Perhaps the most famous theorem of the 1960â€™s, the Atiyah-Singer index theorem bears all the hallmarks of great mathematics: it draws on and relates several fields in mathematics, explicating andâ€¦ (More)

This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via aâ€¦ (More)

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficientâ€¦ (More)

where b(n) are the Bernoulli numbers. When p is an integral polytope, the existence of such operators is the combinatorial counterpart of a homological property of the associated toric variety: theâ€¦ (More)

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75â€¦ (More)

Velleda Baldoni, Nicole Berline, MichÃ¨le Vergne. Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of a rational polytope. Matthias Beck, Christian Haase, Bruce Reznick,â€¦ (More)

For a given sequence Î± = [Î±1, Î±2, . . . , Î±N+1] of N + 1 positive integers, we consider the combinatorial function E(Î±)(t) that counts the non-negative integer solutions of the equation Î±1x1 +Î±2x2 +â€¦ (More)

We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torusâ€¦ (More)