歷史演講 PAST TALKS

摘要： We consider maximum likelihood estimation (MLE) of a regression function using sieves defined by Bernstein polynomials, in terms of their order and coefficients. In case we know that the regression function satisfies certain shape- restriction like monotonicity or convexity, we can impose corresponding restriction through the coefficients of the Bernstein polynomials in the sieves so that the estimate also satisfies the shape-restriction. For sieve MLE of this type, we establish its consistency when the regression function is continuous and its rate of convergence when its derivative satisfies Lipschitz condition. Under the same condition, we also show that the integral of the estimate converges weakly to that of the regression function. Simulation studies are presented to evaluate its numerical performance. In addition to excellent confidence interval estimates of area under the regression function, sieve MLE performs better than the Bayesian method based on Bernstein polynomials and density-regression method, reported in Chang et al. (2007).

摘要： In this talk we consider the following asymptotic behavior of solutions for regularized equation to some nonlinear hyperbolic balance laws arising from the following topics: the viscous gas flow through discontinuous nozzle, viscous traffic flow model, and the atmosphere hydrodynamic escape model. Through the dynamical system theory approach, we can transfer our steady-state problem into a singularly perturbed system. By analyzing the system in different scales, we are able to construct singular stationary wave solutions. By using geometric singular perturbation theory, we can show there exist true stationary solution for our problem shadowing the singular stationary wave solutions. For some special degenerate singular solutions, we apply more extended theory from geometric singular perturbation to prove that they also persists under the perturbation. Moreover, in the first topics, we also analyze the stability of stationary wave solutions, and in the second topics, we introduce a new entropy condition to ensure the uniqueness of the stationary solutions.

摘要： In this talk, I will give the motivation to introduce BSDEs (Backward Stochastic Differential Equations). First of all, I will show that the existence and uniqueness of BSDEs. Secondly, we consider the linear BSDE. Thirdly, we will mention about the comparison principles. Lastly, We can further discuss the connection between control and BSDE. We follow Chap 7 in Pham and Chap 1 and 2 in Ma. 參考： Huyen Pham, Continuous-time Stochastic Control and Optimization with Financial Applications Jin Ma and Jiongmim Yong, Forwand-Backward Stochastic Differential Equations and their Applications

參考: Hanspeter Schmidli, Stochastic Control in Insurance

摘要 : We consider a family of nonlinear eigenvalue problems for viscous Halmiton-Jacobi equations arising in stochastic ergodic control. Quantitative properties of principal eigenvalues and corresponding eigenfunctions are discussed. Such analysis plays the central role in studying the recurrence and transience of feedback diffusions of the associated stochastic ergodic control. Our results can be regarded as a nonlinear extension of the criticality theory for linear Schrodinger equations with decaying potentials.

摘要(詳如附件)

摘要 Consider the following hypothetical scenario as a motivating example. A company wish to market a new product. The company has at hand a description of the social network G formed among a sample of potential customers, where the vertices represent customers and edges connect people to their friends. The company wants to target key potential customers S of the social network and persuade them into adopting the new product by handing out free samples. We assume that individuals in S will be convinced to adopt the new product after they receive a free sample, and the friends of customers in S would be persuaded into buying the new product, which in turn will recommend the product to other friends. The company hopes that by word-of-mouth effects, convinced vertices in S can trigger a cascade of further adoptions, and finally all potential customers will be persuaded to buy the product. A social network (G,θ) is usually modeled by a graph G together with a threshold function θ: V(G) →N such that 1≦θ(v)≦ d(v) for each vertex v in G. Given a vertex subset S of a connected social network (G,θ). Consider the following repetitive process played on (G,θ). At round 0 (the beginning of the game), the vertices of S are colored black and the other vertices are colored white. After that, at each round t > 0, all white vertices v that have at least θ(v) black neighbors at the previous round t-1 are colored black. The colors of the other vertices do not change. The process runs until no more white vertices can update colors from white to black. The set S is called a target set for (G,θ). We are interested in the following optimization problem: finding a target set S of smallest possible size such that all vertices of V(G) \ S are black in the end.

