歷史演講 PAST TALKS

We study Hitchin's equations and Higgs bundles over a non-orientable manifold whose oriented cover is compact K\"ahler. Using the involution induced by the deck transformation, we show that the moduli space of Higgs bundles is Langrangian/complex with respect to the hyper-K\"ahler structure on Hitchin's moduli space associated to the oriented cover.We then establish a Donaldson-Corlette type correspondence and relate Hitchin's moduli space to representation varieties. This is a joint work with N.-K.~Ho and G.~Wilkin.

摘要： 圖形分解是圖論中的一個非常重要課題。因為它可以連接組合，代數和其他數學結構。另一方面，圖分解的結果可以應用在編碼理論，實驗設計，計算機和通信網絡等領域。反魔圖是圖形的一種標號。它是將圖形內的頂點或邊給予對應的一個整數值的標號，或兩者兼而有之。圖形標號推出在20世紀60年代中後期,在這幾十年,圖形標號的研究論文已超過1500篇。圖形標號結果以應用在計算機和通信網絡，應用統計學的研究，與一些設計科學等領域。 本篇論文分為兩個部分(Parts)。第一部分(Part I)為圖形分解(Graph decompositions),第二部分為反魔圖。第一部分包括前三章;第一章為圖形分解的一些基本的定義。第二章為二重邊路徑與二重迴路 (Multipaths and Multicycles with Multiplicity 2)的2分解( 2-split)問題。第三章探討完全二部圖(complete bipartite graph)與皇冠圖(crown)的蜘蛛分解( spider decomposition)。第二部分(Part II)為反魔圖,包括四、五、六等三章;第四章介紹反魔圖的一些基本的定義。第五章是證明:迴圈與正規圖的強乘積(Strong product graphs)是反魔圖。在第六章,證得:球型上的經緯線(globes)及半球型上的經緯線(hemispheres)與莫比烏斯帶上的格子(Mobius grids)都是反魔圖。

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

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摘要： Kernel-based estimators have been widely used for estimating unknown density for a very long time. Consistency properties of such estimators depend on proving convergence results. Different forms of convergence results have been proved by various authors under various conditions. Weid and Weissbach (2010) give a very good account of the known convergence results. However, most of these results are for kernel density estimators with non-random bandwidths. However, the practical applications of kernel-density estimation depend very crucially on efficient algorithms for data-based choice of the bandwidth., thus making random bandwidth selection a very important part of kernel density estimation. However, proving convergence results for such estimators with random bandwidth have proved to be generally harder. In this talk, some new convergence results for the random bandwidth case will be discussed.

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摘要： We study the boundary singularity for the stationary solutions of the linearized Boltzmann equation with hard-sphere potential. An asymptotic formula is established for the gradient of the moments which shows the logarithmic singularity near the boundary. Our formula holds for the solutions of the Milne and Kramer problems obtained by the classical, functional analytic existence theory. Our theorem requires the Holder continuity of the boundary data and applies, in particular, to the complete condensation problem for half space.

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

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摘要： We will introduce some active research topics in the filed of additive combinatorics, which has been known as an area in the interplay of number theory, combinatorics and harmonic analysis. We will then focus on the sum-product, expanders and Erdos distances problems in finite fields.

摘要： We construct a generalized solution of the Riemann problem for systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a'G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is weaker than weak" in the sense that it is weaker than a distributional solution. Thus we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on weaker than weak" solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method.

摘要: Ito’s formula plays an important role in stochastic analysis. However, in most textbook Ito’s formula holds only for semimartingales, which is a summation of local martingales and bounded variation processes. Dirichlet process, which is a summation of local martingale and continuous additive functionals with zero energy, is not necessarily a semimartingale. In this topic, we will first extend Nakao’s stochastic integrals for continuous additive functionals of zero energy from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Then we obtain Itˆo’s formula in terms of the extended stochastic integrals.

摘要： Multiscale analysis using wavelet transforms has been applied to many areas of scientific research and provides very useful tools for science and technology. Various wavelet techniques and applications will be presented such as numerical solutions of integral equations and partial differential equations, applications in medical data, bioinformatics as well as biomathematics modeling and simulation.

