45Minut Invited Talks
The Ricci Flow on Kähler
Manifolds
HuaiDong 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 900951555, 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 NonOscillatory (ENO) schemes for
numerical shock capturing. ENOwavelet 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
DerChen 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 secondorder subelliptic
operators. This class contains degenerate elliptic and
hypoelliptic operators (such as the Grusin operator, the
BaouendiGoulaouic operator and the subLaplacian 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
MeiChu Chang
Department of Mathematics
University of California, Riverside
Riverside, CA 925210135, 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_{1} + a_{2}  a_{i} Î A }, 
 

{a_{1} a_{2}  a_{i} Î 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 n^{2} 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
FourthOrder Flow in Conformal Geometry^{f}
ShuCheng Chang
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan
scchang@math.nthu.edu.tw
Abstract : Let (M,[g_{0}]) be a
closed smooth Riemannian 3manifold with the fixed conformal
class [g_{0}]. In this talk, we first review the socalled
Bondimass type estimates for the Calabi flow on (M,[g_{0}]). With its applications, we show that the solution of the Qcurvature flow exists on M×[0,¥. Moreover, if we
assume that the Q_{0}curvature with respect to g_{0} is
positive, then
the solution converges smoothly to a metric of constant positive Qcurvature. As a consequence, we show that there exists a
positive constant Qcurvature metric in such a fixed conformal
class [g_{0}] with the Paneitz operator of the negative base
eigenvalue. Finally we will deal with the Qcurvature flow on a
CR 3manifold.
^{f} This program is proposed by S.T. Yau during his
visiting at the National Tsing Hua University at Hsinchu in
199192. The author would like to thank him for the constant
encouragement and inspirations during the work.
Normal Dilations
ManDuen 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
Semidiscrete, 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
ChoHo 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.
Nonstationary 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 nonstationary 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 tightframe
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 followup
investigation of that of orthonormal wavelets on the realline (by
introducing certain wavelet functions near the endpoints 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 byproduct (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 ChowMumford 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 biomembrane 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
AiHua Fan
Faculté de Mathématiques et d'Informatique
Université de Picardie Jules Verne
33, rue Saint Leu
80039 Amiens CEDEX 1, France
aihua.fan@upicardie.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.
Numbertheoretic Methods in
Experimental Designs
KaiTai 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 Lowdiscrepancy set in the sense of
numbertheoretic methods (or quasiMonte 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 KaiTai 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 numbertheoretic 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 Lfunctions 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
ChatYin Ho
Department of Mathematics
University of Florida
358 Little
Hall, PO Box 118105
Gainesville, FL 326118105, 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 quasisimple 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 HeringOstrom 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 NonBlowup 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 blowup 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
(nonflat) infinite dimensional spaces. They are equipped with a
distinguished probability measure (Wiener measure) determined by
the LaplaceBeltrami 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 LeviCivita 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 85721¡V0089, 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^{*} ×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 socalled
