歷史演講 PAST TALKS

The overall goal of the lectures is to explain the proofs of two fundamental results about holomorphic curves in Abelian varieties. The first result is the Bloch-Ochiai theorem that the Zariski closure of the image of a holomorphic curve in an Abelian variety is the translate of an Abelian subvariety, and the second result is Siu and Yeung's proof of a conjecture of Lang's that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. In the process, I will introduce two important techniques: Nevanlinna theory and jet spaces. I will also make comparisons to parallel results in arithmetic geometry and the Kawamata structure theorem. In Lecture 1 I will give an introduction to the lecture series, introduce Nevanlinna theory, and in particular the Lemma on the Logarithmic derivative, and I will introduce the concept of jets. In Lecture 2 I will give Bloch's proof, made rigorous by Ochiai, that the Zariski closure of a holomorphic curve in an Abelian variety must be the translate of an Abelian subvariety. Time permitting, I will also discuss McQuillan's proof to this theorem,which was motivated by Faltings's theorem on rational points on subvarieties of Abelian varieties. In Lecture 3 I will discuss a construction of Siu and Yeung using theta functions to construct a logarithmic jet differential with poles along an ample divisor and explain how the existence of such a jet differential implies that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. I will conclude with some remarks about more recent results extending the results I will discuss in detail and on limitations that the methods introduced here may have in attacking related problems on general type varieties. A good reference for my lectures is: Chapter 6 of Min Ru, Nevanlinna Theory and Its Relation of Diophantine Approximation, World Scientific, 2001.

We introduce a feed-forward coupling identical systems acting as a type of network motif. With periodic forcing being imposed and each system being tuned near a Hopf bifurcation point, the network motif performs as a band pass filter and leads to amplification of the incoming signal.We describe that a single Hopf oscillator with periodic forcing is a good candidate for mathematical modeling to develop an understanding of auditory system. We also show the generic behavior of such oscillators which have not been previously studied on auditory system models.

The overall goal of the lectures is to explain the proofs of two fundamental results about holomorphic curves in Abelian varieties. The first result is the Bloch-Ochiai theorem that the Zariski closure of the image of a holomorphic curve in an Abelian variety is the translate of an Abelian subvariety, and the second result is Siu and Yeung's proof of a conjecture of Lang's that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. In the process, I will introduce two important techniques: Nevanlinna theory and jet spaces. I will also make comparisons to parallel results in arithmetic geometry and the Kawamata structure theorem. In Lecture 1 I will give an introduction to the lecture series,introduce Nevanlinna theory, and in particular the Lemma on the Logarithmic derivative, and I will introduce the concept of jets. In Lecture 2 I will give Bloch's proof, made rigorous by Ochiai, that the Zariski closure of a holomorphic curve in an Abelian variety must be the translate of an Abelian subvariety. Time permitting, I will also discuss McQuillan's proof to this theorem,which was motivated by Faltings's theorem on rational points on subvarieties of Abelian varieties. In Lecture 3 I will discuss a construction of Siu and Yeung using theta functions to construct a logarithmic jet differential with poles along an ample divisor and explain how the existence of such a jet differential implies that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. I will conclude with some remarks about more recent results extending the results I will discuss in detail and on limitations that the methods introduced here may have in attacking related problems on general type varieties. A good reference for my lectures is: Chapter 6 of Min Ru, Nevanlinna Theory and Its Relation of Diophantine Approximation, World Scientific, 2001.

The overall goal of the lectures is to explain the proofs of two fundamental results about holomorphic curves in Abelian varieties. The first result is the Bloch-Ochiai theorem that the Zariski closure of the image of a holomorphic curve in an Abelian variety is the translate of an Abelian subvariety, and the second result is Siu and Yeung's proof of a conjecture of Lang's that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. In the process, I will introduce two important techniques: Nevanlinna theory and jet spaces. I will also make comparisons to parallel results in arithmetic geometry and the Kawamata structure theorem. In Lecture 1 I will give an introduction to the lecture series, introduce Nevanlinna theory, and in particular the Lemma on the Logarithmic derivative, and I will introduce the concept of jets. In Lecture 2 I will give Bloch's proof, made rigorous by Ochiai, that the Zariski closure of a holomorphic curve in an Abelian variety must be the translate of an Abelian subvariety. Time permitting, I will also discuss McQuillan's proof to this theorem,which was motivated by Faltings's theorem on rational points on subvarieties of Abelian varieties. In Lecture 3 I will discuss a construction of Siu and Yeung using theta functions to construct a logarithmic jet differential with poles along an ample divisor and explain how the existence of such a jet differential implies that a holomorphic curve omitting an ample divisor in an Abelian variety must be constant. I will conclude with some remarks about more recent results extending the results I will discuss in detail and on limitations that the methods introduced here may have in attacking related problems on general type varieties. A good reference for my lectures is: Chapter 6 of Min Ru, Nevanlinna Theory and Its Relation of Diophantine Approximation, World Scientific, 2001.

