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Characteristic Classes, Lattice Points, and Euler-MacLaurin Formulae

In: Proceedings of the International Congress of Mathematicians

Author

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  • Julius L. Shaneson

    (University of Pennsylvania, Department of Mathematics)

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.

Suggested Citation

  • Julius L. Shaneson, 1995. "Characteristic Classes, Lattice Points, and Euler-MacLaurin Formulae," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 612-624, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_55
    DOI: 10.1007/978-3-0348-9078-6_55
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