摘要： In this talk, I will give the motivation to introduce BSDEs (Backward Stochastic Differential Equations). First of all, I will show that the existence and uniqueness of BSDEs. Secondly, we consider the linear BSDE. Thirdly, we will mention about the comparison principles. Lastly, We can further discuss the connection between control and BSDE. We follow Chap 7 in Pham and Chap 1 and 2 in Ma. 參考： Huyen Pham, Continuous-time Stochastic Control and Optimization with Financial Applications Jin Ma and Jiongmim Yong, Forwand-Backward Stochastic Differential Equations and their Applications

參考: Hanspeter Schmidli, Stochastic Control in Insurance

摘要： We study the roles of adaptation currents played in neuronal oscillations via a simple two-dimensional neuron model. The network behaviors can therefore be predicated based on neuronal behaviors in pairs, either with or without the influence of synaptic delays and coupling inhomogeneity. To validate the significance of their biophysical relevance, we show the results are consistent with a more complex and realistic neuron model.

摘要： In this talk, I will present the joint work with Matt Papanikolas on algebraic independence of Drinfeld logarithms. Methods of transcendental Galois theory will be also introduced.

摘要： Chaos is regarded as a nonlinear phenomenon in general. However, in this talk, we will investigate linear translation operators on locally compact groups, and give the sufficient and necessary condition for such operators to be chaotic. In particular, we show that the density of periodic points of a weighted translation implies automatically hypercyclicity, which is one of the ingredients for a chaotic dynamic system. We also characterize topologically mixing translation operators.

參考: Hanspeter Schmidli, Stochastic Control in Insurance

摘要： In this talk, we will use primitive forms and primitive cohomology to characterize some properties of sympletic manifolds. We will also show that primitive cohomology has Nomizu type theorem for symplectic nilmanifolds.

摘要： As compared with many scientific disciplines, biostatistics is a quite relatively new applied science and has a very bright future in career. The main purpose of this talk is to introduce students the useful and important roles biostatistics plays in scientific research. This talk will include many real examples, including AIDS investigations in early phases when the cause of HIV was not recognized or understood, the major concern based on a surveillance system to monitor epidemics due to the time lag of case reporting, the limitations in using surveillance data to study the effectiveness of safety belt usage in automobile crashes, the limitation of commonly-used indirectly- and directly-adjusted rates in targeting hot areas, the quantification of influenza impacts over years, the small area estimation problem, the basic concept of confounding variables, the interim analysis and the bias of noncompliance in randomized clinical trials.

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

摘要： Bleher, Ott, and Grebogi numerically studied a chaotic phenomenon in 1989 by investigating a particle scattered by a planar potential field. When the energy E of the particle is slightly less than the peak Ec of the potential, their study shows that there exists an invariant Cantor set in the phase space on which hyperbolic chaotic scattering occurs; otherwise the system is either regular or non-hyperbolic. The bifurcation of dynamics happened at E = Ec is called the abrupt bifurcation and is related to a change in the topology of the energy surface. One aim of this talk is to provide an explanation how the chaotic orbits occur after the bifurcation.

Abstract: Hanspeter Schmidli, Stochastic Control in Insurance

摘要： 近些年，隨機Navier-Stokes方程的研究得到快速的發展，我們將針對其中的遍歷性問題做簡單的回顧，並介紹我們在這些方面的一些研究工作。

摘要： We will consider a Dirichlet type scattering problem for which the solution of the wave equation is in some domain which is a subset of the Minkowski space-time with a 3 dimensional boundary surface. The data is prescribed on the surface. We will derive an integral representation of the solution in terms of some unknown density function and then relate it to the given data function, including the farfield equation. We will discuss a special case of small surfaces and derive the first order correction term for the three related equations. We will also discuss the static surface.

摘要： Many problems in biological, physical and material sciences involve solving partial differential equations in complex domains or deformable interfaces. In particular, the underlying material quantities in the bulk domain may couple with the one in the interface through adsorption of mass from the bulk to the interface and the desorption of mass from the interface to the bulk. In this talk, we present a conservative scheme for solving the coupled bulk-surface concentration equations and test it for different setups. The total mass is well conserved numerically comparing with the literature. The effect of solubility of surfactant on drop deformation in a shear flow is investigated in detail.