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

摘要: In this seminar, we will first show the uniqueness theorem of Laplace transforms. To prove this, it suffices to prove Widder's inversion formula since the uniqueness is readily a consequence of Widder's inversion formula. This seminar also contains the convergence theorem of Laplace transforms which parallels the convergence theorem of Fourier transforms. Besides, we will prove Bernstein's Theorem and Feller's Tauberian Theorem, the first one says that a function is the Laplace transform of a measure if and only if it is completely monotone, the second one is about the asymptotic behavior of the distribution function as t goes to infinity. The motivation for choosing this topic is because it has an application in potential theory.

摘要： In this talk, I review some recent progress on our data-driven time-frequency analysis method for studying trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions. This problem can be formulated as a nonlinear optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD/EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data.

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

摘要： In this talk, we analyze an elliptic equation arising in the study of the gauged $O(3)$ sigma model with the Chern-Simons term.We study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in $\mathbb{R}^2$, which implies the possibility that a stable solution which blows up at a vortex point exists.To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

摘要： 1.課程介紹:對於一般單一執行緒（single thread）的程式，多核心的處理器並沒有辦法提升它的處理效能；不過對於多執行緒（multi thread）的程式，就可以透過不同的核心同時計算， 來達到加速的目的。本課程將介紹目前最廣泛使用OpenMP應用程式介面（Application Program Interface, API）在共享記憶體計算環境上如何撰寫平行程式， OpenMP 是一種能透過高階指令，很簡單地將程式平行化、多執行緒化的API。 OpenMP支援 Fortran 和 C/C++ 兩種語言來做平行化， 本課程將以C/C++為基礎， 適合理工背景正從事平行計算的初學者，透過實例說明與實機練習瞭解平行程式撰寫，並且分享平行程式實際經驗。 2.課程內容安排： (1)簡介國網中心計算資源 (2)使用OpenMP應用程式介面撰寫平行程式

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

摘要： In this seminar, we will first show the uniqueness theorem of Laplace transforms. To prove this, it suffices to prove Widder's inversion formula since the uniqueness is readily a consequence of Widder's inversion formula. This seminar also contains the convergence theorem of Laplace transforms which parallels the convergence theorem of Fourier transforms. Besides, we will prove Bernstein's Theorem and Feller's Tauberian Theorem, the first one says that a function is the Laplace transform of a measure if and only if it is completely monotone, the second one is about the asymptotic behavior of the distribution function as t goes to infinity. The motivation for choosing this topic is because it has an application in potential theory.

摘要： The identifying codes were first introduced by Karpovsky, Chakrabarty, and Levitin. The codes can determine the fault diagnosis of computer and communication systems. There are many literatures studying on identifying codes and related problems. In this talk, we will survey results of identifying codes of graphs and introduce identifying colorings of graphs.

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摘要：如附件 歡迎參加!!

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摘要： In this talk, we analyze an elliptic equation arising in the study of the gauged $O(3)$ sigma model with the Chern-Simons term.We study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in $\mathbb{R}^2$, which implies the possibility that a stable solution which blows up at a vortex point exists.To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

摘要： We study the wave structures of the linearized Boltzmann equation in a periodic box for non-smooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable, and the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The iterated averaging lemma plays an important role in constructing the kinetic-like waves. We also supply some new ideas of this lemma.

摘要： Turbulent combustion is a nonlinear multiscale dynamical process. By level set formulation, G-equations are popular models that describe the motion of the thin flame front by both advection term and chemistry term. As the fundamental issue in combustion theory, we aim to study the asymptotic flame propagation speed modeled by G-equations. Few approaches will be introduced, including control theory, homogenization theory and numerical method.

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摘要： We suggest a generic first passage time model with random barrier and information feedback for default times of a portfolio of correlated names, which includes well-known structural and intensity models and opens up new research problems. We illustrate the idea with a copula contagion mixture model and show that contagion and correlation have enormous joint pricing impact on portfolio credit derivatives and CVAs, especially on senior tranches.

