Physics and mathematics of dispersive shocks: An introduction
Prof. Anatoly Kamchatnov ( Institute of Spectroscopy, Russian Academy of Science )2008 - 04 - 02 (Wed.)
15:30 - 17:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)In this lecture, we shall introduce basic notions of Whitham modulation theory. The method will be illustrated by a simple example of derivation of Whitham equations for a modulated periodic wave which evolution is governed by linear Klein-Gordon equation. After that we shall proceed to derivation of Whitham equations for the KdV case.

Anatoly Kamchatnov 2008-04-02
15:30:00 - 17:20:00 Physics and mathematics of dispersive shocks: An introduction405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Black holes and compact Einstein manifolds
Prof. Yoshitake Hashimoto ( Osaka City University, Japan )2008 - 04 - 01 (Tue.)
11:10 - 12:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)This is a joint work with M. Sakaguchi and Y.Yasui. We construct an infinite family of inhomogeneous Einstein metrics on a 3-sphere bundle on a 2-sphere. They appear as limits of Kerr-de Sitter balck holes. Our construction is a higher dimesional version of the method of D. Page, by which he constructed an inhomogeneous compact Einstein 4-manifold.

Differential Geometry of Lagrangian submanifolds and related variational problems
Prof. Yoshihiro Ohnita ( Osaka City University, Japan & Osaka City University Advanced Mathematical Institute, Japan )2008 - 04 - 01 (Tue.)
09:00 - 09:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)In this talk I will give a survey of my recent works and their environs on Lagrangian submanifolds in specific K\"ahler manifolds such as complex projective spaces, complex space forms and Hermitian symmmetric spaces and so on. Especially I shall discuss (1) the Hamiltonian stability problem of Lagrangian submanifolds, (2) the classification problems of homogeneous Lagrangian submanifolds, (3) the relationship between certain minimal Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. This talk is mainly based on joint works with Amartuvshin Amarzaya(Natonal Univ. Mongolia), Hui Ma (Tsinghua Univ., Peking).

Surfaces of prescribed $p$-mean curvature in the Heisenberg group
Prof. Jih-Hsin Cheng ( Academia Sinica )2008 - 04 - 01 (Tue.)
14:00 - 14:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)I will report on the study of the prescribed $p$% -mean curvature equation in recent years. We interpreted the $p$-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve (in dimension $2$), and as a quantity in terms of calibration geometry. As a differential equation, the prescribed $p$-mean curvature equation is degenerate (hyperbolic and elliptic) in dimension 2 while subelliptic in the nonsingular domain for higher dimensions. Also in dimension 2 with zero $p$-mean curvature, it is a special type of scalar conservation law in the variable of the so-called characteristic angle. We analyzed the singular set and formulated an extension theorem. This allowed us to classify the entire solutions to this equation and to solve a Bernstein-type problem. As a geometric application, we proved the nonexistence of $C^{2}$ smooth hyperbolic surfaces having bounded $p$-mean curvature, immersed in a pseudohermitian 3-manifold. From the variational formulation of the equation, we studied the Dirichlet problem by proving the existence and the uniqueness of the ($p$-)minimizers. An intriguing point is that, in dimension 2, a $C^{2}$ smooth solution from the PDE viewpoint may not be a minimizer. As an application to neuro-biology, we could use such a $p$-minimal surface theory as an image completion model for missing visual data in the first layer of the visual cortex. About the regularity of a $C^{1}$ smooth surface with continuous prescribed $p$-mean curvature, we proved that the horizontal normal gains one more derivative by invoking the jump formulas along characteristic curves. Near a singular (or shock) curve we depicted the Rankine-Hugoniot jump condition as characteristic incident and reflected angles being equal. In the last section we reviewed the second variation formula for a $C^{2}$ smooth $p$-minimal surface in an arbitrary $3$-dimensional pseudohermitian manifold.

Special solutions to Lagrangian mean curvature flow
Prof. Yng-Ing Lee ( Department of Mathematics, National Taiwan University )2008 - 04 - 01 (Tue.)
10:00 - 10:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)I have been interested in and working on the existence of special Lagrangian/Lagrangian minimal. There are several different approaches we have taken and obtained various results. In recent years, we took the approach of mean curvature flow. Singularities may occur in the flow. They usually model on self-similar solutions and translating solutions. Hence it is very important to study these solutions in studying the singularities. We have constructed many such solutions in a joint work with Joyce, and Tsui. Among our examples, there are three types that are of particular interests. One family is Lagrangian self-expanders that are asymptotic to pair of Lagrangian planes. The Lagrangian angle can be made arbitrarily small, and these examples can serve as local model for surgery. Another family is translating solutions with arbitrarily small Lagrangian angle. Our example is rather surprising and indicates that the flow does not behave well even if under the almost calibrated assumption, which is different from the expectation in the past. Some other families of our examples give external Brakke solutions.