摘要： In this talk, we will review Morse theory and see how it is used to study the topology of the moduli space of flat connections.

摘要: Let K be a function field over the fnite field Fq. Let A be the ring of elements of K that are integral outside a fixed place of K. The simplest,, but already highly nontrivial example is A being the polynomial ring Fq[t] and K the rational function field Fq(t). The Drinfeld modular group GL2(A) and its action on its Bruhat-Tits tree T play a similarly important role in function field arithmetic as the action of the modular group SL2(Z) on the complex upper halfplane plays in classical number theory. For finite index subgroups H of GL2(A) the quotient H\T is essentially a finite graph. It encodes information about the group-theoretic structure of H, about the Drinfeld modular curve associated to H, and about the classification of elliptic curves over K.

摘要： The Laplace-Beltrami operator is the most important second order differential operator on a smooth surface. It is well-known that the convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this talk we shall discuss a simple convergent algorithm to compute discrete Laplace-Beltrami operators acting on functions over smooth surfaces.

Abstract: This short-course aims at a quick but self-contained introduction of the quantum information theory . The contents are as follows: ◆[First Lecture] *Chapter 1. Quantum Mechanics Section 1.1 Dirac’s Terminology Section 1.2 Quantum Bit Section 1.3 Mixed Quantum State Section 1.4 Purification of Quantum State Section 1.5 Entangled State Section 1.6 Schmidt Decomposition ◆[Second Lecture] Section 1.7 Quantum Teleportation Section 1.8 Quantum Dense Coding Section 1.9 Trace Distance Section 1.10 Fidelity *Chapter 2. Entropy Section 2.1 von Neumann Entropy Section 2.2 Quantum Relative Entropy Section 2.3 Basic Properties of Entropy ◆[Third Lecture] Section 2.4 Subadditivity Section 2.5 Concavity of the Entropy Section 2.6 The Entropy of a Mixture of Quantum States Section 2.7 Strong Subadditivity ◆[Fourth Lecture] *Chapter 3. Generalized Entropy Section 3.1 Quantum Tsallis Entropy Section 3.2 Quantum Tsallis Relative Entropy Section 3.3 Tsallis Relative Operator Entropy

摘要: We consider random variables defined on the Wiener space. We consider smooth random variables such that we can define their derivatives. Then we defined the derivative of a L2 random variable by extension. This will be a closed unbounded operator define on L2 space.We plan to meet on every Friday for several weeks to learn the basic idea from the following book.姜祖恕、須上苑、許順吉will share for the reading and thepresentation.

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

摘要： A picture of abstract Wiener space is the R infinite random variable with component being iid standard normal. The classical Wiener space is a particular case. The L2 space has simple structure by Wiener chaos decomposition. In classical Wiener space, the functions in level n is the n- multiple Wiener integrals. This is the basic for the Malliavin calculus. We use this to construct more “regular” (smooth) functions that the nth order derivative can be defined. The Sobolev space will be defined as in finite dimensional Euclidean space.We plan to meet on every Friday for several weeks to learn the basic idea from the following book. 姜祖恕、須上苑、許順吉will share for the reading and the presentation.

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

Abstract: We are concerned with stochastic control problems of finite time horizon whose running cost function is of superlinear growth with respect to the control variable. We prove that, as the time horizon tends to infinity, the value function converges to a function of variable separation type which is characterized by an ergodic stochastic control problem. Asymptotic problems of this type arise in utility maximization problems in mathematical finance. From the PDE viewpoint, our results concern the large time behavior of solutions to semilinear parabolic equations with superlinear nonlinearity in gradients.

