BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Richard Garner (Macquarie University) DTSTART:20200811T001500Z DTEND:20200811T011500Z DTSTAMP:20260422T212709Z UID:operad/1 DESCRIPTION:Title: S weedler duals for monads\nby Richard Garner (Macquarie University) as part of operad pop-up\n\n\nAbstract\nWhile the linear dual of any coalgebr a is an algebra\, the converse is not true\; however\, there is an adjoint to the coalgebra-to-algebra functor\, given by the so-called Sweedler dua l.\n\nThere is a notion of “linear dual” for an endofunctor of Set\, g iven by homming into the identity functor for the Day convolution structur e. Again\, this sends comonads to monads\, but not vice versa\; but again\ , there is an adjoint. This “Sweedler dual” comonad of a monad was int roduced by Katsumata\, Rivas and Uustalu in 2019.\n\nThe purpose of this t alk is to give an explicit construction of the Sweedler dual comonad of an y monad on Set. The category of coalgebras for the Sweedler dual turns out to be a presheaf category\, whose indexing category can be described expl icitly in terms of a kind of computational dynamics of the monad. If time permits\, we also describe the source-etale topological category which cla ssifies the topological Sweedler dual comonad of a monad on Set\; in parti cular\, this recovers all kinds of etale topological groupoids of interest in the study of combinatorial $C^*$-algebras.\n LOCATION:https://researchseminars.org/talk/operad/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Raynor (Macquarie University) DTSTART:20200811T030000Z DTEND:20200811T040000Z DTSTAMP:20260422T212709Z UID:operad/2 DESCRIPTION:Title: U npacking the combinatorics of modular operads\nby Sophie Raynor (Macqu arie University) as part of operad pop-up\n\n\nAbstract\nWhilst operads ar e governed by trees\, undirected graphs of arbitrary genus are needed in o rder to describe modular operads. And this can get complicated. Especially if we're interested in understanding notions of modular operads\, such as Joyal and Kock's compact symmetric multicategories\, where the com bination of the contraction operation and a unital operadic composition pr esents particular challenges.\n\nI'll describe how to first break the prob lem into its constituent parts\, and then use the classical theory of dist ributive laws to put the pieces back together. The decomposition allows us to apply Weber's theory to get a fully faithful nerve via completely abst ract methods. More interestingly\, the proof method makes the combinatoric s of modular operads\, and especially the fiddly stuff\, completely explic it. Hence it provides a roadmap for developing the theory\, and the possib ility for gaining new conceptual insights into the structures described.\n LOCATION:https://researchseminars.org/talk/operad/2/ END:VEVENT BEGIN:VEVENT SUMMARY:María Ronco (Universidad de Talca) DTSTART:20200811T213000Z DTEND:20200811T223000Z DTSTAMP:20260422T212709Z UID:operad/3 DESCRIPTION:Title: O rder and substitution on graph associahedra\nby María Ronco (Universi dad de Talca) as part of operad pop-up\n\n\nAbstract\nM. Carr and S. Devad oss introduced in [1] associated a finite partially ordered set to any sim ple finite graph\, whose geometric realization is a convex polytope ${\\ma thcal K}\\Gamma$\, the graph-associahedron. Their construction include man y well-known families of polytopes\, liked permutahedra\, associahedra\, c yclohedra and the standard simplexes.\n\nThe goal of the present work is t o give an algebraic description of graph associahedra. We introduce a substitution operation on Carr and Devadoss tubings\, which allows us t o describe graph associahedra as a free object on the set of all connected simple graphs\, for a type of colored operad generated by pairs of a fini te connected graph and a connected subgraph of it.\n\nWe show that substit ution of tubings may be understood in the context of M. Batanin and M. Mar kl's operadic categories. We describe an order on the faces of graph-assoc iahedra\, different from the one given by Carr and Devadoss\, which allows us to construct a standard triangulation of graph associahedra\, followin g [2].\n\n(joint work with Stefan Forcey)\n\n[1] M. Carr\, S. Devadoss\, < i>Coxeter complexes and graph associahedra\, Topol. and its Applic. 15 3 (1-2) (2006) 2155–2168.
\n[2] J.-L. Loday\, Parking functions and triangulation of the associahedron\, Proceedings of the Street’s fest 2006\, Contemp. Math. AMS 431 (2007)\, 327–340.\n LOCATION:https://researchseminars.org/talk/operad/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Ching (Amherst College) DTSTART:20200811T184500Z DTEND:20200811T194500Z DTSTAMP:20260422T212709Z UID:operad/4 DESCRIPTION:Title: G oodwillie calculus and operads\nby Michael Ching (Amherst College) as part of operad pop-up\n\n\nAbstract\nThe goal of this talk is to survey th e role of operads in Goodwillie’s calculus of functors. A key observatio n is that the derivatives of the identity functor\, on a suitable pointed $\\infty$-category $C$\, admit an operad structure which in the case of po inted spaces recovers a spectral version of the Lie operad. I will give a couple different ways to construct the operad structure in general\, and t hen focus on the case where $C$ is itself the $\\infty$-category of algebr as over some (stable\, non-unital) operad $P$. In that case\, the derivati ves of the identity functor on $C$ recover\, in some form\, the operad $P$ .