摘要： In this seminar, we will first show the uniqueness theorem of Laplace transforms. To prove this, it suffices to prove Widder's inversion formula since the uniqueness is readily a consequence of Widder's inversion formula. This seminar also contains the convergence theorem of Laplace transforms which parallels the convergence theorem of Fourier transforms. Besides, we will prove Bernstein's Theorem and Feller's Tauberian Theorem, the first one says that a function is the Laplace transform of a measure if and only if it is completely monotone, the second one is about the asymptotic behavior of the distribution function as t goes to infinity. The motivation for choosing this topic is because it has an application in potential theory.

摘要： In this talk, I will establish several connections of the Poisson weight function to overdispersion and underdispersion. Specifically, I will show that the logconvexity (logconcavity) of the mean weight function is a necessary and sufficient condition for overdispersion (underdispersion) when the Poisson weight function does not depend on the original Poisson parameter. I will also discuss some properties of the weighted Poisson distributions (WPDs). I will then introduce a notion of pointwise duality between two WPDs and discuss some associated properties. Next, after presenting some illustrative examples and providing a discussion on various Poisson weight functions used in practice, I will make some concluding remarks. Finally, I will use these results to introduce and discuss over/under-dispersed Poisson processes.

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摘要： A numerical method for predicting realistically the occurrence of harbor resonance The long wave trains that induce harbor resonance are almost always transient. Therefore, it is unrealistic to predict the occurrence of harbor resonance based on the numerical computations in which the wave field is assumed to be stationary. However if strong resonance can be induced by a stationary wave train due to large incoming and outgoing energy fluxes through the entrance, but not due to energy trapping by the harbor, we may assume that this resonance will also occur when the incident wave train is transient. It is because unlike the accumulation of large energy in the harbor by wave trapping , the large incoming and outgoing energy fluxes through the entrance, which involves no accumulation effects, should occur very soon after the arrival of the incident wave train. As previous numerical techniques cannot be utilized to determine the amount of the incoming and outgoing energy fluxes, and since most harbor resonances are caused by large incoming and outgoing energy fluxes through the entrance, a new numerical technique has been developed by taking into account the energy flux. This technique, though applicable only to the situation when the wave field is stationary so that the variations in the time domain can be neglected to save the computer memories, can, however, be utilized to achieve the same purpose as that when a transient wave field is under consideration. Propagation of deep-water linear gravity waves on three-dimensional, strongly sheared currents Deep-water linear gravity waves propagating obliquely on a steady three-dimensional, strongly sheared current are studied for the case where the current varies slowly in the horizontal directions and deviates slightly from a linear profile in the vertical direction. In this case, a WKBJ description of wave propagation can be developed by solving the boundary-value problem in a local co-ordinate system in which the rotational perturbation velocity can easily be separated from the irrotational one. Without this separation, a perturbation scheme was applied by Voronovich (1976) to successfully derive the action conservation equation in an even more complicated situation. In that case, the variation of the current in the vertical direction can be arbitrary and the variation in the density of the fluid is permitted. This equation, though valid in theory, cannot be implemented in practice. More importantly, this equation, after reduction, cannot be applied to the present case. It is because in this case, the unknown rotational perturbation velocity occurring implicitly in the second-order inhomogeneous boundary-value problem arising from a perturbation scheme can itself be determined independently from another inhomogeneous boundary-value problem in which there is no need to fulfill a secular condition. Therefore this rotational perturbation velocity should be eliminated from the original second-order boundary-value problem before using the secular condition of the latter to derive the modulation equation. This explains why the modulation equation derived here by using a different method is inconsistent with the action conservation equation, implying that the wave action is not conserved in the present case. The numerical computations of the fully nonlinear waves on a strongly sheared current The fast computational method derived by Clamond & Grue (2001) for two-dimensional fully nonlinear non-overturning waves is extended for simulation of these waves propagating on a unidirectional, vertically shear current in the direction parallel to this current. To achieve this purpose, the wave field is decomposed into an irrotational component and a rotational one, and a special relation between these two components is subsequently derived. This relation can serve as a closure equation so that the irrotational component can be solved independently by using the Clamond-Grue method, after which the rotational component can also be estimated at the free surface. This approach involves fewer assumptions than that developed by Nwogu (2009), so that only the assumption that the current profile in the vertical direction is nearly linear has been made here. By using this numerical scheme, we clearly demonstrate that even nonlinear waves can be blocked and reflected by a vertically shear current. Since this phenomenon is stable, it is not unusual to be seen in the ocean and may therefore be related to freak waves. The above-mentioned relation between the irrotational and rotational components of the wave field can result in a solution of the wave field which is accurate at least to the third order in the wave slope, ensuring that the above conclusions are quite reliable. Whether this relation is also valid for fully nonlinear waves will be verified in the near future. 敬請公告，歡迎參加

