Topological methods in singularity theory

该项目将为参加将于2015年7月20-24日在英国爱丁堡国际数学科学中心举办的名为“奇点理论中的拓扑学方法”的(http://www.icms.org.uk/workshops/topological)研讨会的美国参与者提供差旅支持。它的目的是聚集来自世界各地的奇点理论和拓扑学专家，让他们展示奇点理论和拓扑学在奇点理论中的新发展和新前景，并将这些新方向介绍给年轻的研究人员。奇点理论研究一般对象退化为特殊对象并导致对象发生根本变化的数学现象。对这类“灾变”的研究涉及经济学、生物学、化学、视觉与医学成像、控制理论、工程与理论物理等多个领域，奇异性理论与拓扑学相互作用，并与数学物理、辛及接触拓扑学、外科理论等领域产生了丰富的联系。最近，出现了许多新的联系。例如，低维拓扑和新的同调理论给奇点理论带来了新的不变量，外科理论技术有了新的应用，度量几何的应用也形成了一个非常富有成效的新方向。研讨会将集中于五个特定的领域，在这些领域之间已经有了很强的相互作用：1.奇点、同调理论和相关不变量，2.奇异空间的Bilipschitz几何，3.交同调，4.Morse 2-函数，5.外科理论。

The project will provide travel support for US participants in a workshop ``Topological methods in singularity theory'' (http://www.icms.org.uk/workshops/topological) which will take place at the International Centre for Mathematical Sciences in Edinburgh, U.K., from 20--24 July 2015. Its purpose is to bring together specialists in singularity theory and topology from around the world, to have them present new developments and perspectives in singularity theory and new applications of topology to singularity theory, and to introduce these new directions to young researchers. Singularity theory studies mathematical phenomena associated with degenerations of generic objects into special ones and which results in radical changes of the objects. The study of such ``catastrophes" concerns a wide range of areas, ranging through economics, biology, chemistry, vision and medial imaging, control theory, to engineering and theoretical physics.Singularity theory and topology work with each other, and the collaboration has led to many rich connections with various fields such as mathematical physics, symplectic and contact topology, surgery theory. Recently, many new connections have appeared. For example, low dimensional topology and new homology theories have brought new invariants to singularity theory, there are new applications of surgery theory techniques, and applications of metric geometry also form a very fruitful new direction. The workshop will concentrate on five particular particular areas among which there have already been strong interactions: 1. Singularities, homology theories and related invariants, 2. Bilipschitz geometry of singular spaces, 3. Intersection homology, 4. Morse 2-functions, 5. Surgery theory.