Characteristic Classes, Lattice Points, and Euler-MacLaurin Formulae

Julius L. Shaneson
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Julius L. Shaneson: University of Pennsylvania, Department of Mathematics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 612-624 from Springer

Abstract: Abstract In this paper I would like to survey and strengthen some of the connections between topology and other areas of mathematics, pure and applied, and areas beyond mathematics as well. One type of problem that appears in many areas of mathematics, applied mathematics, physics, economics, and probably other sciences concerns the summing of the values of a function over a discrete set of points in a prescribed region of space. Slightly more precisely, let L be a lattice in Euclidean space ℝ n , such as the set of points with integral co-ordinates. Let S ⊂ ℝ n be a region, and let f be a (reasonable) function. Then how can one write (or approximate) the sum ∑n∊L∩S f(x) in terms of quantities that are continuous or continuously computed from f and the (static) geometry of the region S? For example, if f ≡ 1, one is asking for a computation of #(L ∩ S), the number of lattice points in S, in terms of its geometry.

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_55

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