活动地点：

活动类型：学术报告

主讲人：韩老师 高老师

活动时间：

活动内容：

 

  An International Summer school in geometric topology

 

1.     Supported by: Zhejiang University, Peking University and Liao-Ning Normal University

 

2. Time: from July 16 (Sunday, arrival day) to July 29 (Saturday, departure day). Lectures start on July 17 to July 22 and July 24 to July 28. Sunday, July 23 is free.

 

　　3. Organizers: Luo Feng (Rutgers University and Zhejiang University), Wang Shicheng (Peking University), Lin Xiao-song (University of California and Peking University), Liu Kefeng (University of California and Zhejiang University), Jean-Marc Schlenker (Universite Paul Sabatier, France), Han Youfa (Liao-Ning Normal University), Qiu Rui-feng (Dalian University of Technology), Lei Fengchun (Harbin Institute of Technology).

 

　　 4. The schedule of the lectures will be posted in this web soon.

5. The list of titles and abstracts of the lectures

(1). (10-hour-lecture) Zhang Xing-ru, State University of New York, Buffalo
      Title: Introduction Of Three Dimensional Manifold Topology
      Abstract: This is a short course in which we give a quick overview of 3-manifold topology. Starting from very beginning, we shall describe and explain some of basic notions, known results and open problems, such as incompressible surfaces, loop theorem, sphere theorem, connected sum decomposition, tori decomposition, Dehn surgery, Heegaard splitting, Seifert fibered spaces, hyperbolic 3-manifolds, Thurston's geometrization conjecture, Waldhausen's virtual Haken conjecture.
References:
1. 3-Manifolds, by John Hempel.
2. Lectures On Three-Manifold Topology, by William Jaco.
3. The Geometry And Topology of Three Manifolds, by William Thurston

 

(2) (10-hour-lecture) Lei Feng-chun, Harbin Institute of Technology and Qiu Rui-feng, Dalian University of Technology
      Title: Surfaces in 3-Manifolds
      Abstract: This is an introduction course on 3-manifold topology from the point view of surfaces such as Heegaard surface, incompressible surfaces, and normal surfaces in 3-manifolds. We shall describe some of  basic notions, fundemental results on Haken 3-manifolds and Heegaard  splittings. We shall explain the fundamental idea of Scharlemann-Thompson's general Heegaard splitting theory, along the line we describe proofs of  Haken's Lemma, Jaco's handle addition Lemma, and Casson-Gordon's  theorem on weakly reducible Heegaard splittings.

  (3). (10-hour-lecture) Li Weiping, Oklahoma State University,  (in the first week)

      title: An introduction to the volume conjecture
      Abstract: Volume conjecture for hyperbolic knots is an important conjecture to related the colored Jones polynomials. I will explain the basics of hyperbolic 3-manifolds, volume calculations from the Lobatchevsky function and dilogarithmic  functions. The volume and Chern-Simons are related together with the eta  invariants from the Atiyah-Patodi-Singer index. Related algebraic approach to the volume class and Cheeger-Chern-Simons class will be studied to understood the Dupont, Yoshida and Neumann's articles. Colored Jones polynomial from the quantum matrix and quantum determinant will be presented. Some recent approach to the conjecture will also be discussed, included the $L^2$-invariants, Bloch regulators and  Deligne's cohomology as well as the $SL_2(C)$ character variety role in the volume conjecture from the Gukov's mathematical physics point of view.

(4). (5-hour-lectures) Luo Feng, Rutgers University and Zhejiang University

        Title : variational principles on triangulated surfaces

 

Abstract: In the discrete approach to smooth metrics on surfaces, the basic building blocks are sometimes taken to be triangles in constant curvature spaces. In this setting edge lengths and inner angles of triangles correspond to the metrics and its curvatures. For triangles in hyperbolic, spherical and Euclidean geometries, edge lengths and inner angles are related by the cosine law. Thus cosine law should be considered as the metric-curvature relation. From this point of view, the derivative of the cosine law is an analogy of the Bianchi identity in Riemannian geometry. This lecture series is focus on many applications of the derivative cosine law. It provides a unification of many known approaches of constructing constant curvature metrics on surfaces. These include the work of Colin de Vedierer, Igor Rivin, Greg Leibon, Bragger, Chow and myself on variational approaches and discrete curvature flows on triangulated surfaces.

 

 

 

 

　　(5). (5-hour-lecture) Lu Zhi, Fudan University
Title : An introduction to transformation groups
The abstract of the talks roughly contains the following contexts:

the basic notion for G-manifolds and examples;  equivariant cohomology for G-manifolds; equivariant cobordism for G-manifolds; the connection between G-manifolds and combinatorics.
    
