Liouville Type Equations

Chiun-Chuan Chen
Department of Mathematics
National Taiwan University
Taipei 106, Taiwan

Abstract : Liouville type equations arise from problems in different fields. In this talk, we are going to discuss some recent development of these equations.

Arnold Diffusion in a Priori Hyperbolic Hamiltonian Systems

Chong-Qing Cheng*
Department of Mathematics
Nanjing University
Nanjing 210093, China

Jun Yan
Institute of Mathematics
Fudan University
Shanghai 200422, China

Abstract : We consider a priori hyperbolic and time-periodic Hamiltonian systems with arbitrary n+1 degrees of freedom. The Hamiltonian has the form

H(u,v,t) = h1(p)+h2(x,y)+P(u,v,t)
where u = (q,x), v = (p,y), (p,q) \Bbb R×\Bbb T, (x,y) \Bbb Tn×\Bbb Rn, P is a time-1-periodic small perturbation. H Cr (r 3) is assumed to satisfy the following hypothesis:

H1, h1+h2 is a convex function in v, i.e., Hessian matrix 2vv(h1+h2) is positive definite. It is finite everywhere and has superlinear growth in v, i.e., (h1+h2)/||v|| as ||v||.

H2, it is a priori hyperbolic in the sense that the Hamiltonian flow Fth2 determined by h2 has a non-degenerate hyperbolic fixed point (x,y) = (0,0), the function h2(x,0):\Bbb Tn\Bbb R attains its strict maximum at x = 0 mod 2p. We set h2(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., Ws(0,0) Wu(0,0). Such condition appears not natural when n > 1.

Let Be,K denote a ball in the function space Cr({(u,v,t) \Bbb Tn+1×\Bbb Rn+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 Se,K Be,K such that for each P Se,K there exists an orbit of the Hamiltonian flow which connects the region with p < A to the region with p > B.

Recent Development of Nonparametric Methods in Financial Econometrics

Jianqing Fan
Department of Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544, U.S.A

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, time-dependent 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 state-domain methods for volatility estimation. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.

Saddlepoint Approximations for Random Fields and their Applications

Tze Leung Lai
Department of Statistics
Stanford University
Stanford, CA 94305-4065, U.S.A

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 multi-dimensional indices, multivariate empirical processes, and scan statistics in change-point 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.

Period Integrals on Shimura Varietiesf

Jian-Shu Li
Department of Mathematics, HKUST
Clear Water Bay, Hong Kong
Zhejiang University

Abstract : Let SH 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

lH(f) =

Such integrals have been used to characterize the type of representations arising from f. For example it is used in the study of base change, where it is sufficient to know whether the period integral is identically zero.

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

|lH(f)|2 = L(f, s0)     (*)
(up to a constant). In fact with a suitable choice of H one hopes to obtain the value of the L-function at the center of symmetry. Such an expression can be viewed as a first step towards a ``Gross Zagier formula'' for S.

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, RGC-CERG grants HKUST6126/00P, HKUST6115/02P, and the Cheung Kong Scholars Programme

Kolmogorov Complexity and Its Applications

Ming Li
School of Computer Science
University of Waterloo
Waterloo, Ontario N2L 3G1, U.S.A.

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 average-case 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, Springer-Verlag, 1997, 2n Edition; and Charles B. Bennett, Ming Li, and Bin Ma, Chain letters and evolutionary history, Scientific American, June 2003, 76-81; Dana MacKenzie, On a roll, New Scientist, Nov. 6, 1999, 44-47.

Rigidity and Structure of Complete Manifolds

Peter Li
Department of Mathematics
University of California, Irvine
CA 92697-3875 U.S.A

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.

Classical and Complex Incompressible Fluids

Fanghua Lin
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012, U.S.A.

Abstract : The classical incompressible fluids are described by the (Incompressible) Euler and navier-Stokes 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 multi-scales 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.

