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2008-04-01TIMS Computational Science Software Development Workshop 2008Surfaces 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. -
2008-04-01TIMS Computational Science Software Development Workshop 2008Special 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. -
2008-04-01TIMS Computational Science Software Development Workshop 2008The 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 $n>1$. -
2008-04-01Computational Methods for microstructure of Liquid crystals and Liquid crystal flowsProf. Ping Lin ( University of Dundee, UK )
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2008-04-01TIMS Computational Science Software Development Workshop 2008Weakly 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. -
2008-04-01One-Day Workshop on Differential GeometryApril 1, 2008
Room 308 , Mathematics Research Center Building (ori. New Math. Bldg.)
Website http://140.112.50.155/workshop/WorkshopNTU.pdf -
2008-04-01Differential Geometry of Lagrangian submanifolds and related variational problemsYoshihiro Ohnita
2008-04-01
09:00 - 09:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01Special solutions to Lagrangian mean curvature flowYng-Ing Lee
2008-04-01
10:00- 10:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01Black holes and compact Einstein manifoldsYoshitake Hashimoto 2008-04-01
11:10 - 12:00 308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01Surfaces of prescribed $p$-mean curvature in the Heisenberg groupJih-Hsin Cheng 2008-04-01
14:00 - 14:50 308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01The Li-Yau Eigenvalue Estimates and CR Obata Theorem in Pseudohermitian ManifoldsShu-Cheng Chang 2008-04-01
15:00 - 15:50 308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01Weakly reflective orbits and tangentially degenerate orbits of $s$-representationsTakashi Sakai 2008-04-01
16:10 - 17:00 308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-04-01Computational Methods for microstructure of Liquid crystals and Liquid crystal flowsPing 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.) -
2008-03-28Mathematical modeling and Scientifical ComputingVortex 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.) -
2008-03-28Computational Science on Medicine-Modelling, Simulation and ApplicationMulti-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. -
2008-03-28Multi-scale Modeling, Analysis and Simulation for Solid MaterialsPing 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.) -
2008-03-28Vortex dynamics in meteorologyHung-Chi Kuo 2008-03-28
15:30:00 - 17:00:00 Vortex dynamics in meteorology308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-03-27Seminar on Graph ColoringCircular 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.) -
2008-03-27Seminar on Graph ColoringCircular 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.) -
2008-03-27Student seminar on differential geometryComparison Theorems in Riemannian Geometry2008 - 03 - 27 (Thu.)
19:00 - 20:30
405, Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-03-27Circular consecutive choosability of graphs (V)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.) -
2008-03-27Circular consecutive choosability of graphs (VI)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.) -
2008-03-27Comparison Theorems in Riemannian Geometry2008-03-27
19:00:00 - 20:30:00 Comparison Theorems in Riemannian Geometry405 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-03-26Physics and mathematics of dispersive shock waves: An introductionPhysics 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. -
2008-03-26Physics and mathematics of dispersive shocks: An introductionAnatoly 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.) -
2008-03-25NCTS/TPE & TIMS Joint Student Geometry Seminar (2008)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.) -
2008-03-25NCTS/TPE & TIMS Joint Geometry Seminar (2009-2010)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.) -
2008-03-25Vafa-Witten theory for non-simply laced gauge groupsSiye Wu 2008-03-25
14:00:00 - 15:00:00 Vafa-Witten theory for non-simply laced gauge groups308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-03-25Vafa-Witten theory for non-simply laced gauge groupsSiye Wu 2008-03-25
14:00:00 - 15:00:00 Vafa-Witten theory for non-simply laced gauge groups308 , Mathematics Research Center Building (ori. New Math. Bldg.) -
2008-03-21Workshop on Statistical Methodology of Microarray DataSmall volume imaging
Prof. Hyeonbae Kang ( Inha University, South Korea )2008 - 03 - 21 (Fri.)
10:00 - 11:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)Lecture 1. Method of Small Volume Expansions and its Applications to Medical Imaging I will explain the mathematical theory of the small volume expansions in the context of the conductivity equation, which is the simplest possible model. I then discuss how this mathematical theory can be applied to an electrical impedance tomography problem to detect small inclusions buried inside a body. Lecture 2. Mathematical Analysis for MRElastography and Applications MRElastography is a recent modality to image the internal part of body using the internal measurements of the displacement vectors. One of the advantages of the imaging method using MRE over other methods is that it can detect the stiffness parameter of the tissue, by which we can distinguish malign cancer from benign cancer. In this lecture I will explain a new method of reconstruction using MRE data. Lecture 3. New reconstruction methods for Magneto-acoustic Imaging. A new and promising technique in the medical imaging is to excite a local part of body using acoustic or ultrasonic focusing and to measure changes due to the excitation. I will explain new reconstruction methods for the Magneto-acoustic imaging and the vibration potential tomography.