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
JingSong 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 Sarithmetic 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 Ltheory. There are also
assembly maps in algebraic Ktheory 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 Sarithmetic 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 BorelSerre 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
BorelSerre compactification [`X]^{BS} is needed. For an
Sarithmetic subgroup G of a semisimple linear algebraic
group G, the product of X together with some
BruhatTits 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
RongQing 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 LowDimensional Dynamical Systems
Yunping Jiang
Department of Mathematics
CUNY Graduate Center and Queens College
6530 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 lowdimensional
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 noncritical 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 Ulamvon Neumann transformations,
which are geometrically finite maps conjugating to the map f(x) = x^{2}+2 of [2, 2], will also mentioned. We will show that
the conjugacy between two Ulamvon 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 onedimensional maps, which
contains all geometrically finite maps. The theorem says that the
topological conjugacy between two such maps is piecewise C^{1}
if and only if it is differentiable at one general point with
bound, in addition, it is piecewise diffeomorphic if the
exponents at corresponding power law singularities are also the
same.
We used to call the space of Ulamvon Neumann transformations as
the boundary of hyperbolicity. I will explain the reason. That is,
I will show the deformation from hyperbolic systems to a
nonhyperbolic 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 EulerPiosson equations with applications to modulated
electron beams in Klystrons, and general symmetric hyperbolic
systems will be discussed.
Results and Open Problems on
SemiLinear Elliptic Equations
Man Kam Kwong
Lucent Technologies
kwong@nwsgpa.ih.lucent.com
Abstract : We survey some
conjectures and open problems in the study of semilinear elliptic
equations of the form Du+f(u) = 0 either in a bounded
domain or the entire R^{n} . 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.
On an Invariant
Complementation Property of the Group Von Neumann Algebra of a
Locally Compact
Anthony ToMing 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.
G_{2} Geometry
Conan NaiChung 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
G_{2}manifolds using vector cross product and octonion structure.
We will also discuss its relationships with mirror symmetry of
CalabiYau threefolds, Mtheory, SeibergWritten theory,
triality and so on.
The Implications of the Third
International Mathematics and Science Study for Mathematics
Curriculum Reform in Chinese Communities
Frederick KoonShing 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 withincountry
differences in achievement failed to explain acrosscountry
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 wholeclass interaction. Teachers talked most of the
time, while students were found solving procedural problems
unrelated to reallife 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 943052125, 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 Pseudoconvex Pseudohermitian CR
manifolds
SongYing Li
Department of Mathematics
University of California, Irvine
Irvine, CA 926973875, U.S.A.
sli@math.uci.edu
Abstract : In this talk, I present
some recent results on strictly pseudoconvex pseudohermitian CR
manifolds of hypersurface type. It includes some joint works with
D.C. Chang, and with H.K. Luk. It contains estimating eigenvalues
for subLaplacian, expression for Riemann zeta function of
subLaplacian, formula for the Webster Ricci curvatures and
characterizations of balls in C^{n}.
Symplectic Topology in Dimensions 4 and 6
TianJun 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
ChangShou Lin
Department of Mathematics
National Chung Cheng University
160 SanHsing, MinHsiung, ChiaYi 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
Du_{i} + 
N å
j = 1

