歷史演講 PAST TALKS

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

參考： Probability Theory: An Analytic View, 2nd Ed., Daniel W. Stroock

參考： Probability Theory: An Analytic View, 2nd Ed., Daniel W. Stroock

摘要： In this talk, we will construct a geometrically invariant class of pseudolocal pseudodifferential operators when a distribution of hyperplanes is given on $mathbb{R}^N$. These contain pseudodifferential operators of mixed homogeneities, and we have in mind applications to several complex variables. This is joint work with E. Stein.

歡迎參加，謝謝!

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

歡迎參加，謝謝!

09:20-09:30 Opening 09:30-11:00 Hai-liang Liu(Iowa State University) Title: Critical Thresholds in Eulerian Dynamics 11:00-12:30 Chi-Hin Chan(National Chiao Tung University) Title: About the Regularity of Solutions to Parabolic Type Equations with Diffusion Term Represented by a Nonlocal Integral Operator 13: 30-15:00 Jin-Cheng Jiang (National Tsing Hua University) Title: One Dimensional Quantum Zakharov System 15:00-16:30 Tien-Tsan Shieh (National Chiao Tung University) Title: Some Topics in Interface Dynamics 16:30-18:00 Ching-Hsiao Cheng (National Central University) Title: Interesting Topics Related to Compressible Fluids & Free Bomdary Problems

摘要: Random walk的研究已有長久的歷史，它是最基本的離散時間隨機過程，這如同Brownian motion是連續時間的最基本的隨機過程一般，在隨機過程的發展史，是累積有很豐富的理論的一個題材，至今其重要性依然存在，它能夠在很多研究領域找到新的生命，在保險理論，在風險管理(Levy Process)，在隨機環境的相關理論，都可以看到Random walk的影子。Spitzer的經典作品，他一方面有不同一般數學書的寫作風格，另一方面在書中少用近代機率論的語言，因此是初者絕佳的一本入門書，但並非容易閱讀的一本書，因此借助於輪流的讀書報告，讓我們慢慢能夠學習Random walk的一些基本理論。以這個為基礎，我們可以研讀近代的Random walk理論 (G.F. Lawler andV. Limic), 隨機環境下的Random walk理論，Levy Process理論。 參考： Frank Spitzer, Principles of Random Walk 2/E

題目: 1.Sharp Range of Flux and the Structure of Solutions for Elliptic Equations Arising from Self-Dual Chern-Simons CP(1) Model 2.Boundary Conditions for the Invisicd Shallow Water Equations 3.Temporal Oscillations in Evolution Equation and Some Related Problems

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

摘要： This is a joint work with Ming-Cheng Shiue. For several decades, concerning the long time behavior of fluid motion, the time periodic flows have become an important type of flow patterns. In 1959, Serrin proposed a very heuristic method for proving the existence of asymptotic stable periodic solutions of the Navier-Stokes equations with small periodic forcing terms under suitable assumptions. Namely, in such case, one may prove that every small ( in a suitable sense ) solution would converge to a time periodic solution ( with the same period as the non-trivial forcing term ). In this article, we prove the existence of time periodic solution for the 3-D primitive equation with suitable time periodic forcing condition. Some related asymptotic behaviors of the solutions are also demonstrated.

摘要: Random walk的研究已有長久的歷史，它是最基本的離散時間隨機過程，這如同Brownian motion是連續時間的最基本的隨機過程一般，在隨機過程的發展史，是累積有很豐富的理論的一個題材，至今其重要性依然存在，它能夠在很多研究領域找到新的生命，在保險理論，在風險管理(Levy Process)，在隨機環境的相關理論，都可以看到Random walk的影子。Spitzer的經典作品，他一方面有不同一般數學書的寫作風格，另一方面在書中少用近代機率論的語言，因此是初者絕佳的一本入門書，但並非容易閱讀的一本書，因此借助於輪流的讀書報告，讓我們慢慢能夠學習Random walk的一些基本理論。以這個為基礎，我們可以研讀近代的Random walk理論 (G.F. Lawler andV. Limic), 隨機環境下的Random walk理論，Levy Process理論。 參考： Frank Spitzer, Principles of Random Walk 2/E