摘要： Calcium ion (Ca2+) plays an important role in the heart functionality, which is the linker between electrical activity and mechanical muscle contraction. The sarcoplasmic reticulum (SR) is an important Ca2+ storage compartment inside the cardiac myocyte to maintain low Ca2+ concentration. SR consists of several components: the uptake channel “sarco/endoplasmic reticulum Ca2+-ATPase (SERCA pump)”, the release channel “ryanodine receptor 2 (RyR2)”, and the Ca2+ buffer inside SR “calsequestrin 2 (CASQ2)”. Lately, a genetic disease caused by mutations of RyR2 and CASQ2 has been found to induce heart arrhythmia named catecholaminergic polymorphic ventricular tachycardia (CPVT) and is a very hot issue. Recently, we have published a mathematical model of the RyR2 mutation and successfully reproduced some key features of CPVT. Furthermore, we have indicated some targets for clinical drug treatments. Recent findings pointed out that the class I anti-arrhythmia drug (sodium (Na+) channel blocker) “flecainide” is also a very good candidate for CPVT treatment; however, a controversial issue is about its pharmacological mechanism. One research group claimed that flecainide exerts its effect mainly via the blockade of sodium channel and there is no effect on the SR Ca2+ release channel RyR2. In contrast, another group attributed the effectiveness of flecainide to its blockade on RyR2. We are using our RyR2 mutation model and a newly published subcellular stochastic local control model of SR Ca2+ handling to clarify this controversy and some progress will be reported.

Abstract: A new approach to large deviations controls with long horizon is proposed. Starting from quantile hedging result in complete market, and computing its long time asymptotics, we re-derive solutions to two types of large deviations controls problems in a unified way: i.e., upside-chance maximization and downside-risk minimization.

Bezout's theorem tells us that two polynomials f(x,y) and g(x,y) have (deg f)(deg g) common zeros in the projective plane. What about the number of common zeros in the affine plane? Interestingly, this number is related to the Newton polygons of f and g. We will begin with this first glimpse of connection between convex geometry and algebraic geometry, and proceed to introduce a recently developed generalization of this theory.

Abstract: The geometric L\'evy process (GLP) models play very important roles in the field of mathematical finance. They are incomplete market models. It is well-known that the distributions of the log returns from the market data of the asset prices have the asymmetry and the fat tail. The GLP models have those properties in general. Among them the geometric stable process (GSP) model is probably the most important model. The importance of the stable process was discussed by Fama('63). In this talk we shall overview the GLP option pricing models, and explain the properties of these models. After that we shall see that the [GSP \&MEMM] model has many advantages among the models for incomplete markets. We also discuss several applications of GLP models to other problems in the mathematical finance, for example the analysis of credit risk, optimal portfolio theory, etc.

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

參考： Hanspeter Schmidli, Stochastic Control in Insurance

摘要： There are two common approaches in stochastic partial differential equations (SPDE). One is to think of SPDE as a stochastic differential equation with values in a Hilbert space, the other one introduced by J. Walsh in 1986 is more probabilistic. Nowadays SPDE is one of the most important topics in probability. We consider the following stochastic heat equation: u_t=ku_xx+sigma(u)F, where sigma is globally Lipschitz continuous and F is a Gaussian noise. In this talk, I will make sense solutions and we will discuss mathematical intermittency and how do initial data, k and F effect the fluctuations. This is joint work with Daniel Conus, Matthew Joseph and Davar Khoshnevisan.

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

This is a study seminar on Malliavin calculus. The theory of Malliavin calculus is a differential calculus on the Wiener space. This is the probability space where the Brownian motion ( and other diffusion processes) lives. It is known that there is a close connection of a diffusion process and a second order parabolic equation. The transition probability distribution of this diffusion process satisfies a second order parabolic equation. As a consequence, the transition density is the fundamental solution of a parabolic equation which exists as a result of the theory of partial differential equation. Malliavin calculus, invented by Malliavin in 1970’s, is a probabilistic approach to prove this result. He went further to use his new idea to prove that Hormander condition implies the existence of a density for the diffusion process. Hormander original proof was based on the theory of partial differential equation. Since the invention of the theory, it has found many applications not only to partial differential equations., but also to stochastic partial differential equations,mathematical finance, to name a few. We plan to meet on every Friday for several weeks to learn the basic idea in the following book. 姜祖恕、須上苑、許順吉will share the presentation.