a_{ij} r_{j} 
æ è


h_{j}e^{uj}

 1 
ö ø

= 0 in S 

for
i = 1, 2, ¼, N, where
(a_{ij}) = 
æ ç ç ç ç
ç ç ç è


 
ö ÷ ÷ ÷ ÷
÷ ÷ ÷ ø



is the Cartan Matrix, r_{j} Î R and h_{j} (x)
is a positive C^{1} function on S. The Toda System is a
natural generalization of the wellknow Liouvill equation. Here,
we want to talk about the existence of minimizers at critical
values of r_{j} 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
TaiChia 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
BoseEinstein condensates.
Patterns Generation and Spatial
Entropy in MultiDimensional Lattice Models
SongSun Lin^{*}
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
sslin@math.nctu.edu.tw
JungChao Ban
National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu 300, Taiwan
jcban@math.cts.nthu.edu.tw
YinHeng 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 multidimensional models.
Patterns may be realized as stationary solutions of infinite
dimensional lattice dynamical system on integer lattices
Z^{d}. We first review the problems and associated
transition matrices H_{n} on Z_{2×n}
which have been obtained earlier. To study spatial entropy
h = h(H_{2}), we need to identify all admissible patterns
on Z_{(m+1)×n}. By introducing a Reduction
operator R_{m}, H_{n+1}^{m} can be expressed
in terms of H_{n}^{m}. R_{m} can also be
used to given a lower bound of h. We then introduce trace
operator T_{m}, and show
h = limsup_{m®¥}[(logr(T_{m}))/ m], where
r(T_{m}) is the maximum eigenvalue of T_{m}. When
H_{2} is symmetric, [(logr(T_{2}m))/ 2m] are
upper bounds of h.
Knot
Adjacency and Classification of Knots
XiaoSong Lin
Department of Mathematics
University of California, Riverside
Riverside, CA 92521, U.S.A.
xl@math.ucr.edu
Abstract : Two knots in the 3space
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
3manifolds, Dehn surgery and the theory of geometric structures
of 3manifolds to answer the question of the extent to which
nonisotopic 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
Multiparameter Stable Process
Zhengyan Lin
Department of Mathematics
Zhejian University
Hangzhou 310028, China
zlin@zju.edu.cn
Abstract : Let X = {X(t),t Î R_{+}^{N}} be a ddimension stable process with Nparameter 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
HarveyMoore Problem
AiKo Liu
Department of Mathematics
University of California, Berkeley
Berkeley, CA 947203840 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 SeibergWitten Invariants. Based on the algebraic proof
of the universality theorem, we will discuss the enumeration
problem of immersed nodal curves on K3 fibered CalabiYau
threefolds and the resolution of HarveyMoore conjecture on the
GopakumarVafa numbers.
MarinoVafa Formula of Onepartition
Hodge Integrals
ChiuChu 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 onepartition Hodge
integrals based on duality between open GromovWitten theory and
ChernSimons 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 NavierStokes Dynamics
JianGuo Liu
Department of Mathematics
University of Maryland
College Park, MD 207424015, U.S.A.
jliu@math.umd.edu
Abstract : The pressure term has
always created difficulties in treating the NavierStokes
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 noslip boundary conditions, the NavierStokes
pressure can be determined in a such way that it is strictly
dominated by viscosity. As a consequence, in a general domain with
noslip boundary conditions, we can treat the NavierStokes
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 timedifferencing 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
LandauLifshitz 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 LandauLifshitz system of the ferromagnetic spin chain from a
mdimensional manifold M into the unit sphere S^{2} of R^{3}.
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
u_{k}® u weakly in W^{1,2}(M×R_{+}, S^{2}) and that
S^{t} is the blow up set for fixed t. In the present paper
we first prove that S^{t} is a H^{m2}rectifiable
set for almost all t Î R_{+}. And then we verify that
S^{t} moves by the quasimean curvature under some
assumptions, which is a new 2codimension 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
JiangHua 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 3manifolds
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 3manifold 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 nonnegative
curvature. The later means that one can realize each 3simplex in
the triangulation by a Euclidean 3simplex 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
YatSun 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 particularmodel 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 405060027, U.S.A.
shenz@ms.uky.edu
Abstract : In this talk we will
discuss a new approach to the boundary value problems with L^{p}
data, via L^{2} estimates. The approach may be used for elliptic
systems as well as higherorder 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.
Nonslip 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 twophase flow
in the vicinity of the contact line (CL), where the fluidfluid
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 noslip 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
molecularscale 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 GaltonWatson Tree
NarnRueih 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 GaltonWatson 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
ManKeung 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 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
 Rational curves in S U_{C}(r,L) and
applications.
 Factorizations of generalized theta functions
on U_{C}(r,d).
 Degeneration of the moduli spaces when
C degenerates.
 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 FokkerPlanck
equations and BoseEinstein 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 CheegerGromoll 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
MuTao 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 CalabiYau
manifolds are called special Lagrangians. Several important
conjectures on CalabiYau 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 YuenKong 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
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 Ricciflat
system.
On Piece Algebraic
Variety
RenHong 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 socalled ``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 Selffocusing
of
SuperGaussian 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
superGaussian 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 303320160, 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 L^{1}. 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
Dumu + u^{[(N+2)/( N2)]} = 0, u > 0 in W, 
¶u
¶n

