Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.Comment: 31 page

Knot homology via derived categories of coherent sheaves II, sl(m) case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.Comment: 51 pages, 9 figure

Generalized McKay quivers of rank three

For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{G} in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi-definiteness as in the classical case of affine Cartan matrices (the case of SL(2,C)). The complete McKay quivers for SL(3,C) are explicitly described and classified based on representation theory

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct "exotic" t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A

Network-aware search in social tagging applications: instance optimality versus efficiency

We consider in this paper top-k query answering in social applications, with a focus on social tagging. This problem requires a significant departure from socially agnostic techniques. In a network- aware context, one can (and should) exploit the social links, which can indicate how users relate to the seeker and how much weight their tagging actions should have in the result build-up. We propose algorithms that have the potential to scale to current applications. While the problem has already been considered in previous literature, this was done either under strong simplifying assumptions or under choices that cannot scale to even moderate-size real-world applications. We first revisit a key aspect of the problem, which is accessing the closest or most relevant users for a given seeker. We describe how this can be done on the fly (without any pre- computations) for several possible choices -- arguably the most natural ones -- of proximity computation in a user network. Based on this, our top-k algorithm is sound and complete, addressing the applicability issues of the existing ones. Moreover, it performs significantly better in general and is instance optimal in the case when the search relies exclusively on the social weight of tagging actions. To further address the efficiency needs of online applications, for which the exact search, albeit optimal, may still be expensive, we then consider approximate algorithms. Specifically, these rely on concise statistics about the social network or on approximate shortest-paths computations. Extensive experiments on real-world data from Twitter show that our techniques can drastically improve response time, without sacrificing precision

Recherche Top-k Dépendant D'un Contexte À L'aide De Vues

National audienc

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct "exotic" t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A