From Rational Curves to Complex Structures on Fano Manifolds

Ngaiming Mok
Department of Mathematics
The University of Hong Kong
Pokfulam Road, Hong Kong

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 K-rational 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 X be a general point and Ux be the normalization of the Chow space of minimal rational curves marked at x. Then, there is a tangent map tx: Ux \Bbb PTx(X) which assigns to each element [C] Ux the tangent direction [Tx(C)] at the given marking at x, and the variety of minimal rational tangents Cx \Bbb PTx(X) is the strict transform of Ux under the rational map tx. The dimension dimCx for a general point x X agrees with the number p of direct summands of degree 1 in the Grothendieck decomposition of the tangent bundle TX over a standard minimal rational curve. We call p = p(K) the positivity index. Together with Jun-Muk 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 algebro-geometric 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 K-curves. 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 set-up and illustrate its use by high points of various applications. These include

  1. rigidity of rational homogenenous spaces G/P of Picard number 1 under Kähler deformation;

  2. local rigidity theorems for finite holomorphic maps onto Fano manifolds of Picard number 1;

  3. solutions to Lazarsfeld-type problems on rational homogeneous spaces;

  4. reconstruction of certain rational homogeneous spaces from their varieties of minimal rational tangents;

  5. bounds on dimensions and vanishing orders of holomorphic vector fields on uniruled projective manifolds; and

  6. Severi-type theorems for finite holomorphic maps onto Fano manifolds of Picard number 1 with positivity index 0.

Discontinuous Galerkin Methods for Convection Dominated Partial Differential Equations

Chi-Wang Shu
Division of Applied Mathematics
Brown University
Providence, Rhode Island 02912, USA

Abstract : Convection dominated partial differential equations include hyperbolic equations as well as convection dominated convection-diffusion equations, convection dominated convection-diffusion-dispersion 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.

The Role of [`()] Estimates in Algebraic and Complex Geometry

Yum-Tong Siu
Department of Mathematics
Harvard University
One Oxford Street Cambridge, MA 02138, U.S.A.

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.

On the Structure of Complete Kähler Manifolds with Nonnegative Curvature

Luen-Fai Tam
Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T., Hong Kong

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 Kähler-Ricci 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 Kähler-Ricci flow.

The Role of Chern Classes in Birational Geometry

Chin-Lung Wang
Department of Mathematics
National Central University
Chung-Li, Taiwan 32054

Abstract : A birational class of projective manifolds is naturally partial-ordered 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.

Singularity Behavior of the Mean Curvature Flow

Xu-Jia Wang
Centre for Mathematics and its Applications
Mathematical Sciences Institute
Australian National University
Canberra, ACT 0200, Australia

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 Huisken-Sinestrari and White that the blow-up 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 blow-down of any convex solution must be a shrinking sphere or cylinder. We also prove that, for convex solutions in the Euclidean 3-space (and true only in 3-space), 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 k-non-collapsing solutions of nonnegative sectional curvatures to the 3-dim Ricci flow. We will also mention some related open problems.

From Limit Cycles to Strange Attractors

Lai-Sang Young
Courant Institute of Mathematical Sciences
New York University
251 Mercer St., New York

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 rank-one attractors, one of the relatively few classes of strange attractors that are amenable to analysis.

Periods, Heights, and L-series

Shou-Wu Zhang
Department of Mathematics
Columbia University
2990 Broadway 509 Mathematics Building
4406 New York, NY 10027, U.S.A

Abstract : In this talk, I will survey some recent progress on the special value (or derivative) formula for L-series 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.

Localization and Duality

Jian Zhou
Department of Mathematical Science
Tsinghua University
Beijing China

Abstract : In a series of remarkable papers, Vafa and his various collaborators (Aganagic, Klemm, and Marino) gave physical calculations of Gromov-Witten invariants of some open Calabi-Yau threefolds based on duality with Chern-Simons 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 Chiu-Chu Melissa Liu and Kefeng Liu. We will also discuss the duality with 4D Yang-Mills theory and its consequences. Our main technical tool is localization techniques combined with combinatorial theory of partitions and symmetric functions.

Uniformization Theorems For Complete Non-compact Kaehler Surfaces

Xi-Ping Zhu
School of Mathematics and Computational Science
Zhongshan University
Guangzhou, 510275 China

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 \BbbR2n for n > 1. From the point of view of differential geometry, one consequence of the uniformization theorem is that a positively curved compact or non-compact 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 Siu-Yau independently. In this talk, we are thus interested in complete non-compact Kaehler manifolds with positive holomorphic bisectional curvature. The Yau conjecture, which states that a complete non-compact 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.

Modular Famlies of Projective Manifolds

Kang Zuo
Faculty of Mathematics & Computer Science
Johannes Gutenberg Universitat Mainz
Staudingerweg 9 D-55128 Mainz, Germany

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 Arakelov-Yau 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 non-rigid famliy of Calabi-Yau 3-folds.

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On 07 Dec 2004, 16:53.