     (6). (5-hour-lecture) Ding Fan, Peking University
       Title : Introduction to contract 3-manifolds
     Abstract: First, we will introduce some basic concepts in contact 3-manifolds such as characteristic foliation, tight and overtwisted contact  structures, convex surface. Then we will introduce some classification results for contact 3-manifolds.

             
      (7). (5-hour-lecture) Liu Xiao-bo, Columbia University
      Title : An introduction to quantum Teichmuller space
      Abstract: We investigate the representation theory of the polynomial  core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of  ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.

      (8). (5-hour-lecture) Ni Yi, Princeton University
      Title: Introduction to Heegaard Floer homology
      Abstract:In these lectures, I will give a brief introduction to Heegaard Floer homology, which was introduced by Ozsv\'ath and Szab\'o. The topics I would like to cover are: Morse theory and handle decomposition, the  definition of Heegaard Floer homology and basic properties, Alexander polynomial and knot Floer homology.

      (9). (5-hour-lecture) Zheng Hao, Zhong Shang University
    Title: "Knot Invariants and Volume Conjecture".
    Abstract: The talks will cover the definition and compuatation of the colored Jones polynomials, the statement and significance of the Volume Conjecture, possible approaches to solving the problem and their difficulties.

 

　(10) (5-hour-lecture) Wu Jie, National University of Singapore
Title: Braid groups? Abstract: "The notion of a braid as "anything plaited, interwoven, or entwined" goes back many centuries, and braids have been used universally for decoration, art and fastening surposes. Only recently have mathematicians tried to describe braids by means of abstract theory. Fortuitously, as the theory has developed, it has enabled applications to outstanding problems in physics, chemistry and biology. In topology, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is "do the first braid on a set of strings, and then follow it with a second on the twisted strings". Such groups may be described by explicit presentations, as was shown by E. Artin in 1925. A braid with n strands can also be thought of as paths of n distinct particles moving through time, and which do not collide (with variations involving particles which can collide). Braids may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces, comprising configurations of distinct points in a manifold X. When X is the plane, the braid can be closed, that is, corresponding ends can be connected in pairs, to form a link, a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link depends on the permutation of strands determined by the link. J.W. Alexander (1928) observed that every link can be obtained in this way from a braid (see also work by Markov). Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. The Jones polynomial of a link (1987) is defined, a priori, as a braid invariant and then shown to depend only on the class of the closed braid. Until recently, the study of braids has been regarded as a topic within knot theory, a major branch of low-dimensional topology. However, recent development has shown that the study of Brunnian braids has application to longstanding problems in homotopy theory, and in particular the fundamental question of the homotopy groups of spheres. (Brunnian braids [Brunn, 1892] are those which reduce to the trivial, untwisted braid when any strand is removed. The familiar Borromean rings are the link obtained by closing up one such Brunnian braid.) .To date, most mathematical interest in braids has come from algebraists, topologists and mathematical physicists. As well, braids are also engaging the attention of computer scientists, as a basis for public-key cryptosystems. Probabilistic algorithms are being employed to search for solutions to word problems in the braid group. In this short course, we will talk braid group and its recent connections with algebraic topology. The lectures will start from very beginning by going through the definitions and basic notions on configuration spaces and braid groups. Then the lectures will move to some classical results such as the Artin representations of braid groups. Finally, we will talk the simplicial structure on braids and the most recent development on the connections between braid groups and homotopy groups.? The concepts and notions will be reviewed during the lectures. Thus the audience are NOT required to have strong background in topology.

    

 

 

 

5. 友情提示

（1）报到地点：大连 辽宁师范大学 国际文化交流中心5号楼。在大连火车站乘101路电车到兰玉街下车，前走20米向右拐，再走200多米，打的15元人民币；大连周水机场打的16元人民币。

（2）该学习班主要针对对低维拓扑感兴趣的研究生，欢迎研究生参加，来参加学习的研究生我们解决旅费（限乘火车）和食宿费。凡是要参加的研究生请填写下面的申请表。报名截止到：7月10日.

（3）    联 系 人：韩老师（0411-84258626，email: hanyoufa@dl.cn）

高老师（0411-84258356，0411-84258626，gl12102002@yahoo.com.cn）

凡有意学习者请填写下列回执报名，请寄到：大连 辽宁师范大学数学学院办公室 高琳收，邮编：116029.

也可以通过e-mail报名，请发到：gl12102002@yahoo.com.cn

 

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