Research in Topology and Applications

朱利叶斯·L·沙纳森大多数建议涉及高维连续求和与离散求和之间关系的应用。这种关系被称为Euler-MacLaurin公式，它描述了如何根据函数及其导数在区域和部分边界上的积分来对几何区域中的一组平面点上的函数求和。对于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，作为他们关于奇异簇的特征类的结果的结果，存在这样做的精确的、封闭的公式。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，其系数与实线上经典的Bernoulli多项式的推广有关。处理优势函数的能力使得使用Euler-MacLaurin公式来研究光滑区域(如圆或双曲面下的区域)中格点的数量随着图形的扩展而增长成为可能。晶格点可以被认为是多维网格或晶格图中的顶点。将这些点上的求和与某种连续求和过程(即某种类型的积分方法)相关联的结果可能会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目的问题，特别是某些非常对称的问题，如圆或部分双曲面，都是众所周知的，在数论的中心部分，对这些问题的良好估计可能会导致关于素数分布的新结果和其他重要问题。最后，这些计数方法可能是逻辑/计算机科学问题(“P=NP”)的很好测试用例，这些问题涉及执行某些类型的计算所需的最少时间。

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.