= 0 on ¶W 

where W is a bounded smooth domain in R^{N}. 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 ³ 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
ZhiYing Wen
Department of Mathematics
Tsinghua University
Beijing, P. R. China
wenzy@mail.tsinghua.edu.cn
Abstract : (A joint work with
XILiFeng) 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 tquasi arc with Hausdorff dimension
< t [N2], and selfsimilar 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 tquasi arc and selfconformal arc.
Some applications to arcs which are attractors of Iterated
function System are discussed, including the selfconformal arc,
selfsimilar arc and selfaffine arc. Finally, we obtain an
example of critical arc such that any subarc of it fails to be a
tquasiarc 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
NgaiChing Wong
Department of Applied Mathematics
National Sun Yatsen 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
PitMann 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 pseudodistance, now known as the
Kobayashi pseudodistance. The infinitesimal version of this,
known as the Kobayashi/Royden pseudometric, was first formulated
by H. Royden in 1971. A complex manifold M is said to be
Kobayashi hyperbolic if the Kobayashi pseudodistance is indeed a
distance. Equivalently, M is Kobayashi hyperbolic if the
Kobayashi/Royden pseudometric 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 nonconstant holomorphic map f : C ® M. It
is known that the two concepts of hyperbolicity are equivalent if
M is compact. For noncompact 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 ³ 5 in P^{3} is Kobayashi hyperbolic.
The second result is the noncompact case.
Theorem 2. For a generic curve C of degree
d ³ 5 in P^{2} the complement P^{2} \C is
Kobayashi hyperbolic.
On Developing a Nonlinear Recursive Predictor
YauShu Wong
Department of Math & Stat Sciences
University of Alberta
Edmonton, Alberta, Canada, T6G 2G1
yaushu.wong@ualberta.ca
Abstract : Developing techniques for longterm
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 shortterm 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 longterm 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 GerberShiu'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 òD^{k}s(x)^{2}dx (D
denote the differential operator) subjects to some interpolatory
condition. We find that the solutions are piecewise polynomials of
degree 2k1 (order 2k). Later the spline is generalized to the
piecewise polynomials of any order, or satisfying D^{n}s(x) = 0
piecewisely.
Now we define the generalized spline to be the function minimizes
the generalized energy òQ(D)s(x)^{2}dx 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.
^{
f}Supported 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 KacMoody algebras of type
A_{1}^{(1)}, (4) Quantum groups of type A_{1}^{(1)}, and (5)
Affine canonical bases of type A_{1}^{(1)}. Finally I will
introduce an algebraic method to realize the canonical bases of
the quantized enveloping algebras of all symmetric affine
KacMoody Lie algebras, by using the RingelHall 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 787120257, 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 multifrequency
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 builtin 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 quasiMonte
Carlo methods are presented.
ContinuousTime MeanRisk Portfolio Selection
JiaAn Yan
Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing, China
jayan@mail.amt.ac.cn
Abstract : This paper is concerned
with continuoustime 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 meanvariance
problem, as the weight on the upside variance goes to zero, is the
meansemivariance model which is shown to admit no optimal
solution. This negative result is further generalized to a
meandownsiderisk portfolio selection problem where the risk has
nonzero 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 NavierStokes 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 BoseEinstein Condensate
HorngTzer Yau
Department of Mathematics
Stanford University
Stanford, CA 943052125, 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 N^{2} V(N(x_{i}x_{j})), where x = (x_{1}, ¼,x_{N}) denotes the positions of the particles. Suppose that the
initial data y_{N,0} satisfies the condition (y_{N,0},H_{N}^{2} y_{N,0}) £ C N^{2}, where H_{N} is the Hamiltonian of
the Bose system. Let y_{N,t} denote the solution to the
Schrödinger equation with Hamiltonian H_{N}. Gross and
Pitaevskii proposed to model the dynamics of such system by a
nonlinear Schrödinger equation, the GrossPitaevskii (GP)
equation. The GP hierarchy is an infinite BBGKY hierarchy of
equations so that if u_{t} solves the GP equation, then the family
of kparticle density matrices {Ä_{k} u_{t}, k ³ 1 }
solves the GP hierarchy. We prove that as N® ¥ the limit
points of the kparticle density matrices of y_{N,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 606077045, 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
BuchweitzMilsonMiyajima 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 multidimensional
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
multidimensional 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 CourantFriedrichs 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, timeinconsistent preferences, and
(forwardbackward) stochastic differential utility theory.
Recent Mathematical Development on
Boltzmann Equation
ShihHsien 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 particlelike and wavelike 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
Yaxiang 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
zigzag, giving very slow convergence, particularly when the
function is illconditioned. 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 boundarylayer flux provides the solvability condition
that is required to solve the higherorder 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

ó õ

V

( u ·Ñ^{2} u ) dV º 0, 

where u is the velocity of any threedimensional inviscid
inertial waves or oscillations in an oblate spheroid of arbitrary
eccentricity and ò_{V} denotes threedimensional integration
over the volume of the spheroidal cavity.
The C^{a} 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 F44322 Nantes Cedex 3, France
wang@math.univnantes.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 pdimensional 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 firstmover advantage and the
information asymmetry in Stackelberg duopoly and show that in most
cases the secondmover will gain more profit than the firstmover.
Only when the firstmover's information is quite accurate and the
secondmover's is much more accurate, the firstmover will
outperform the secondmover. We also compare the outcomes from
Stackelberg duopoly with those from Cournot duopoly and show that
the firstmover's (secondmover'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 926973875, 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 L^{2} method could play an important role.
Inexact Preconditioned Uzawa
Algorithms and Domain Decomposition Methods for Threedimensional
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 saddlepoint systems arising from the discretization of
Maxwell system and NavierStokes 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
nonoverlapping domain decomposition methods for solving the
threedimensional Maxwell equations in nonhomogeneous media. The
work was supported by Hong Kong RGC grants (project CUHK4048/02P
and project 403403).