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

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

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

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

An investigation of latency prediction for NoC-based communication architectures using machine learning techniques

© 2019, Springer Science+Business Media, LLC, part of Springer Nature. Due to the increasing number of cores in Systems on Chip (SoCs), bus architectures have suffered with limitations regarding performance. As applications demand higher bandwidth and lower latencies, buses have not been able to comply with such requirements due to longer wires and increased capacitance. Facing this scenario, Networks on Chip (NoCs) emerged as a way to overcome the limitations found in bus-based systems. Fully exploring all possible NoC characteristics settings is unfeasible due to the vast design space to cover. Therefore, some methods which aim to speed up the design process are needed. In this work, we propose the use of machine learning techniques to optimise NoC architecture components during the design phase. We have investigated the performance of several different ML techniques and selected the Random Forest one targeting audio/video applications. The results have shown an accuracy of up to 90% and 85% for prediction involving arbitration and routing protocols, respectively, and in terms of applications inference, audio/video achieved up to 99%. After this step, we have evaluated other classifiers for each application individually, aiming at finding the adequate one for each situation. The best class of classifiers found was the Tree-based one (Random Forest, Random Tree, and M5P) which is very encouraging, and it points to different approaches from the current state of the art for NoCs latency prediction

Extraction du parallélisme à l'exécution pour la syntghèse d'applications basées sur un NoC

National audienc