29,101 research outputs found
Quantum Schubert Calculus
This paper works out the versions of the classical Giambelli and Pieri
formulas in the context of quantum cohomology of a complex Grassmannian.Comment: 20 pages (LaTeX). To appear in Advances in Mathematics. The quantum
Pieri formula in the original version has been corrected (see also
alg-geom/9705024), and the Title has been ``quantized'
Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors
We define a simplicial differential calculus by generalizing divided
differences from the case of curves to the case of general maps, defined on
general topological vector spaces, or even on modules over a topological ring
K. This calculus has the advantage that the number of evaluation points growths
linearly with the degree, and not exponentially as in the classical, "cubic"
approach. In particular, it is better adapted to the case of positive
characteristic, where it permits to define Weil functors corresponding to
scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).Comment: V2: minor changes, and chapter 3: new results included; to appear in
Forum Mathematicu
Weil Spaces and Weil-Lie Groups
We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups
over an arbitrary commutative base ring K (in particular, over discrete rings
such as the integers), and we develop the basic theory of such spaces, leading
up the definition of a Lie algebra attached to a Weil Lie group. By definition,
the category of Weil spaces is the category of functors from K-Weil algebras to
sets; thus our notion of Weil space is similar to, but weaker than the one of
Weil topos defined by E. Dubuc (1979). In view of recent result on Weil
functors for manifolds over general topological base fields or rings by A.
Souvay, this generality is the suitable context to formulate and to prove
general results of infinitesimal differential geometry, as started by the
approach developed in Bertram, Mem. AMS 900
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