歷史演講 PAST TALKS

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Abstract: Everybody knows George Boole and his theory of mathematical logic for sure, but not many know that he actually discovered a measure-preserving map (called Boole's map) way before Lebesgue's integration theory came out in 1902. We will discuss a noncommutative probability approach to the infinite ergodic theory of Boole's map. The main result here is a sufficient condition for the recurrence and ergodicity of Boole type transformations on the real line.

Abstract: Operator algebras come in two types: von Neumann algebras and C*-algebras. Although they look similar at first sight, von Neumann algebras have a measure-theoretic flavor, while C*-algebras are more topological. In 1970s Alain Connes made stunning breakthroughs in understanding the structure and symmetries of simple injective von Neumann algebras. In the last couple of decades, an army of mathematicians have been working on C*-analogues of Connes' results, and we are now witnessing remarkable advances. I will discuss some of the highlights, including the recent development of non-commutative Rokhlin type theorems. This talk will be accessible to first- year graduate students. No prior knowledge of operator algebras.is needed.

Abstract: In this work, we propose a simple and novel prediction-correction direct-forcing immersed boundary method, which is combined with the projection methods, for simulating the dynamics of fluid-solid interaction problems. The immersed solid object can be stationary or moving in the fluid with a prescribed velocity. As usual, an Eulerian description is used for the fluid dynamics, while the Lagrangian representation is employed for the immersed solid boundary. The proposed approach can be categorized as a discrete forcing method with a prediction-correction strategy, in which a virtual force of discrete type distributed on the whole solid body is introduced and added to the fluid momentum equations to accommodate the no-slip boundary condition at the immersed solid boundary. More precisely, based on the rate of moment changes of the solid body, the virtual force at the grid points can be first predicted by using the difference between the prescribed solid velocities and the computed velocities which are obtained by the projection method for the incompressible Navier-Stokes equations without adding any virtual forcing term. Such predicted virtual force is then put into the momentum equations for the correction step to update the velocity field, pressure and virtual force. This prediction-correction process can be iterated to generate a more general method, if necessary. Numerical experiments of several benchmark problems are performed to illustrate the simplicity and high performance of the proposed prediction-correction approach. We find that our numerical results are in very good agreement with the previous works in the literature and in most cases, one correction step is good enough. This is joint work with Tzyy-Leng Horng, Po-Wen Hsieh and Suh-Yuh Yang.

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COSINE Conference 專題演講 演講時間：3:00 p.m. ~ 3:45 p.m. 延伸討論：3:45 p.m. ~ 4:50 p.m. Abstract: 資料同化技術為結合數值天氣預報(NWP)模式及氣象觀測之重要橋樑，其主要目的為最佳化結合模式與觀測資訊以提供NWP模式一個接近真實大氣的分析(初始)場。初始場越準，越能提供更準確的天氣預報。而在決定此分析場時，不僅使用短期模式預報及觀測資訊，並須考慮兩者之誤差特性。近年來，資料同化技術大幅躍進，如使用系集卡爾曼濾波器進行資料同化，並透過系集及非線性動力模式考慮模式動力誤差隨時間變化的特性，進而能提高觀測的使用效益。 本次演講中將介紹在台灣針對劇烈天氣預報所建立的區域系集卡爾曼濾波器資料同化系統，及其針對颱風及梅雨豪大雨預報上的資料同化技術改進。

COSINE Conference 專題演講 演講時間：9:20 a.m. ~ 10:05 a.m. 延伸討論：10:05 a.m. ~ 11:30 a.m. Abstract: 演講的第一個部分，從計算力學祖師爺西方的尤拉 (Leonhard Euler) 與東方的孫子介紹開始，讓大家了解數學力學與計算的歷史進程。接著再介紹有關計算力學的軟體(有限元素法與邊界元素法)以及硬體(計算尺與求積儀)來串連數學力學與計算，並從中分享高斯的智慧。最後利用了動畫模擬以及各種實例來演繹，將大學時期所學習過的力學及工程數學知識用動畫呈現，以達到眼見為憑的效果，也可說是「有圖見真相，動畫即現象」，進而從單純的數字及文字中跳脫出來，賦予更多樣化的意象。最後以「There is nothing more practical than the right theory. Whether the theory is right or not depends on the experiment」和「To be an engineer with mathematical background. To be a mathematician with physical concept and engineering judgement」總結數學力學與計算三位一體的重要性。

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： .Healthcare benefit s individuals by slowing the natural growth of mortality, indirectly increasing utility through consumption over a longer lifetime. This paper solves the problem of household dynamic healthcare, consumption, and saving when natural mortality grows exponentially to reflect the Gompertz law, while both utility and healthcare are isoelastic. The optimization problem reduces to a nonlinear ordinary diff erential equation with a unique solution, which is constructed using Perron’s method for viscosity solutions. Optimal consumption and health-care spending policies depend on mortality and imply an endogenous curve similar to the Gompertz law. Absolute health expenditure increases with age; relative to consumption, it increase for the typical adult life and slowly declines in the old age.