Recent remarkable results of Bourgain, Katz and Tao on the sum-product problem in finite fields have stimulated a series of studies and the results consequently have been quantified and generalized by some people, including myself. A related problem raised by Bourgain and Widgerson asks to find two variables polynomials which satisfy expanding property in a certain sense. In this talk we will give a solution of this problem and show that in certain cases our bounds are sharp. We also address that in reals one can get Elekes type bound. At the end, we will discuss a version of Balog-Szemeredi-Gowers type estimate established by Bourgain and Garaev which plays a crucial role in our proof.

In this talk, I will consider two dimensional shock waves with solid boundary condition. We use partial hodograph transformation to rewrite our equation as second order quasilinear wave equations, and study the short time existence.

In a population model with Allee effect, threshold dynamic exists such that extinction occurs when the initial population is below the threshold, and persistence occurs when it is above the threshold. In a structured population model such as reaction-diffusion equation or nonlinear matrix equation, the threshold is related to the order of the phase space, and very often, the exact number of equilibrium states of the system. I will review the development of related theories on 1) global bifurcation diagrams of equilibrium solutions; 2) existence and properties of threshold manifold; and various examples from population biology.

全球暖化危機迫在眉睫，嚴重威脅人類及地球上其他生物的生存，我們將討論以下內容，來了解全球暖化目前的現況與我們該如何因應： 1.地球表面暖化證據 2.何謂溫室氣體，與大氣溫度有何關係 3.兩三百年來溫室氣體含量的急速增加來自產也革命後石化能源的大量使用 4.全球暖化對人類和地球生態環境將造成的影響與衝擊 5.石化能源與各種再生能源的比較 6.畜牧業，人類飲食與全球暖化關係 7.因應全球暖化，我們的對策及自救之道-危機或轉機？

Buddha teaches one to transcend the human sufferings and future states of rebirth to reach nirvana by practicing the eightfold way. In this talk, I will look at The Eightfold Way, the sculpture by Helaman Ferguson standing in front of MSRI, which is the artist's Euclidean rendition of the celebrated Klein quartic curve of genus 3, that achieves the maximum symmetry of automorphisms (168 in number) on a surface of genus 3. I hope at the end of the talk folks can be convinced why to us mathematicians, we can reach the ultimate enlightenment when we, as we mimic monks' chant "Nan Woo, Nan Woo, ...", complete the eightfold turn of the dharma wheel.

In this talk I will present some new developments in applications of Cartan's equivalence method to symmetry analysis of differential equations. First, I will describe the method and use it to solve the equivalence problem for equations arising in nonlinear reflection of shock waves. Second, I will discuss the interplay between Cartan's structure theory of Lie pseudo-groups and coverings of integrable (2+1) dispersionless equations.

To describe a multicomponent mixture with the large number of components we use formal transition from multicomponent mixture to infinitecomponent mixture. The discrete set of concentrations dependent on a discrete index k is replaced by distribution function of continuous parameter (sort parameter). Similarly, the kinetic coefficients characterizing a mixture (diffusion coefficients, mobility coefficients, etc.) are replaced by distribution functions of sort parameter. In this case the system of equations is replaced by only one equation dependent on sort parameter. For correct replacement it is required the constructing of S-space with additive measure (space of sort parameter). ),(txuk),,(txkuk Several examples solved with help such approach to problem are presented. 1. Construction the Riemann invariants for the hyperbolic equations (mass transfer under action of an electric field). 2. The problem of hydrodynamic stability (concentration gravitational convection in a layer of a liquid). 3. Stationary distribution of concentration in infinitecomponent mixture

We present an approach for using kinetic theory to capture first and second order statistics of neuronal activity. We coarse grain neuronal networks into populations of neurons and calculate the population average firing rate and output cross-correlation in response to time varying correlated input. We derive coupling equations for the populations based on first and second order statistics of the network connectivity. This coupling scheme is based on the hypothesis that second order statistics of the network connectivity are sufficient to determine second order statistics of neuronal activity. We implement a kinetic theory representation of a simple feed-forward network and demonstrate that the kinetic theory model captures key aspects of the emergence and propagation of correlations in the network, as long as the correlations do not become too strong. By analyzing the correlated activity of feed-forward networks with a variety of connectivity patterns, we provide evidence supporting our hypothesis of the sufficiency of second order connectivity statistics.

Hedging long term commitments with short-dated futures contracts has been a hot topic in the finance industry. A new measurement called $\mu$-total risk was introduced. Two main theorems are established for the relationship between the optimal hedging strategies and projected funding situation.

We characterize non-negative measures μ on the unit disk D for which the area operator Aμ is bounded or compact from Hardy space Hp to Lq( D) spaces.

For a class of general quasilinear equations on S1; we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution.

The class group of the ring of integers in a number field or a function field is a finite abelian group that measures how far this ring is from having unique factorization into prime elements. Class groups are a central object in Algebraic Number Theory, with applications ranging from Fermat's Last Theorem to cryptography. We discuss some open questions concerning the existence of class groups with high m-rank and give some partial answers in the function field case.