\n LOCATION:https://researchseminars.org/talk/operad/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Syunji Moriya (Osaka Prefecture University) DTSTART:20200811T041500Z DTEND:20200811T051500Z DTSTAMP:20260422T212709Z UID:operad/5 DESCRIPTION:Title: A spectral sequence for cohomology of knot spaces\nby Syunji Moriya (Os aka Prefecture University) as part of operad pop-up\n\n\nAbstract\nThis ta lk is based on the preprint arXiv:2003.03815.\nLet $Emb(S^1\,M)$ be the sp ace of embeddings from $S^1$ to a closed manifold $M$ (space of knots in $ M$). Recently\, this space is studied by Arone-Szymik\, Budney-Gabai\, and Kupers\, using Goodwillie-Weiss embedding calculus. In this talk\, we int roduce a spectral sequence for cohomology of $Emb(S^1\,M)$ whose $E_2$-ter m has an algebraic presentation\, using Sinha's cosimplicial model which i s derived from the calculus. This converges to the correct target if $M$ i s simply connected and of dimension $\\geq 4$ for general coefficient rin g. Using this\, we see a computation of $H^*(Emb(S^1\,S^k\\times S^l))$ i n low degrees under some assumption on $k\,l$ and an isomorphism \n $\\pi_ 1(Emb(S^1\,M))\\cong H_2(M\,\\mathbb{Z})$ for some simply connected $4$-di mensional $M$. \n\nOur main idea of the construction is to replace conf iguration spaces in the cosimplicial model with fat diagonals via Poincar é Lefschetz duality. To do this\, we use a notion of a (co)module over an operad. A somewhat curious point is that we need spectra (in stable homot opy) even though our concern is singular cohomology.\n LOCATION:https://researchseminars.org/talk/operad/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Petersen (Stockholms universitet) DTSTART:20200811T090000Z DTEND:20200811T100000Z DTSTAMP:20260422T212709Z UID:operad/6 DESCRIPTION:Title: L ie\, associative and commutative quasi-isomorphism\nby Dan Petersen (S tockholms universitet) as part of operad pop-up\n\n\nAbstract\nLet A and A ' be commutative dg algebras over Q. There are two a priori different noti ons of what it means for them to be quasi-isomorphic: one could ask for a zig-zag of quasi-isomorphisms in the category of commutative dg algebras\, or a zig-zag in the larger category of not necessarily commutative dg alg ebras. Our first main result is that these two notions coincide. The secon d main result is Koszul dual to the first\, and states that if two dg Lie algebras over Q have quasi-isomorphic universal enveloping algebras\, then the derived completions of the two dg Lie algebras are quasi-isomorphic. The latter result is new even for classical Lie algebras concentrated in d egree zero. Both results have immediate consequences in rational homotopy theory. (Joint with Campos\, Robert-Nicoud\, Wierstra)\n LOCATION:https://researchseminars.org/talk/operad/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Luciana Basualdo Bonatto (University of Oxford) DTSTART:20200811T101500Z DTEND:20200811T111500Z DTSTAMP:20260422T212709Z UID:operad/7 DESCRIPTION:Title: A n infinity operad of normalized cacti\nby Luciana Basualdo Bonatto (Un iversity of Oxford) as part of operad pop-up\n\n\nAbstract\nNormalized cac ti are a graphical model for the moduli space of genus 0 oriented surfaces . They are endowed with a composition that corresponds to glueing surfaces along their boundaries\, but this composition is not associative. By intr oducing a new topological operad of bracketed trees\, we show that this op eration is associative up-to all higher homotopies and that normalized cac ti form an $\\infty$-operad in the form of a dendroidal space satisfying a weak Segal condition. In particular\, this provides one of the few exampl es in the literature of infinity operads that are not a nerve of an actual operad.\n LOCATION:https://researchseminars.org/talk/operad/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Joost Nuiten (Université de Montpellier) DTSTART:20200811T114500Z DTEND:20200811T124500Z DTSTAMP:20260422T212709Z UID:operad/8 DESCRIPTION:Title: M oduli problems for operadic algebras\nby Joost Nuiten (Université de Montpellier) as part of operad pop-up\n\n\nAbstract\nA classical principle in deformation theory asserts that any formal deformation problem over a field of characteristic zero is classified by a differential graded Lie al gebra. This principle has been described more precisely by Lurie and Pridh am\, who establish an equivalence between dg-Lie algebras and formal modul i problems indexed by Artin commutative dg-algebras. I will discuss an ext ension of this result to more general pairs of Koszul dual operads over a field of characteristic zero. For example\, there is an equivalence of inf inity-categories between pre-Lie algebras and formal moduli problems index ed by permutative algebras. Under this equivalence\, permutative deformati ons of a trivial algebra are classified by the usual pre-Lie structure on its deformation complex. In the case of the coloured operad for nonunital operads\, a relative version of Koszul duality yields an equivalence betwe en nonunital operads and certain kinds of operadic formal moduli problems. This is joint work with D. Calaque and R. Campos.