Research in Topology and Applications

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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.