Abstract: In this talk I shall present our recent results about CR analysis and geometry.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： This reading seminar is based on the first chapter of the book “Quantum Computation and Quantum Information” by Michael Nielsen and Issac Chuang.

Abstract: Firn is a type of snow and is at an intermediate stage between snow and glacial ice. Most importantly, its structure reveals the climate information in certain ways. We use computational topology, which is a natural way to describe the shape of an object, to explore the firn structure. The main difficulty is that the data is noisy, but with the topological tool, in particular persistent homology, the robust features can be extracted from the noisy data. In this talk, we will give a brief introduction to computational topology and persistence homology, and analyze the firn structure by these topological tools . This is joint work with Sarah Day, and Kaitlin Keegan.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： This reading seminar is based on the first chapter of the book “Quantum Computation and Quantum Information” by Michael Nielsen and Issac Chuang.

Abstract: When a population grows old, it is always interesting to know what happened to it in the past. The coalescence problem provides a way to understand the structure and the ancestry of the individuals in the population. We consider a rapidly-growing Galton-Watson branching process and pick two individuals in the current generation by simple random sampling without replacement and trace their lines of descent backward in time till they meet for the first time. We call the common ancestor of these chosen individuals at the coalescent time their most recent common ancestor. The coalescence problem is to investigate the limit behaviors of some characteristics of this most recent common ancestor such as its death time and its generation number. In this talk, we will introduce branching random walks by imposing movement structures to the above processes and we will see what happens to the limit distribution of the positions of the particles in the branching random walks by means of the coalescence theory.

Abstract: It is known that the algebro-geometric properties of toric varieties can be translated into the combinatorial properties of lattice polytopes and fans. In this talk, I will explain how one can understand the arithmetic of toric varieties using convex geometry.

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參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： We consider the one-dimensional stochastic heat equation on [0,1] with Dirichlet boundary condition driven by a space time white noise with constant initial data. In this talk I will show a result of Mueller and Nualart that the solution has negative moment of all orders. This is on the paper “regularity of the density for the stochastic heat equation.”

Abstract: In 1966, Marc Kac posed the question ``Can one hear the shape of a drum?'' It can be rephrased as ``Does the spectrum of the Laplacian on a compact Riemannian manifold determine the manifold up to isometry?'' This problem had attracted a lot of attention. To this day, interesting pairs of isospectral but nonisometric manifolds, graphs and complexes have been constructed. Many constructions are based on Sunada's algebraic criterion published in 1985. In this talk we shall discuss isospectrality in the context of number theory, geometry and combinatorics, the role played by Sunada's criterion, and connections to zeta functions.

Abstract: In this talk, we will demonstrate the relation between the vortex filament equation( VFE) and nonlinear Schr\"odinger equation( NLS), including the Hashimoto transform. We also use this relation to obtain the existence of solutions to the Cauchy problems of VFE for periodic boundary conditions.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: We consider the one-dimensional stochastic heat equation on [0,1] with Dirichlet boundary condition driven by a space time white noise with constant initial data. In this talk I will show a result of Mueller and Nualart that the solution has negative moment of all orders. This is on the paper “regularity of the density for the stochastic heat equation.”

Abstract: Super-massive binary black holes are believed to exist in many galaxies. We study the dynamical systems with super-massive binary black holes in centers of galaxies. Assuming black holes have equal mass, the existence and stability of equilibrium points are investigated. The important equi-potential surfaces are also presented and discussed.

