Geometry and categorification

We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform categorical calculations, and readily relate classical constructions of geometric representation theory to categorical ones.Comment: 23 pages. an expository article to appear in "Perspectives on Categorification.

Unfurling Khovanov-Lauda-Rouquier algebras

In this paper, we study the behavior of categorical actions of a Lie algebra $\mathfrak{g}$ under the deformation of their spectra. We give conditions under which the general point of a family of categorical actions of $\mathfrak{g}$ carry an action of a larger Lie algebra $\mathfrak{\tilde{g}}$, which we call an {\bf unfurling} of $\mathfrak{g}$. This is closely related to the folding of Dynkin diagrams, but to avoid confusion, we think it is better to use a different term. Our motivation for studying this topic is the difficulty of proving that explicitly presented algebras and categories in the theory of higher representation theory have the "expected size." Deformation is a powerful technique for showing this because of the upper semicontinuity of dimension under deformation. In particular, we'll use this to show the non-degeneracy (in the sense of Khovanov-Lauda) of the 2-quantum group $\mathcal U$ for an arbitrary Cartan datum and any homogeneous choice of parameters.Comment: 32 pages; v2: added a more direct argument for sl_2 relations in deformed category (Thm. 3.13

Studies of the Habits and Development of Neocerata rhodophaga Coquillett

About the year 1897, in the vicinity of Chicago, Illinois, certain varieties of roses grown under glass, notably the Meteor, were attacked by great numbers of minute cecidomyian larvae which destroyed the terminal leaf and blossom buds. In the greenhouses of one extensive rose-grower, the injury was so severe as to render the production of the Meteor unprofitable, and he stopped growing it for a time, until the pest seemed to have disappeared. Strangely enough, another grower, whose houses were separated from those of the first only by a narrow alley, did not at that time suffer at all from the ravages of the insect, but continued to grow the Meteor in his rose-houses without difficulty until sometime after, when he, too, began to experience severe losses on account of its depredations. The species was not definitely determined at that time, and it is impossible in the light of later investigations to say with certainty whether or not more than one was engaged in these attacks. Since then, however, a number of extensive rose-growers about Chicago have been obliged to abandon the growing of this particular variety of rose on account of its extreme liability to attack from these larvae.Ope

China: development, change and engagement

This Editorial introduces the Local Economy special issue on China, guest edited by David Webster.It gives the background to the main themes considered in the special issue: China's rapid urbanization and associated problems of rural to urban migration; issues involved in planning the country's infrastructure; problems of urban regeneration and renewal; adjustment to marketization in industry and housing; and the question how economic development specialists in the UK can relate to China

Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems

In this paper, we consider how the approach of Bezrukavnikov and Kaledin to understanding the categories of coherent sheaves on symplectic resolutions can be applied to the Coulomb branches introduced by Braverman, Finkelberg and Nakajima. In particular, we construct tilting generators on resolved Coulomb branches, and give explicit quiver presentations of categories of coherent sheaves on these varieties, with the wall-crossing functors described by natural bimodules.Comment: 46 pages. v2: added discussion of connection to physics, as well as many other improvements to exposition and technical detail

A Lattice Isomorphism Theorem for Cluster Groups of Mutation-Dynkin Type AnA_{n}

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type $A_{n}$ case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.Comment: 99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratu