Mathematical Sciences: Mapping Class Groups and TeichmuellerSpaces

9401284伊万诺夫首席研究员尼古拉·伊万诺夫打算继续他在曲面映射类群、曲面辫子群、曲面曲线复形和Teichmueller空间方面的工作。研究者打算集中于这些概念之间的相互关系，以便一方面揭示曲面映射类群的代数结构，另一方面揭示Teichmueller空间的整体几何。这里一个重要的指导原则是，在映射类群和Teichmueller空间的一面与算术群和对称空间的另一面之间仍然有一种相当神秘的类比，这在主要研究者和其他数学家的早期工作中建立了。特别地，主要研究人员试图证明曲面的辫子群只有明显的自同构，并且从阶至少为2的算术群到映射类群不存在非平凡同态。主要研究者打算继续研究映射类群和Teichmueller空间的刚性性质。他还打算给出在亏格上表现稳定的曲面拓扑中的各种现象的一般解释，包括由生成元和关系表示的映射类群以及量子场论表示。最后，作者将继续研究泛Teichmueller空间中的刚性和稳定性现象。映射类群和Teichmueller空间是数学的几个主要分支的交汇点，包括拓扑学、复分析和代数几何。最近，它们在理论物理的一些问题上也被证明是重要的。这些对象以一种微妙的方式既类似又不同于更好地理解的算术群和对称空间。进行这种类比的各种方法(以及不同！)精确和明确构成了该项目的核心。虽然对称空间形成了数学上研究的可能的空间图像的良好池(算术群在很大程度上控制了它们的结构)，但Teichmueller空间逐渐成为一种新的、更丰富和同样重要的模式(映射类群同样被期望控制它们的结构)。首席研究员打算集中研究这些物体最有趣和最引人注目的两个特性：刚性和稳定性。由于刚性，这些物体的结构预计将完全由它们的形状决定。稳定性告诉我们，当这些物体随着自然参数(属)的增长而变得越来越复杂时，它们的主要特征在某一点上停止变化。稳定性现象不仅与数学有关，而且与映射类群和Teichmueller空间的量子物理应用有关。***

9401284 Ivanov The principal investigator, Nikolai Ivanov, intends to continue his work on the mapping class groups of surfaces, braid groups of surfaces, complexes of curves of surfaces, and Teichmueller spaces. The investigator intends to concentrate on the interrelationships between these notions in order to shed new light on the algebraic structure of the mapping class groups of surfaces on the one hand, and the global geometry of the Teichmueller spaces on the other hand. An important guiding principle here is a still fairly mysterious analogy between the mapping class groups and Teichmueller spaces on the one side and arithmetic groups and symmetric spaces on the other side, established in earlier works of the principal investigator and other mathematicians. In particular, the principal investigator intends to prove that braid groups of surfaces have only the obvious automorphisms, and that there are no nontrivial homomorphisms from the arithmetic groups of rank at least two into mapping class groups. The principal investigator intends to continue his study of the rigidity properties of the mapping class groups and Teichmueller spaces. He also intends to give a general explanation of various phenomena in the topology of surfaces exhibiting stabilization over the genus, including presentations of the mapping class groups by generators and relations and the quantum field theory representations. Finally, the investigator will continue his investigation of rigidity and stability phenomena in the universal Teichmueller spaces. Mapping class groups and Teichmueller spaces serve as a meeting ground for several of the main branches of mathematics, including topology, complex analysis, and algebraic geometry. Recently they have also proved to be important in some questions of theoretical physics. These objects in a subtle way both resemble and differ from the much more well understood arithmetic groups and symmetric spaces. Various approaches to m ade this analogy (and difference!) precise and explicit form the core of the project. While symmetric spaces form a well established pool of possible spatial images investigated in mathematics (and arithmetic groups to a big extent control their structure), Teichmueller spaces gradually emerge as a new, richer and equally important pattern (and mapping class groups are similarly expected to control their structure). The principal investigator intends to concentate on two of the most interesting and striking properties of these objects: rigidity and stability. Because of rigidity, the structure of these objects is expected to be completely determined by their shape only. Stability tells us that when these objects become more and more complicated with a natural parameter (genus) growing, their main characteristics cease to change at a certain point. The stability phenomena is expected to be relevant not only to the mathematical but also to the quantum physics applications of the mapping class groups and Teichmueller spaces. ***