Mathematical Sciences: Mapping Class Groups and TeichmuellerSpaces

小行星9401284 首席研究员尼古拉·伊万诺夫打算继续他的工作，对映射类组的表面，编织群的表面，复合物的曲线的表面，和Teichmueller空间。 调查人员打算集中在这些概念之间的相互关系，以揭示新的光的代数结构的映射类群体的表面一方面，和全球几何的Teichmueller空间的另一方面。 这里一个重要的指导原则是映射类群和Teichmueller空间之间的一个仍然相当神秘的类比，算术群和对称空间在另一边，在早期的工作中建立的主要研究者和其他数学家。 特别是，主要研究人员打算证明，辫子群的表面只有明显的自同构，并没有非平凡的同态从算术群的秩至少为2到映射类群。 首席研究员打算继续他的研究刚性性质的映射类组和Teichmueller空间。 他还打算给一个一般性的解释各种现象的拓扑结构的表面表现出稳定的属，包括介绍的映射类组的发电机和关系和量子场论表示。 最后，研究人员将继续他的调查刚性和稳定性现象的普遍Teichmueller空间。 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，它们在理论物理的某些问题上也被证明是重要的。 这些对象在一个微妙的方式既相似又不同于更好地理解算术群和对称空间。 各种方法来制造这种类比（和差异！）精确和明确是项目的核心。 虽然对称空间形成了一个在数学中研究的可能空间图像的良好池（算术群在很大程度上控制了它们的结构），但Teichmueller空间逐渐成为一个新的，更丰富的和同样重要的模式（映射类群也同样被期望控制它们的结构）。 首席研究员打算集中在这些物体的两个最有趣和最引人注目的特性：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，当这些对象随着一个自然参数（属）的增长而变得越来越复杂时，它们的主要特征在某个点上停止变化。 稳定性现象不仅与映射类群和Teichmueller空间的数学应用有关，而且与量子物理应用有关。 ***

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. ***