Abstract: By tensor multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simple connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope. Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970’s, around 1999 Knutson and Tao invented their hive model for the type A cases, which led to the solution of the saturation conjecture. Outside type A, Berenstein and Zelevinsky’s models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model. In this talk, I will explain how to use cluster algebras, an interesting class of commutative algebras introduced by Fomin and Zelevinsky, to discover new polyhedral models for all Dynkin types. Those new models improve Berenstein-Zelevinsky’s models, or in some sense generalize the hive model. In the first hour of this 2-hour talk I will explain the main idea and result; In the second hour I will introduce the main machinary – the upper cluster algebra and its quiver with potential model. The talk is based on the preprint arXiv:1603.02521. The talk is friendly to a general audience.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: It is an important technique to introduce zeros in the desired entries of a matrix by performing nonsingular transformations. Two well known examples are Gaussian eliminations and orthogonal diagonalization of symmetric matrices. In this talk, we present two research problems solved by two types of creating zeros. One is a numerical algorithm to triangularize a quadratic matrix polynomial. The other is to precondition the advective operator.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: Painlevé VI equation is a 2nd order nonlinear complex ODE. Hitchin in 1995 proved a beautiful theorem so called the Hitchin theorem which gives explicit expression for a class of solutions to PVI((1/8),((-1)/8),(1/8),(3/8)) (the Hitchin equation). In this talk, I will talk about a deep connection between Hitchin equation and mean field equation and apply it to study the smoothness of solutions to the Hitchin equation. Second, we will generalize the Hitchin theorem to the case PVI((1/2)(n+(1/2))²,((-1)/8),(1/8),(3/8)) where n∈N. As an important application of this generalization, we proved the pre-modular form obtained by Chai, Lin and Wang has simple zeros only and then confirm the Dahmen-Beukers conjecture.

Abstract: In-home Smart Grid, the integration of the renewable power systems with the conventional power systems, calls for cost-effective management for the electricity usages of end users’ household appliances. In this talk, we will first describe how to use the queuing theory to develop an analytical model for the electricity consumption within a community. In view of the fact that the conventional power system has sufficient energy to supply power demands from the end users consistently, whereas the power supplied by the renewable power system is intermittent, depending on the availability of renewable energy sources (e.g., solar or wind energy). By applying the theory of finite-horizon continuous-time Markov decision processes, we obtain a threshold policy. According to this threshold policy, the in-home smart meter can determine whether the power demands should be served by the conventional power system or the renewable power system. Numerical results show that the threshold policy can manage the electricity usage very effectively.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: Fluid-structure interaction problems have been a challenging subject for computational fluid dynamics, not to mention if the problem involves a free surface between two fluids such as water and air. While level set methods have been celebrating huge success than ever in simulating two-phase flows, various novel and efficient direct-forcing immersed boundary (DFIB) methods have been recently applied successfully to solve difficult fluid-structure interaction problems. However, few methods were reported to solve the fluid-structure interaction problems involving free surface, which can be seen as a kind of three-phase flow problems. Those methods are difficult to implement, which motivated us to develop an efficient joint method of level set and DFIB for it. Here incompressible Navier-Stokes equations together with the level set equation are discretized by finite difference method in simple staggered Cartesian grids, with the accuracy of order in space is 2nd order, featuring central difference for the viscous term and 2nd-order upwind scheme for convection term. Projection method is employed here with pressure Poisson equation to replace divergence-free condition. 2nd-order Adams-Bashforthmethod is used for the time-integration of convective and viscous terms, and forward Euler method for pressure and force terms. A 5th-order WENO scheme is used to solve the level set equation to track the interface of two fluids with surface tension that requests very high accuracy. As to the solid part, a DFIB method is applied to simulate a solid object moving in two fluids. Volume of solid interpolation is employed to obtain further accuracy in space for flow near the solid. The whole method is validated with a 2D uniform flow past a circular cylinder with vortex shedding. Coefficients of drag and lift, Strouhalnumber were computed and are all very accurate compared with literatures. Several cases of fluid-structure interaction computational examples are shown to demonstrate the excellence of the current method. In conclusion, our current method is efficient, straight-forward, robust and successful in solving fluid-structure interaction problems with free surface involved.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Self-avoiding walk (SAW) on $\mathbb{Z}^d$ is often used as a model for linear polymers in $d$ dimensions. It is known that the susceptibility ( = the sum of the two-point function) diverges as the energy-cost $h$ for each bond goes down to the critical point $-\log\mu$, where $\mu$ is the connective constant. Suppose that the energy-cost is not homogeneous, but it is perturbed by i.i.d. random variables, so that the susceptibility becomes random and loses translation invariance. A natural question would be to ask how the randomness affects the phase transition and critical behavior of SAW. In the talk, I will explain the recent results on the quenched critical point and future problems.

In this elementary talk we will introduce Lie algebras, Lie superalgebras, and their representations. We shall explain the constructions of the fermionic and bosonic Fock spaces and explain the the boson-fermion correspondence. We shall derive the Jacobi triple product identity as an application.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.