Character formulas for the operad of two compatible brackets and for the bihamiltonian operad

We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.Comment: 24 pages, accepted by Functional Analysis and its Applications, a few typos correcte

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander and strict polynomial functors in the sense of Friedlander and Suslin. Applications to functor cohomology are discussed.Comment: published version, 24 pages. Section 2.7 reorganized, and notational distinction between left and right tensor product reinstalle

A model structure for coloured operads in symmetric spectra

We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give suficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.Comment: 16 page

Manin products, Koszul duality, Loday algebras and Deligne conjecture

In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne's conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne's conjecture.Comment: Final version, a few references adde

Notes on factorization algebras, factorization homology and applications

These notes are an expanded version of two series of lectures given at the winter school in mathematical physics at les Houches and at the Vietnamese Institute for Mathematical Sciences. They are an introduction to factorization algebras, factorization homology and some of their applications, notably for studying $E_n$-algebras. We give an account of homology theory for manifolds (and spaces), which give invariant of manifolds but also invariant of $E_n$-algebras. We particularly emphasize the point of view of factorization algebras (a structure originating from quantum field theory) which plays, with respect to homology theory for manifolds, the role of sheaves with respect to singular cohomology. We mention some applications to the study of mapping spaces and study several examples, including some over stratified spaces.Comment: 122 pages. A few examples adde

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat

Derived coisotropic structures I: affine case

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a $P_n$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.Comment: 49 pages. v2: many proofs rewritten and the paper is split into two part

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the $(0,1,2)$-dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ we obtain in this way an aspect of topologically twisted 4-dimensional ${\mathcal N}=4$ super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of $G$-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to $U_q(\mathfrak g)$, and from the punctured torus we recover the algebra of quantum differential operators associated to $U_q(\mathfrak g)$. From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum $\mathcal D$-modules.Comment: 57 page, 5 figures. Final version, to appear in J. To

Single high-dose erythropoietin administration immediately after reperfusion in patients with ST-segment elevation myocardial infarction: results of the Erythropoietin in Myocardial Infarction Trial

Background Preclinical studies and pilot clinical trials have shown that high-dose erythropoietin (EPO) reduces infarct size in acute myocardial infarction. We investigated whether a single high-dose of EPO administered immediately after reperfusion in patients with ST-segment elevation myocardial infarction (STEMI) would limit infarct size. Methods A total of 110 patients undergoing successful primary coronary intervention for a first STEMI was randomized to receive standard care either alone (n = 57) or combined with intravenous administration of 1,000 U/kg of epoetin β immediately after reperfusion (n = 53). The primary end point was infarct size assessed by gadolinium-enhanced cardiac magnetic resonance after 3 months. Secondary end points included left ventricular (LV) volume and function at 5-day and 3-month follow-up, incidence of microvascular obstruction (MVO), and safety. Results Erythropoietin significantly decreased the incidence of MVO (43.4% vs 65.3% in the control group, P = .03) and reduced LV volume, mass, and function impairment at 5-day follow-up (all P < .05). After 3 months, median infarct size (interquartile range) was 17.5 g (7.6-26.1 g) in the EPO group and 16.0 g (9.4-28.2 g) in the control group (P = .64); LV mass, volume, and function were not significantly different between the 2 groups. The same number of major adverse cardiac events occurred in both groups. Conclusions Single high-dose EPO administered immediately after successful reperfusion in patients with STEMI did not reduce infarct size at 3-month follow-up. However, this regimen decreased the incidence of MVO and was associated with transient favorable effects on LV volume and function