Mathematical Sciences: Topology and its Applications

小行星9626817 技术总结：在本项目中，研究者将研究 关于奇异簇拓扑不变量的几个问题 它们与辛几何和代数几何的关系。 奇异品种的不变量的一般公式将被开发和应用，使用复曲面品种，以获得精确的比较格求和和集成。 分析，组合和数论方面的这些比较将探讨，以期许多可能的应用。 另一条调查线将使用辛几何方法在研究的代表性理论不变量的三个流形。 这将包括一系列的研究马斯洛夫指数和分析公式卡森的SU（2）不变量以及研究SU（n）不变量。 第三个领域的调查将是发展一个令人满意的一般理论的拓扑分类有限群行动的流形上，这将包括这样的基本问题作为适当的概念functorality这一分类问题在一个给定的同伦类型。 非技术总结：在这个项目中， 从拓扑学和几何学中得出的精确公式 比较多维积分和格求和。 等 几个世纪以来，一维积分和求和的比较公式一直是可用的，即，在Euler-MacLaurin展开中，但在许多应用所需的高维设置中没有足够的一般性。 通过一系列的中间步骤，这些问题已经被转化为关于具有奇点（即不一致）的空间的拓扑的问题，以及这些奇点如何有助于计算这些空间的不变量的测量。 现代拓扑学的深层方法将被应用于解决这些几何问题，从而最终解决积分与求和的精确比较的具体问题。 设想的应用包括快速集成和离散优化的新方法。 ***

9626817 Cappell Technical summary: In this project the investigator will study several problems about topological invariants of singular varieties and their relation to symplectic geometry and algebraic geometry. General formulae for invariants of singular varieties will be developed and will be applied, using toric varieties, to obtain precise comparisons of lattice summation and integration. Analytical, combinatorial and number theoretical aspects of these comparisons will be explored with a view to many possible applications. Another line of investigation will be the use of symplectic geometrical methods in the study of representation-theoretic invariants of three-manifolds. This will include a series of studies of Maslov indices and analytical formulae for Casson's SU(2) invariant as well as a study of SU(n) invariants. A third area of investigation will be the development of a satisfactory general theory for the topological classification of finite group actions on manifolds; this will encompass such foundational questions as the appropriate notion of functoriality for this classification problem in a given homotopy type. Non-technical summary: In this project, constructions and ideas drawn from topology and geometry will be used to get precise formulae that compare multidimensional integration with lattice summations. Such comparison formulae have been available for several centuries for one-dimensional integrals and sums, i.e., in the Euler-MacLaurin expansion, but have not been available in adequate generality in the higher dimensional settings needed for many applications. Through a series of intermediate steps, such problems have been translated into questions about the topology of spaces which have singularities (that is, which are not uniform) and the measurement of how such singularities contribute to the computation of invariants of such spaces. The deep methods of modern topology will be applied to resolve these geometrical ques tions and hence ultimately to solve the concrete problems of precise comparisons of integrals and sums. Envisioned applications include new approaches to rapid integration and to discrete optimization. ***