Research in Topology and Applications

DMS-0071202 Julius L. Shaneson大多数的建议涉及应用之间的关系连续和离散求和在更高的维度。 这样的关系称为欧拉-麦克劳林公式，它描述了如何根据几何区域中一组平面点上的函数及其在区域和部分边界上的导数的积分来对该函数求和。 对于在具有格点顶点的多面体的格点上求和的多项式函数，由于提出者和S。Cappell，作为他们的结果的后果特征类的奇异品种。 对于一般的（超越）函数，有给定长度的近似公式，以及涉及微分算子的余项，其系数与经典伯努利多项式在真实的直线上的推广有关。 处理trascendental函数的能力使得有可能使用欧拉-麦克劳林公式研究在光滑区域（例如圆或双曲面下的区域）中的晶格点的数量随着图的扩展而增长。晶格点可以被认为是多维网格或晶格中的顶点。将这些点上的求和与某些连续求和过程（即某种类型的积分方法）相关的结果可以影响许多应用数学问题，包括快速积分方案、网络容量问题以及航运和通信中的优化问题。 此外，关于光滑区域中格点的数量的问题，特别是某些非常对称的区域，如圆或双曲面的一部分，众所周知，在数论的中心部分出现，对这些问题的良好估计可能会导致素数分布和其他重要问题的新结果。最后，这些计数方法可能是逻辑/计算机科学中关于执行某些计算所需的最小时间的问题（“P = NP”）的很好的测试案例。

DMS-0071202Julius L. Shaneson Most of the proposal concerns applications of relationsbetween continuous and discrete summation in higher dimensions. Such a relation is called an Euler-MacLaurin formula, and describes how to sum a function over a set oflattice points in a geometric region in terms of integrals of the function and its derivatives on the region and parts of the boundary. For polynomial functions summed on lattice points of a polytope with lattice point vertices, there are exact, closed formulae for doing this, due to the proposer and S. Cappell, obtained as consequences of their results on characteristic classes of singular varieties. For general (transcendental) functions, there are approximate formulae of given length, together with remainder terms involving differential operators whose coefficients are related to generalizations of the classical Bernoulli polynomials on the real line. The capability to handle trascendental functions makes possible the use Euler-MacLaurin formulae the study the growth of the number of lattice points in smooth regions, such as circles or the regions under a hyperboloid, as the figure expands. A lattice point can be thought of as a vertex point in a multidimensional grid or lattice-work. Results relating summation over such points to some continuous summation process, i.e. some type of integration method, can impact numerous applied mathematics problems, including rapid integration schemes, problems in network capacity, and optimization issues in shipping and communications. In addition, questions concerning the number of lattice points in smooth regions, especially certain very symmetric ones like circles or parts of hyperboloids, are well-known to arise in central parts of number theory, and good estimates on these could lead to new results on the distribution of prime numbers and other important questions. Finally, these counting methods might be good test cases for the problems ("P = NP") of logic/computer science concerning the miminum time needed to perform certain kinds of computations.