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

摘要： I will present several different exact interacting solutions of systems of conservation laws, and in particular the compressible Euler equations. These present a nice platform for studying new and surprising behaviors in a relatively simple context. Along the way, we'll discuss shock formation and cancellation, some questions of uniqueness and persistence of waves. I'll also put these in the context of work on periodic and shock-free solutions of the Euler equations.

摘要: Random walk的研究已有長久的歷史，它是最基本的離散時間隨機過程，這如同Brownian motion是連續時間的最基本的隨機過程一般，在隨機過程的發展史，是累積有很豐富的理論的一個題材，至今其重要性依然存在，它能夠在很多研究領域找到新的生命，在保險理論，在風險管理(Levy Process)，在隨機環境的相關理論，都可以看到Random walk的影子。Spitzer的經典作品，他一方面有不同一般數學書的寫作風格，另一方面在書中少用近代機率論的語言，因此是初者絕佳的一本入門書，但並非容易閱讀的一本書，因此借助於輪流的讀書報告，讓我們慢慢能夠學習Random walk的一些基本理論。以這個為基礎，我們可以研讀近代的Random walk理論 (G.F. Lawler andV. Limic), 隨機環境下的Random walk理論，Levy Process理論。 參考: Frank Spitzer, Principles of Random Walk 2/E

Abstract: Extremal set theory is a subject of studying the extremal sizes or the structure of the families of subsets satisfying the given properties. We are interested in how large a family of subsets of the set [n] = {1,2,…, n} that avoids a given partially ordered set (poset) P. The foundational result of this sort, Sperner's Theorem (1928), solves this problem for families that contain no two-element chain: A B (that is, for antichains), determining that the maximum size is the binomial coefficient . Let La (n,P) be the largest size of families of subsets of [n] that do not contain the poset P. A conjecture posed by Griggs and Lu (2009), independently by Bukh (2009), is that La (n,P) is asymptotically equal to e , where e is an integer determined by P when n is large enough. Although it is generally believed to be true, the conjecture is still open even for the poset on four elements. We will see how to use a combinatorial counting method to determine the values of La (n,P) for some types of posets P, which consequently proves that the conjecture holds for those posets.

摘要： The main issue in cooperative games is how to distribute among all players the fortune that they produce. Many interesting concepts have been invented. Like core, bargaining set, nucleous and Shapley value. Note that Shapley got the Nobel prize in Economics this year.

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

摘要： Random walk的研究已有長久的歷史，它是最基本的離散時間隨機過程，這如同Brownian motion是連續時間的最基本的隨機過程一般，在隨機過程的發展史，是累積有很豐富的理論的一個題材，至今其重要性依然存在，它能夠在很多研究領域找到新的生命，在保險理論，在風險管理(Levy Process)，在隨機環境的相關理論，都可以看到Random walk的影子。Spitzer的經典作品，他一方面有不同一般數學書的寫作風格，另一方面在書中少用近代機率論的語言，因此是初者絕佳的一本入門書，但並非容易閱讀的一本書，因此借助於輪流的讀書報告，讓我們慢慢能夠學習Random walk的一些基本理論。以這個為基礎，我們可以研讀近代的Random walk理論 (G.F. Lawler andV. Limic), 隨機環境下的Random walk理論，Levy Process理論。 參考： Frank Spitzer, Principles of Random Walk 2/E,

Abstract: We will illustrate through two areas of continuum physics, the kinetic theory and the gas dynamics, the important of physical motivations for mathematical analysis. We do this from both the historical perspectives and the Hilbert Sixth Problem on the relation between different mathematical formulations of the gas dynamics. Multi-variable calculus is the only pre-requisite.

