北京大学特别数学讲座
第十一期(2008)
课程安排
2008年《北京大学特别数学讲座》第十一期
课 程 安 排
“特别数学讲座”第十一期将于2008年6月下旬至7月底继续举行，讲座课程和主讲教授基本信息如下：
1.课程名称：Mathematical Aspects of String Theory
主讲教授：Prof. He Yanghui（Oxford University, UK）
授课时间：June 23 – July 12
Title：Mathematical Aspects of String Theory
Prof. He Yanghui（Oxford University, UK）
| Time: | June 24、June25、June30、July1、July8 | A.M 9:00-11:00 | |
| June 24、June 30、July7 | P.M 2:30-4:30 |
Abstract:The purpose of this minicourse is to introduce the basics as well as some contemporary topics in string theory to the mathematically oriented beginning graduate student. In week one we will cover the rudiments and formalism of bosonic and supersymmetric string theory. In week two we will see why algebraic geometry is crucial in relating the 10-dimensional superstring to our 4-dimensional world, or what has traditionally been called string phenomenology. Finally, in week three, we move onto some more advanced topics such as D-branes and singularity resolutions.
2.课程名称：Sobolev inequalities, heat kernel estimates and some applications to Ricci flow
主讲教授：Prof. Zhang Qi (University of California, Riverside)
授课时间：June 30 – July 25
Title：Sobolev inequalities, heat kernel estimates and some applications to Ricci flow
Prof. Zhang Qi (University of California, Riverside)
| Time: | July4、July10、July11、July14、July16 | A.M 9:00-11:00 | |
| July18、July21、July23、July24 | A.M 9:00-11:00 | ||
| July1、July 3、July8 | P.M 2:30-4:30 |
Abstract: The course is centered around Sobolev inequalities and their applications to analysis on manifolds and in particular to Ricci flow. Roughly speaking, a Sobolev inequality states that if the derivative of a function is integrable in certain sense ($L^p$, etc), then the function itself has better integrability. It lies in the foundation of modern analysis. For example, Sobolev imbedding is an essential tool in study partial differential equations since the goal of solving a differential equation is to integrate out the derivatives to recover the unknown function.
On the other hand a Sobolev inequality will also yield interesting partial differential equations via minimizing the Sobolev constants. It can also reveal useful information on the underlying space or manifold.
This lecture notes is divided into three parts. In the first part we will present basic materials on Sobolev inequalities in the Euclidean case.
These includes the standard $W^{1, p}({\bf R^n})$ imbedding into $L^{np/(n-p)}$, Orlicz and $C^\alpha$ spaces, when $p \in [1, n)$, $p=n$ and $p>n$ respectively. We will also present the Poincar\'e inequality and Log Sobolev inequality. All these material can be found in standard books, such as [Gilbarg-Trudinger], [Mazya], [Adams], [Lieb-Loss]. The prerequisite for this part is just graduate Real Analysis.
In the second part we discuss Sobolev imbedding on compact or noncompact manifolds. The main theme is to prove several equivalent conditions for the Sobolev imbedding. These include, log Sobolev inequality, certain heat kernel estimates, Poincar\'e inequality and doubling condition, Harnack inequalities etc. We will also show that the validity of certain Sobolev inequalities imply such geometric properties as volume non-collapsing, isoperimetric inequalities.
Much of the material in this part is taken from the books [Hebey] and [Saloff-Coste]. The students needs some basic knowledge of Riemann geometry.
In the third part concerns some recent research. We describe how Sobolev inequalities will change if the underlying manifold undergoes surgeries (cut and paste). We will show some applications of this to the study of Ricci flow and the Poincar\'e and Geometrization conjectures. Basic knowledge about Ricci flow and Perelman's W-entropy is required. If time is short, these part will only be outlined. But detailed writing will be provided.
