gives rise to corresponding partial orders in various geometric constructions attached to the manifolds. We shall discuss recent development on these issues with emphasizes on the c_1 equivalent case. In particular we discuss the equivalence of Chow motives under ordinary flops and equivalence of quantum cohomology under simple ordinary flops. Chern clases play fundamental roles in our disuccion.
XuJia Wang
Centre for Mathematics and its Applications
Mathematical
Sciences Institute
Australian National University
Canberra, ACT 0200, Australia
X.J.Wang@maths.anu.edu.au
Abstract : We consider the motion of a compact hypersurface driven by its mean curvature. The flow will develop singularity in finite time. When the mean curvature of the hypersurface is nonnegative, it is proved by HuiskenSinestrari and White that the blowup of the mean curvature flow at singularity is convex.
To understand the singularity behavior of the mean curvature flow one needs to classify all such convex solutions. An open problem is whether a convex solution is a shrinking sphere or cylinder, or a rotationally symmetric translating solution. We give examples to show that this is not true in general, but prove that the parabolic blowdown of any convex solution must be a shrinking sphere or cylinder. We also prove that, for convex solutions in the Euclidean 3space (and true only in 3space), a translating solution must be rotationally symmetric, and any convex solution is locally a perturbation of a shrinking cylinder except a compact subset. The latter result is analogous to that by Perelman on the classification of ancient knoncollapsing solutions of nonnegative sectional curvatures to the 3dim Ricci flow. We will also mention some related open problems.
LaiSang Young
Courant Institute of Mathematical Sciences
New York University
251 Mercer St., New York
lsy@cims.nyu.edu
Abstract : I will discuss the emergence of chaotic behavior in the form horseshoes and strange attractors when external forcing is applied to very simple dynamical systems at periodic time intervals. Geometric and stochastic properties of the chaotic systems are discussed. These results are deduced from a theory of rankone attractors, one of the relatively few classes of strange attractors that are amenable to analysis.
ShouWu Zhang
Department of Mathematics
Columbia University
2990 Broadway 509 Mathematics Building
4406 New York, NY 10027, U.S.A
szhang@math.columbia.edu
Abstract : In this talk, I will survey some recent progress on the special value (or derivative) formula for Lseries in terms of period integrals (or heights) of cycles in algebraic varieties. I will also explain the applications of these formula on diophantine equations and equidistribution problems.
Jian Zhou
Department of Mathematical Science
Tsinghua University
Beijing China
jzhou@math.tsinghua.edu.cn
Abstract : In a series of remarkable papers, Vafa and his various collaborators (Aganagic, Klemm, and Marino) gave physical calculations of GromovWitten invariants of some open CalabiYau threefolds based on duality with ChernSimons theory. We will explain how to mathematically justify some of their calculations. We will use formulas for some Hodge integrals, first conjectured by Marino and Vafa, proved and generalized in joint works with ChiuChu Melissa Liu and Kefeng Liu. We will also discuss the duality with 4D YangMills theory and its consequences. Our main technical tool is localization techniques combined with combinatorial theory of partitions and symmetric functions.
XiPing Zhu
School of Mathematics and Computational Science
Zhongshan University
Guangzhou, 510275 China
stszxp@zsu.edu.cn
Abstract : One of the most beautiful results in complex analysis of one variable is the classical uniformization theorem of Riemann surfaces. Unfortunately, a direct analog of this result to higher dimensions does not exist. For example, there is a vast variety of biholomorphically distinct complex structures on \BbbR^{2n} for n > 1. From the point of view of differential geometry, one consequence of the uniformization theorem is that a positively curved compact or noncompact Riemann surface must be biholomorphic to the Riemann sphere or the complex line respectively. It is thus natural to ask whether there is similar characterization for higher dimensional complete Kaehler manifold with positive "curvature". That such a characterization exists in the case of compact Kaehler manifold is the famous Frankel conjecture which says that a compact Kaehler manifold of positive holomorphic bisectional curvature is biholomorphic to a complex projective space. This conjecture was completely solved by Mori, and SiuYau independently. In this talk, we are thus interested in complete noncompact Kaehler manifolds with positive holomorphic bisectional curvature. The Yau conjecture, which states that a complete noncompact Kaehler manifold of positive holomorphic bisectional curvature is biholomorphic to a complex Euclidean space, provides the main impetus. I will report some recent progresses on the Yau's conjecture.
Kang Zuo
Faculty of Mathematics & Computer Science
Johannes Gutenberg Universitat Mainz
Staudingerweg 9 D55128 Mainz, Germany
kzuo@mathematik.unimainz.de
Abstract : This is a joint programme with Eckart Viehweg. I shall talk on a geometric characterization of modular famlies of projective manifolds in terms of ArakelovYau type inequality, Higgs field, and Yukawa coupling. Yau's Uniformization Theorem plays an fundamental roll in the case when the base dimension is bigger than 1. I shall also mention that how a modular family can be arised from a nonrigid famliy of CalabiYau 3folds.