摘要： Modular curves and Shimura curves play a central role in modern number theory. In this talk, we will discuss several problems in the theory of modular curves and Shimura curves, including explicit families of elliptic Q-curves of degree 2, families of genus 2 lifts of certain Q-curves, hypergeometric functions as automorphic forms on Shimura curves and Schwarzian differential associated to Shimura curves of genus 0.

摘要： In this talk, I will introduce basic definitions about weighted projective space, complete intersections and quasismoothness. Then, I will illustrate some properties by explicit examples. In the end, we prove the finiteness of those objects for Calabi-Yau and general type cases. 敬請公告，歡迎參加

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摘要 We study the equilibrating effects of the boundary and intermolecular collision for rarefied gases in the spherical domain with variable boundary temperature in $\mathbb{R}^d$, $d=1,2,$ under the Maxwell-type boundary condition. First, we obtain the algebraic convergence rate to the steady state for free molecular flow. And then we construct the steady state solution of the Boltzmann equation for large Knudsen numbers and small variation of boundary temperature. We establish the nonlinear stability, with exponential rate of the stationary Boltzmann solutions. The results hold for the models of Maxwell molecule, hard potential, and hard sphere. Our analysis is based on the explicit formulations of boundary condition for symmetric domain.

摘要： In this talk I want to present some first results on the homogenization of the pseudo differential equation (x)=f(x), where is period in y with period one for all x in .We use the probabilistic approach and prove a limit theorem for the stable-like processes (t) generated by the operator as tends to zero.

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摘要： In this talk, I would like to prove that an integer picked at random from {1,...,n} has about X prime divisors where X is the normal distribution with mean loglog n and variance loglog n. This is a problem from number theory but can be characterized in the form of Central Limit Theorem. The main ideal is apply Lindeberg-Feller Theorem and a well-known result in number theory. The material is basically coveredby the book, “Probability:Theory and Examples”,of Durrett.

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摘要： A frequently used tool to identify an unknown set of defective (positive) elements out of a large collection of elements by group tests is called Group Testing. In theclassic group testing, a "group" test can be any "subset" of the given collection and its outcome is binary under Boolean operations: YES or NO. The former indicates that there is a positive element in this test and the latter implies no positive elements. In this talk, I will introduce a novel but natural generalization of Group Testing, called Multi-Group Testing. Namely, a test can be applied to any "multi-subset" of the given set, where "multi" means that every single element is allowed to be taken more than once in the same test. Obviously, allowing a number of duplicate copies in a test is meaningless under the assumption of Boolean operations. Therefore, we relax the rule of test outcomes in the Multi-Group Testing to arithmetic operations. In this talk, I will propose efficient strategies to solve this problem under some additional conditions.

摘要： Let E be an elliptic curve over a global field k. The conjecture of Birch and Swinnerton-Dyer predicts that the vanishing order of the Hasse-Weil L-function L(E/k,s) at the central critical point s=1 is equal to the rank of the Mordell-Weil group E(k). when the base field k is the rational number field or a global function field, it is known that E(k) is finite if the special value L(E/k,1) is non-zero. On the other hand, the “modularity” of E assures that there exists a unique “weight 2” cusp form f_E such that its L-function coincides with L(E/k,s) In this talk, I will discuss two formulas of central critical L-values coming from these cusp forms. Consequently, these formulas give us criterions for the finiteness of E(k).

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