Showing papers in "Duke Mathematical Journal in 1996"

Geometric categories and o-minimal structures

Abstract: The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain “nice” sets—like { (x, x) : x > 0 } for positive irrational r, and { (x, e−1/x) : x > 0 }—are not subanalytic (at the origin) in R. Here we make available an extension of the category of subanalytic sets that has these sets among its objects, and that behaves much like the category of subanalytic sets. The possibility of doing this emerged in 1991 when Wilkie [27] proved that the real exponential field is “model complete”, followed soon by work of Ressayre, Macintyre, Marker and the authors; see [21], [5], [7] and [19]. However, there are two obstructions to the use by geometers of this development: (i) while the proofs in these articles make essential use of model theory, many results are also stated there (efficiently, but unnecessarily) in model-theoretic terms; (ii) the results of these papers apply directly only to the cartesian spaces R, and not to arbitrary real analytic manifolds. Consequently, in order to carry out our goal we recast here some results in those papers—as well as many of their consequences—in more familiar terms, with emphasis on results of a geometric nature, and allowing arbitrary (real analytic) manifolds as ambient spaces. We thank W. Schmid and K. Vilonen for their suggestion that this would be a useful undertaking; indeed, they gave us a “wish list” (inspired by Chapters 8 and 9 of Kashiwara and Schapira [12]; see also §10 of [22]) which strongly influenced the form and content of this paper. We axiomatize in Section 1 the notion of “behaving like the category of subanalytic sets” by introducing the notion of “analytic-geometric category”. (The category Can of subanalytic sets is the “smallest” analytic-geometric category.) We also state in Section 1 a number of properties shared by all analytic-geometric categories. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of R. For subsets of R, there already exists the theory of “o-minimal structures on the real field” (defined in Section 2); this subject is developed in detail in [4] and is an abstraction of

Stacks of Stable Maps and Gromov-Witten Invariants

Abstract: We correct some errors in the earlier version of this paper. Most importantly, the definition of isogeny of marked stable graphs has changed.

Local statistics for random domino tilings of the Aztec diamond

Abstract: We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.

A universal multicoefficient theorem for the Kasparov groups

Abstract: that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using HomΛ(K(A),K(B)) in place of KK(A,B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K0(A). With this additional structure, the “Λ−module” K(A) becomes a powerful invariant of C*algebras. We show that it is a complete invariant for the class of real-rank-zero AD algebras. The AD algebras are a certain kind of approximately subhomogeneous C∗-algebras which may have torsion in K1 [Ell]. In addition to classifying these algebras, we calculate their automorphism groups up to approximately innerautomorphisms.

Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics

Abstract: This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A precise formulation of this problem was given in [3]; we shall explain it in more detail a bit later. 2. Construct a basis in every polynomial representation of GLr+1, such that the maximal element w0 of the Weyl group Sr+1 (considered as an element of GLr+1) acts on this basis by a permutation (up to a sign), and explicitly compute this permutation. This problem is motivated by recent work by John Stembridge [10] and was brought to our attention by his talk at the Jerusalem Combinatorics Conference, May 1993.

Noninvolutory Hopf algebras and $3$-manifold invariants

Abstract: We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Structure of tilings of the line by a function

Abstract: A function f 2 L 1 (R) tiles the line with a constant weight w using the discrete tile set A if P a2A f(x ? a) = w almost everywhere. A set A is of bounded density if there is a constant C such that #fa 2 A : n a n + 1g C for all integers n. This paper characterizes compactly supported f 2 L 1 (R) that admit a tiling of R of bounded density. It shows that for such functions all tile sets of bounded density A are nite unions of complete arithmetic progressions. The results apply to some noncompactly supported f 2 L 1 (R). The proofs depend on Cohen's theorem characterizing idempotent measures on locally compact abelian groups. We use a result of Meyer which, using Cohen's theorem, characterizes the collections of point masses on the real line whose Fourier transform is a locally nite measure with total variation that grows at most linearly.

Representation theory of Chern-Simons observables

Abstract: Recently we suggested a new quantum algebra, the moduli algebra, which was conjectured to be a quantum algebra of observables of the Hamiltonian Chern Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.

Sheaves with connection on abelian varieties

Abstract: The Fourier-Mukai transform is lifted to the derived category of sheaves with connection on abelian varieties. The case of flat connections (D-modules) is discussed in detail.

Canonical bases and self-evacuating tableaux

Abstract: Let g be a complex semisimple Lie algebra. In this paper, we prove that the action of a certain involution on canonical bases for irreducible g-modules deened by Lusztig agrees with the action of a special element of the associated simply connected Lie group, up to a scalar of unit absolute value. This leads to formulas for the number of xed points of the involution by means of the Weyl character formula. In the g = sl(n) case, Lusztig's involution has been proved by Berenstein and Zelevinsky to coincide with evacuation of semistandard tableaux. Thus we obtain as a corollary formulas for the number of self-evacuating semistandard tableaux of xed shape. We also prove some reenements that keep track of the xed points of the canonical basis belonging to each weight space. In particular, we obtain a formula for the number of self-evacuating standard tableaux of xed shape.