摘要： Random walk的研究已有長久的歷史，它是最基本的離散時間隨機過程，這如同Brownian motion是連續時間的最基本的隨機過程一般，在隨機過程的發展史，是累積有很豐富的理論的一個題材，至今其重要性依然存在，它能夠在很多研究領域找到新的生命，在保險理論，在風險管理(LevyProcess)，在隨機環境的相關理論，都可以看到Random walk的影子。Spitzer的經典作品，他一方面有不同一般數學書的寫作風格，另一方面在書中少用近代機率論的語言，因此是初者絕佳的一本入門書，但並非容易閱讀的一本書，因此借助於輪流的讀書報告，讓我們慢慢能夠學習Random walk的一些基本理論。以這個為基礎，我們可以研讀近代的Random walk理論(G.F. Lawler andV. Limic)，隨機環境下的Random walk理論，Levy Process理論。 參考: Frank Spitzer, Principles of Random Walk 2/E

摘要： The spectral theory is used in many branches of mathematics, the first eigenvalue is the leading term of the spectrum and so plays an important role in theorist research and practical applications. Some of our results on the first (nontrivial) eigenvalue and some powerful mathematical tools are introduced in the talk.

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

摘要： 我將證明我最近的工作(arXiv:1112.5464)中簡化版的結果. 我們只需用到基本的PDE 及C^n上的Bergman kernel 的計算。 從這個簡化版的結果我們就可得到Demailly's weak Morse inequalities and Grauert-Riemenschneider conjecture.

Abstract: This short-course aims at a quick but self-contained introduction of the quantum information theory . The contents are as follows: ◆[Fifth Lecture] Chapter 4. Quantum Information Capacity Section 4.1 Holevo Bound Section 4.2 Capacity ◆[Sixth Lecture] Section 4.3 Concavity of the Auxiliary Function Appearing in Quantum Reliability Function ◆[Seventh Lecture] Chapter 5. Generalized Uncertainty Relations Section 5.1 Heisenberg Type Section 5.2 Shrodinger Type Section 5.3 Wigner-Yanase-Dyson Skew Information Section 5.4 Metric Adjusted Skew Information Section 5.5 Metric Adjusted Correlation Measure Section 5.6 Topics ◆[Eighth Lecture] Chapter 6. Quantum Cryptography Section 6.1 BB84 Protocol Section 6.2 B92 Protocol

(I)首先我將簡介複幾何裡的幾個重要課題: Kodaira vanishing and embedding theorems, Grauert-Riemenschneider conjecture, Demailly's weak and strong Morse inequalities. 接著我會簡介如何從Bergman kernel的角度及我最近的工作(arXiv:1112.5464)來重新詮釋及證明這些問題並得到一些新結果。

The purpose of this mini workshop is to provide a forum for young generations within the dynamical systems community in Taiwan. Speakers are encouraged to present their research interests or projects with ambition, contents need not be finished works of their own. In this regard, the workshop is a meeting for people to share their "mathematical dreams", instead of limited to their research accomplishments. Anyone interested in dynamical systems are welcomed to participate, especially young scholars and graduate students. Program： 10:00-10:10 Opening 10:10-11:00 Kuo-Chang Chen(National Tsing Hua University) Title: Some perspectives of the interplay between dynamical systems and PDE 11:20-12:10 Wen-Guei Hu(National Chiao Tung University) Title: Pattern generation problems arising in multiplicative integer systems 13: 30-14:20 Ting-Hui Yang (Tamkang University) Title: Global dynamics of some ecological model in discrete or continuous habitat 14:40-15:30 Shih-Feng Shieh (National Taiwan Normal University) Title: Structure-preserving curves for symplectic pairs 15:50-16:40 Chih-Hung Chang (Feng Chia University) Title: From multilayer cellular neural networks to symbolic dynamical systems 16:40-17:20 Discussions