3.课程名称：Equidistribution on the homogeneous varieties
主讲教授：Prof. Luo Wenzhi (Ohio State University, USA)
授课时间：June 23 – July 4
Title: Equidistribution on the homogeneous varieties
Prof. Luo Wenzhi(Ohio State University, USA)
| Time: | June23、June26、July2、July3 | A.M 9:00-11:00 |
Abstract:In these lectures, we'd like to survey some recent development on various equidistribution problems concerning the cuspidal holomorphic Hecke-eigenforms on arithmetic hyperbolic surfaces, which are analogues of the ergodicity of Laplacian eigenfunctions on surfaces whose geodesic flow is ergodic, as well as its higher dimensional generalization to arithmetic quotients of Hermitian symmetric spaces of non-compact type, with focus on the Siegel modular varieties and Hilbert-Blumenthal varieties.
Our approach is via the Selberg trace formula, through the Bergman kernel and the Selberg-Godement dimension formula. In the case of modular surface, we'll give applications of a remarkable relation (a la Jacquet, Harris-Kudla and Watson) between the equidistribution of eigenforms and the degree 8 triple product L-functions.
Next we turn to more geometric equidistribution results.
It is well-known that the closed geodesics on the modular surface,
when collected according to the discriminants, are equidistributed
with respect to the hyperbolic measure, by the works of Duke and Iwaniec.
We study and evaluate asymptotically the variance of this distribution on
the unit tangent bundle, and show it is equal to the classic variance
of the geodesic flow as introduced and studied by Ratner, but twisted by
an intriguing arithmetic invariant, the central value of certain L-function.
Our approach is via Weil representation and the theta correspondence.
Some of the above works are based on my various joint works with J.Cogdell, Z.Rudnick and P.Sarnak.
除上述课程外，田刚教授等还将组织一些专题讨论班，另有一些教授短期来访做学术报告，欢迎参加。
1、联系人：余湘辉 北京大学数学科学学院 （邮编100871）
电话: 010-58876691
2、报名时间：4月1日至5月10日(申请表格可从网上下载，非北大生源的同志报名时请寄回由导师推荐信和所在单位加盖公章的申请表，并用E-mail发送一份，以便核查)
3、报名时请写清楚您的详细地址、E-MAIL以及联系电话。
4、各课程的讲课时间可能还会有调整，请及时注意通知。
5、包括各课程内容摘要等在内的有关信息可以到网上查询了解，网址:http://www.math.pku.edu.cn/ or http://www.bicmr.org/
北京大学数学科学学院
北京国际数学研究中心
北京大学数学研究所
北京大学数学及其应用重点实验室
2008年03月18日
学员名单
北京地区：6人
北京理工大学：易瑾、孙丹蒂
首都师范大学：牛艳艳、彭纲、邵红亮、潘丽娜
外地： 26人
鲍尔考(University of Wisconsin-Madison)
李志国（河北工业大学）
尚海锋（大连理工大学）
姚景华、陈立志（兰州大学）
郭芳承、巩军胜（西北师范大学）
蒋利平、郑丽玲、郑苗苗（福州大学）
圣宝建、潘建丹、丁蕾、赵宝俊（南京信息工程大学）
白翠霞、陈宇佳、周黎黎、金灿、缑葵香（天津大学）
江伟、魏美华、任翠萍、张玉、李景荣、李小丽、马翠、（陕西师范大学）
1.Mathematical Aspects of String Theory(He Yanghui 6.23-7.12)11人
鲍尔考(Univeristy of Wisconsin-Madison)
李志国（河北工业大学）
圣宝建、潘建丹、丁蕾、赵宝俊（南京信息工程大学）
白翠霞、陈宇佳、周黎黎、金灿、缑葵香（天津大学）
2.Sobolev inequalities, heat kernel estimates and some applications to Ricci flow(zhang Qi, 6.30-7.25) 22人
北京理工大学：易瑾、孙丹蒂
首都师范大学：牛艳艳、彭纲、邵红亮、潘丽娜
尚海锋（大连理工大学）
姚景华、陈立志（兰州大学）
郭芳承、巩军胜（西北师范大学）
白翠霞、陈宇佳、周黎黎、金灿（天津大学）
蒋利平、郑丽玲、郑苗苗（福州大学）
江伟、魏美华、任翠萍、张玉、李景荣、李小丽、马翠（陕西师范大学）
