We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on .

Let be a link. We study the Heegaard Floer homology of the branched double-cover of , branched along . When is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose term is a suitable variant of Khovanov's homology for the link , converging to the Heegaard Floer homology of .

The set is low for (Martin-Löf) randomness if each random set is already random relative to . is -trivial if the prefix complexity of each initial segment of is minimal, namely . We show that these classes coincide. This answers a question of Ambos-Spies and Kučera in: P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000: each low for Martin-Löf random set is . Our class induces a natural intermediate ideal in the r.e. Turing degrees, which generates the whole class under downward closure. Answering a further question in P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000, we prove that each low for computably random set is computable.

We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and the linking numbers of its components.

Cluster algebras form an axiomatically defined class of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.

We study finite-dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type , we show that Kirillov–Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module. For the current algebra of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the -module structure of the Kac–Moody algebra -module as a semi-infinite fusion product of finite-dimensional -modules.

We study the behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then, we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes (absolutely convex hulls of rows of random matrices). All our results hold with high probability, that is, with probability exponentially (in dimension) close to 1.

For a class of population models of competitive type, we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity. We show that the limiting problem is a remarkable system of differential inequalities, which defines the functional class in (2). By exploiting the regularity theory recently developed in Conti et al. (Indiana Univ. Math. J., to appear) for the elements of functional classes of the form , we provide some qualitative and regularity property of the limiting configurations. Besides, for the case of two competing species, we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence. Finally, we prove some new Liouville-type results which allow to have uniform regularity estimates of the solutions.

This is the second in a series on in an abelian category . Given a finite poset , an - is a finite collection of objects and morphisms or in satisfying some axioms, where . Configurations describe how an object in decomposes into subobjects. The first paper defined configurations and studied moduli spaces of -configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category of coherent sheaves on a projective scheme , or mod- of representations of a quiver . Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of , we define an associative multiplication ∗ on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a - . We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on . Then we generalize all these ideas to , a universal generalization of constructible functions, containing more information. When for all and , or when for a Calabi–Yau 3-fold, we construct ( ) from stack algebras to explicit algebras, which will be important in the sequels on invariants counting -semistable objects in .

We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold and a submanifold and reduces to Kontsevich's theorem if . It states that the DGLA of multivector fields on an infinitesimal neighbourhood of is -quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a D-brane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold).

We develop a new framework for noncommutative differential geometry based on derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra , we define its noncommutative cotangent bundle , which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where is the preprojective algebra associated with a quiver of non-Dynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of in terms of the Lie algebra .

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of . For this, we use -categories (i.e. simplicially enriched categories) as models for certain kind of -categories, and we develop the notions of - , - and over them. We prove in particular, that for an -category endowed with an -topology, there exists a model category of stacks over , generalizing the model category structure on simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between -topologies on an -category , and certain of the model category of pre-stacks on . Based on the above results, we study the notion of introduced by Rezk, and we relate it to our model categories of stacks over -sites. In the second part of the paper, we present a parallel theory where -categories, -topologies and -sites are replaced by , and . We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over -sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of , extending the étale -theory of commutative rings.

We define in a global manner the notion of a connective structure for a gerbe on a space . When the gerbe is endowed with trivializing data with respect to an open cover of , we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J.-L. Brylinski and N. Hitchin. We give a global definition of the 3-curvature of this connective structure as a 3-form on with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebra-valued 3-forms. The Bianchi identity, which the curvature of a connection on a principal bundle satisfies, is replaced here by a more elaborate equation.

We obtain two formulae for the higher Frobenius–Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degrees. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius–Schur (FS-)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FS-exponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FS-exponent of a spherical fusion category is a multiple of its exponent, in the sense of Etingof, by a factor not greater than 2. As applications of these results, we prove that the exponent and the dimension of a semisimple quasi-Hopf algebra have the same prime divisors, which answers two questions of Etingof and Gelaki affirmatively for quasi-Hopf algebras. Moreover, we prove that the FS-exponent of divides . In addition, if is a group-theoretic quasi-Hopf algebra, the FS-exponent of divides , and this upper bound is shown to be tight.

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are:

We study -Schur functions characterized by -tableaux, proving combinatorial properties such as a -Pieri rule and a -conjugation. This new approach relies on developing the theory of -tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender–Knuth involution. This work lays the groundwork needed to prove that the set of -Schur Littlewood–Richardson coefficients contains the 3-point Gromov–Witten invariants; structure constants for the quantum cohomology ring.

Double Hurwitz numbers count branched covers of with fixed branch points, with simple branching required over all but two points 0 and , and the branching over 0 and specified by partitions of the degree (with and parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case (complete branching over one point) and is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given and , as a function of genus. In the case where is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves . For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points.