We consider sequences of graphs and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into ; “right convergence” defined in terms of the densities of homomorphisms from into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs , and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemerédi partitions, sampling and testing of large graphs.
It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann–Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel–Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.
In this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable. The theory we describe follows a classical idea of Cullen, but we use a more geometric formulation to show that it is possible to build a rather complete theory. Our theory allows us to extend some important results for polynomials in the quaternionic variable to the case of power series.
We prove that in a 2-Calabi–Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen–Macaulay modules is 3-Calabi–Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: , Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi–Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: , Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi–Yau duality from 2-Calabi–Yau to -Calabi–Yau categories. We show how to produce many examples of -cluster tilted algebras.
We consider the family of affine conjugacy classes of polynomial maps of one complex variable with degree , and study the map which maps each to the set of fixed-point multipliers of . We show that the local fiber structure of the map around is completely determined by certain two sets and which are subsets of the power set of . Moreover for any , we give an algorithm for counting the number of elements of each fiber only by using and . It can be carried out in finitely many steps, and often by hand.
We introduce the concept of maximal -orthogonal subcategories over Artin algebras and orders, and develop -dimensional Auslander–Reiten theory on them. We give the -Auslander–Reiten translation and the -Auslander–Reiten duality, then show the existence of -almost split sequences and -fundamental sequences. We give some examples.
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 .
This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov–Witten invariants (at all genera). These Gromov–Witten type invariants depend on a Calabi–Yau category, which plays the role of the target in ordinary Gromov–Witten theory. When we use an appropriate version of the derived category of coherent sheaves on a Calabi–Yau variety, this constructs the model at all genera. When the Fukaya category of a compact symplectic manifold is used, it is shown, under certain assumptions, that the usual Gromov–Witten invariants are recovered. The assumptions are that open-closed Gromov–Witten theory can be constructed for , and that the natural map from the Hochschild homology of the Fukaya category of to the ordinary homology of is an isomorphism.
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.
This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good- inequality with two parameters and the other uses Calderón–Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all spaces for . Pointwise estimates are then replaced by appropriate localized – estimates. We obtain weighted estimates for a range of that is different from and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.
We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander–Reiten theory [O. Iyama, Higher dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22–50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195–222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770].
We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace–Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.
In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.
The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2 -dimensional simplicial manifold on vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.
We define, for a regular scheme and a given field of characteristic zero , the notion of -linear mixed Weil cohomology on smooth -schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth -schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over to the derived category of the field . This implies a finiteness theorem and a Poincaré duality theorem for such a cohomology with respect to smooth and projective -schemes (which can be extended to smooth -schemes when is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories.
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 .
In this article we study Cohen–Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite-dimensional symmetric and satisfy . In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces with . Local results for arbitrary initial data are also given.