\n LOCATION:https://researchseminars.org/talk/operad/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Vallette (Université Sorbonne Paris Nord) DTSTART:20200811T130000Z DTEND:20200811T140000Z DTSTAMP:20260422T212709Z UID:operad/9 DESCRIPTION:Title: D eformation theory of cohomological field theories\nby Bruno Vallette ( Université Sorbonne Paris Nord) as part of operad pop-up\n\n\nAbstract\nI will explain how to develop the deformation theory of cohomological field theories as a special case of a general deformation theory of morphisms o f modular operads. Two cases will be considered: a classical and a quantum one. Using ideas of Merkulov–Willwacher based on graphs complexes\, I w ill introduce and develop a new universal deformation group which acts fun ctorially via explicit formulas on the moduli spaces of gauge equivalence classes of morphisms of modular operads. In the classical case\, the actio n is trivial\; but in the quantum case\, this group contains the prounipot ent Grothendieck–Teichmüller group and its action is highly non-trivial even in the simplest case. Then\, I will enrich these graph complexes wit h characteristic classes coming from the geometry of the moduli spaces of curves and obtain in this way (rather surprisingly) a natural homotopy ext ension to Givental group action in the classical case\, and in the quantum case\, a huge group that includes both Givental and Grothendieck–Teichm üller groups. \n\nIt is a joint work with Volodya Dotsenko\, Sergey Shadr in\, and Arkady Vaintrob (arXiv:2006.01649).\n LOCATION:https://researchseminars.org/talk/operad/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathaniel Bottman (University of Southern California) DTSTART:20200811T200000Z DTEND:20200811T210000Z DTSTAMP:20260422T212709Z UID:operad/10 DESCRIPTION:Title: The relative 2-operad of 2-associahedra in symplectic geometry\nby Nat haniel Bottman (University of Southern California) as part of operad pop-u p\n\n\nAbstract\nThe Fukaya A-infinity category $\\mathrm{Fuk}(M)$ is a ri ch invariant of a symplectic manifold $M$\, and its manipulation and compu tation is a core focus of current symplectic geometry. Building on work of Wehrheim and Woodward\, I have proposed that the correct way to encode th e functoriality properties of $\\mathrm{Fuk}$ is by defining an "$(A_\\inf ty\,2)$-category" called Symp\, in which the objects are symplectic manifo lds and hom($M\,N$) is defined to be $\\mathrm{Fuk}(M^-\\times N)$. Underl ying the new notion of an $(A_\\infty\,2)$-category is a family of abstrac t polytopes called 2-associahedra\, which form a "relative 2-operad" (anot her new notion\, which is related to Batanin's theory of higher operads). I will describe all of these constructions from scratch\, without assuming any knowledge of symplectic geometry. This talk is based partly on joint work with Shachar Carmeli\, and I will mention related joint work with Ale xei Oblomkov.\n LOCATION:https://researchseminars.org/talk/operad/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitry Vaintrob (University of California\, Berkeley) DTSTART:20200811T224500Z DTEND:20200811T234500Z DTSTAMP:20260422T212709Z UID:operad/11 DESCRIPTION:Title: The operad of framed formal curves and a program of Kontsevich\nby Dmi try Vaintrob (University of California\, Berkeley) as part of operad pop-u p\n\n\nAbstract\nIt has long been conjectured (originally formalized by Ko ntsevich) that the operad of framed little disks can be enriched to an ope rad in an appropriate category of motives (in the sense of Grothendieck an d Voevodsky). I will explain such a construction\, in a motivic category a ssociated to logarithmic schemes (or more generally\, stratified formal sc hemes) and explain how (via ideas of Kontsevich\, Tamarkin\, Beilinson and others)\, this construction leads to a systematic resolution of several f ormality and deformation theoretic results\, including the Deligne formali ty conjecture and deformation-quantization previously proven via more tran scendental techniques.\n LOCATION:https://researchseminars.org/talk/operad/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Cepek (IBS Center for Geometry and Physics) DTSTART:20200811T013000Z DTEND:20200811T023000Z DTSTAMP:20260422T212709Z UID:operad/12 DESCRIPTION:Title: Configuration spaces of $\\mathbb{R}^n$ and Joyal’s category $\\mathbf{\ \Theta}_n$\nby Anna Cepek (IBS Center for Geometry and Physics) as par t of operad pop-up\n\n\nAbstract\nWe examine configurations of finite subs ets of Euclidean space within the homotopy-theoretic context of $\\infty$- categories by way of stratified spaces. Through these higher categorical m eans\, we identify the homotopy types of these configuration spaces in ter ms of the category $\\mathbf{\\Theta}_n$.\n LOCATION:https://researchseminars.org/talk/operad/12/ END:VEVENT END:VCALENDAR