Topology and its Applications

DMS-0073006Sylvain Cappl在低维和高维拓扑和几何中需要研究的一系列问题中，有些涉及到变换群的研究，即流形和更一般空间的对称性。为了与等变同伦论的方法建立良好的联系，将发展对这类群作用进行分类的新方法；将研究流形上的群作用集的基本“自然性”问题。另一组问题涉及簇的各种拓扑不变量和代数不变量，例如特征类的理论，以及显式计算它们的新方法。要研究的一系列相关问题涉及因子的全局拓扑和奇点的局部几何之间的关系，一般而言，奇点必须被视为“奇点”。映射类群及其Torelli子群在表示的模空间(在代数几何、规范理论、三维流形拓扑和弦理论中是基本的)上的自然作用将被研究，以期在三维拓扑中的应用。研究流形的解析不变量和几何不变量的分解方法以及它们之间的关系，也是为了将这种关系应用于三个流形。环簇的广义特征类的计算将与其他拓扑学、几何学和分析方法相结合，以获得几何组合学的结果和格和问题的应用。本研究项目涉及一系列低维和高维的拓扑和几何问题的研究，以及这些问题在其他数学领域的一些新应用的研究。一些研究工作将涉及流形和更一般空间的不变量，如奇异簇。我们将寻求计算这类自然不变量的有效方法。要研究的模空间和三个流形不变量也出现在理论物理的几何方法中。我们将结合几何、代数和分析的方法来研究奇异簇在格子和与积分比较问题中的可能应用。这种比较在数学科学的许多领域都很有趣。

DMS-0073006Sylvain CappellAmong the range of problems to be investigated in topology and geometry in low and high dimensions, some concern the study of transformation groups, that is the symmetries of manifolds and of more general spaces. New methods of classifying such group actions will be developed with a view to making good connections with methods of equivariant homotopy theory; basic "naturality" questions for the set of group actions on a manifold will be studied. Another set of problems concerns a variety of both topological and algebraic invariants of varieties, e.g. theories of characteristic classes, and new methods of explicitly computing them. A related series of questions to be investigated concerns the relations between the global topology of divisors and the local geometry of their singularities which must, in general, be regarded as "singular knots." The natural actions of mapping class groups and their Torelli subgroups on the moduli spaces of representations (which are foundational in algebraic geometry, in gauge theory, in 3-manifold topology, and in string theory) will be investigated with a view to applications in three dimensional topology. Decomposition methods for studying analytic and geometric invariants of manifolds and the relations between them will be investigated, again with a view of applying such relation to three manifolds. Computations of generalized characteristic classes of toric varieties will be combined with other topological, geometric, and analytical methods to obtain results in geometrical combinatorics and applications to problems concerning lattice sums.This research project involves several investigations in a range of problems in topology and geometry in low and high dimensions and studies of some new applications of these in other areas of mathematics. Some of the research work will involve invariants of manifolds and of more general spaces, such as singular varieties. Effective methods of computing such natural invariants will be sought. The moduli spaces and three manifold invariants to be investigated also arise in geometrical approaches to theoretical physics. A combination of geometrical, algebraic and analytical methods will be used to study possible applications of singular varieties to problems concerning comparisons of lattice sums with integrals. Such comparisons are of interest in many areas of the mathematical sciences.