The Li-Yau Eigenvalue Estimates and CR Obata Theorem in Pseudohermitian Manifolds "
Prof. Shu-Cheng Chang ( Department of Mathematics, National Taiwan University )2008 - 04 - 01 (Tue.)
15:00 - 15:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)In this talk, based on a jointed work with Hung-Lin Chiu, we first have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of psitive first eigenvalue of the sub-Laplacian in pseudohermitian $(2n+1)$-manifolds. Second, we prove the CR Obata's theorem on a closed pseudohermitian manifold with vanishing torsion. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator for any closed pseudohermitian $(2n+1)$-manifold with $n>1$.

Prof. Ping Lin  ( University of Dundee, UK )

Weakly reflective orbits and tangentially degenerate orbits of $s$-representations
Prof. Takashi Sakai ( Tokyo Metropolitan University, Japan & Osaka City University Advanced Mathematical Institute, Japan )2008 - 04 - 01 (Tue.)
16:10 - 17:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)This talk is based on a joint work with Osamu Ikawa and Hiroyuki Tasaki. A linear isotropy representation of a Riemannian symmetric space is called an $s$-representation. In this talk we will study some geometric properties of orbits of $s$-representations as submanifolds in the Euclidean sphere. We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we give the classification of weakly reflective orbits and austere orbits of $s$-representations. We observe that some wealky reflective orbits have a remarkable property, namely, they are tangentially degenerate. Furthermore these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.

April 1, 2008
Room 308 , Mathematics Research Center Building (ori. New Math. Bldg.)
Website  http://140.112.50.155/workshop/WorkshopNTU.pdf

Yoshihiro Ohnita 
2008-04-01 
09:00 - 09:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Yng-Ing Lee 
2008-04-01 
10:00- 10:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Yoshitake Hashimoto 2008-04-01 
11:10 - 12:00 308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Jih-Hsin Cheng 2008-04-01 
14:00 - 14:50 308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Shu-Cheng Chang 2008-04-01 
15:00  - 15:50 308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Takashi Sakai 2008-04-01 
16:10 - 17:00 308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Ping Lin 2008-04-01
10:20:00 - 12:10:00 Computational Methods for microstructure of Liquid crystals and Liquid crystal flows405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Vortex dynamics in meteorology
Prof. Hung-Chi Kuo ( Department of Atmospheric Sciences, National Taiwan University )2008 - 03 - 28 (Fri.)
15:30 - 17:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)

Multi-scale Modeling, Analysis and Simulation for Solid Materials
Prof. Ping Lin ( University of Dundee, UK )2008 - 03 - 28 (Fri.)
10:00 - 12:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)Materials may be modeled by a large number of particles (or atoms) where any particle interacting with any other through a pair potential energy. The equillibrium configuration of the material is the minimizer of the total energy of the system. The computational cost is high since the number of atoms is huge. Recently much attention has been paid to the so-called quasicontinuum (QC) approximation which is a mixed atomistic/continuum model.

Ping Lin 2008-03-28
10:00:00 - 12:00:00 Multi-scale Modeling, Analysis and Simulation for Solid Materials308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Hung-Chi Kuo 2008-03-28
15:30:00 - 17:00:00 Vortex dynamics in meteorology308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Circular consecutive choosability of graphs (VI)
Prof. Xuding Zhu ( Zhejiang Normal University, China )2008 - 03 - 27 (Thu.)
13:30 - 15:30
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Circular consecutive choosability of graphs (V)
Prof. Xuding Zhu ( Zhejiang Normal University, China )2008 - 03 - 27 (Thu.)
10:00 - 12:00
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Comparison Theorems in Riemannian Geometry2008 - 03 - 27 (Thu.)
19:00 - 20:30
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Xuding Zhu 2008-03-27
10:00:00 - 12:00:00 Circular consecutive choosability of graphs (V)405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Xuding Zhu 2008-03-27
13:30:00 - 15:30:00 Circular consecutive choosability of graphs (VI)405 , Mathematics Research Center Building (ori. New Math. Bldg.)

2008-03-27
19:00:00 - 20:30:00 Comparison Theorems in Riemannian Geometry405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Physics and mathematics of dispersive shocks: An introduction
Prof. Anatoly Kamchatnov ( Institute of Spectroscopy, Russian Academy of Science )2008 - 03 - 26 (Wed.)
15:30 - 17:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)In this lecture, we shall introduce the Korteweg-de Vries equation for description of combined action of nonlinear and dispersiveeffects. Periodic solution of this equation will be derived.Basic notions of Whitham modulation theory will be introduced.They will be illustrated by their application to a simple example of modulated linear waves.

Anatoly Kamchatnov 2008-03-26
15:30:00 - 17:20:00 Physics and mathematics of dispersive shocks: An introduction405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Vafa-Witten theory for non-simply laced gauge groups
Prof. Siye Wu ( National Tsing Hua University )2008 - 03 - 25 (Tue.)
14:00 - 15:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)