Differential forms on smooth operadic algebras

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace algebras, respectively. We also prove an HKR theorem for operads obtained from a filtered distributive law, which recovers, in particular, all the aspects of the classical HKR theorem. Finally, we show that this property is Koszul dual to the operadic PBW property defined by V. Dotsenko and the second author (1804.06485).

Joint Event Extraction along Shortest Dependency Paths using Graph Convolutional Networks

Event extraction (EE) is one of the core information extraction tasks, whose purpose is to automatically identify and extract information about incidents and their actors from texts. This may be beneficial to several domains such as knowledge bases, question answering, information retrieval and summarization tasks, to name a few. The problem of extracting event information from texts is longstanding and usually relies on elaborately designed lexical and syntactic features, which, however, take a large amount of human effort and lack generalization. More recently, deep neural network approaches have been adopted as a means to learn underlying features automatically. However, existing networks do not make full use of syntactic features, which play a fundamental role in capturing very long-range dependencies. Also, most approaches extract each argument of an event separately without considering associations between arguments which ultimately leads to low efficiency, especially in sentences with multiple events. To address the two above-referred problems, we propose a novel joint event extraction framework that aims to extract multiple event triggers and arguments simultaneously by introducing shortest dependency path (SDP) in the dependency graph. We do this by eliminating irrelevant words in the sentence, thus capturing long-range dependencies. Also, an attention-based graph convolutional network is proposed, to carry syntactically related information along the shortest paths between argument candidates that captures and aggregates the latent associations between arguments; a problem that has been overlooked by most of the literature. Our results show a substantial improvement over state-of-the-art methods.

Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements.

A model for framed configuration spaces of points

We study configuration spaces of framed points on compact manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for compact smooth manifolds without boundary.