May 16-20, 2022 @ University of British Columbia
|11:30 Seol, 12:00 Pietromonaco
Cubical approach to higher Lie groupoids
Attached to any nilpotent Lie algebra L is a Kan complex NL; this simplicial set, called the nerve of L, is naturally isomorphic to the nerve of the associated Lie group G(L) of L. In higher Lie theory, this construction generalizes: the nerve of a nilpotent differential graded Lie algebra L, or more generally, nilpotent L-infinity algebra, is a certain Kan complex NL. In these talks, we present a new approach to the construction of NL based on cubical instead of simplicial sets.
The construction of the cubical nerve of a nilpotent L-infinity algebra is modeled on the construction of the simplicial nerve, with the Dupont homotopy acting on differential forms on a simplex replaced by a considerably simpler homotopy acting on differential forms on a cube.
Next, we review the homotopy theory of cubical sets and the construction of an explicit functor from cubical Kan complexes to simplicial Kan complexes giving an equivalence of homotopy theories.
Our main result is that applying this functor to the cubical nerve of a nilpotent L-infinity algebra gives its simplicial nerve. The proof relies on Berglund’s homotopical perturbation theory for L-infinity algebras.
In the special case where L is a nilpotent differential graded Lie algebra concentrated in degrees > -2, the nerve is a 2-groupoid. Our main theorem allows us to identify this 2-groupoid with the Deligne 2-groupoid of L, as was proved by Yakutieli.
Monoidal categories, B-infinity structures and singularity categories
After a brief historical introduction to B-infinity structures, we will report on recent work by Lowen-Van den Bergh, who construct such structures on the derived endomorphism algebra of the unit in a suitably structured monoidal category. As an application, we will sketch the preservation of the Hochschild cochain complex (as a B-infinity algebra) under Koszul duality. Finally, if time permits, we will present a conjecture on the agreement of two B-infinity structures on the Hochschild cochain complex of a singularity category (with its canonical differential graded enhancement). These two structures are essentially due to Gerstenhaber (1963) and to Zhengfang Wang (2018).
Higher representations and Heegaard-Floer theory
|I will discuss 2-representation theory for the super-Lie algebra gl(1
|1). In that setting, one can define a tensor product that does not involve the A-infinity constructions required in the Kac-Moody case. I will explain how this construction provides a partial 2-dimensional topological field theory that extends Douglas, Lipshitz and Manolescu’s cornered Heegard-Floer theory. This is expected to extend to a 4-dimensional theory containing Ozsvath and Szabo’s theory. (joint work with Andy Manion)
A cubical approach to higher Lie theory
A bit loosely, by Lie integration we mean a process that associates a group-type object to some Lie algebra-type object, the prototypical example being the integration of ordinary Lie algebras to Lie groups. As another example, to a DG Lie algebra we can associate its Deligne groupoid, which plays a fundamental role in the applications to deformation theory. In the more general case of an L-infinity algebra Getzler showed, working simplicially, how to associate to it a Kan complex, whose higher homotopy groupoids are the appropriate higher generalizations of the (reduced) Deligne groupoid of a DG Lie algebra. After reviewing these constructions, we shall present an alternative approach to the one by Getzler, this time working cubically. This has some computational advantages, and as an application we shall discuss an explicitly computable generalization of the Baker-Campbell-Hausdorff product in any L-infinity algebra. Further applications include an explicit construction of the (reduced) Deligne groupoid and the (reduced) Deligne bigroupoid for any L-infinity algebra.
Variants of the Waldhausen S-construction
The S-construction, first defined in the setting of cofibration categories by Waldhausen, gives a way to define the algebraic K-theory associated to certain kinds of categorical input. It was proved by Galvez-Carrillo, Kock, and Tonks that the result of applying this construction to an exact category is a decomposition space, also called a 2-Segal space, and Dyckerhoff and Kapranov independently proved the same result for the slightly more general input of proto-exact categories. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that these results can be maximally generalized to the input of augmented stable double Segal spaces, so that the S-construction defines an equivalence of homotopy theories. In this talk, we’ll review the S-construction and the reasoning behind these stages of generalization. Time permitting, we’ll discuss work in progress with Stern in which we characterize those augmented stable double Segal spaces that correspond to cyclic spaces.
Infinitesimal deformations of semi-smooth varieties and moduli of Godeaux surfaces
This is a report on joint work with Marco Franciosi and Rita Pardini. A surface is called semi-smooth if its only singularities are a curve of nodes xy=0 and finitely many pinch points. We calculate the tangent and higher tangent sheaf in terms of the normalization map, we then apply Tziolas’ formal smoothability to deduce that semi-smooth Godeaux surfaces are smooth points in the moduli, and in the closure of smooth surfaces.
Mapping spaces for dg Hopf cooperads and homotopy automorphisms of the rationalization of E_n-operads
I will explain a joint work with Thomas Willwacher about the definition of a simplicial enrichment on the category of differential graded Hopf cooperads (dg Hopf cooperads), which serves as a model for the rational homotopy theory of operads in topological spaces. This construction enables us to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. To be specific, I will explain, as a main application, that the spaces of Maurer-Cartan forms on the Kontsevich graph complex Lie algebras are homotopy equivalent, in the category of simplicial monoids, to the homotopy automorphism spaces of the rationalization of the operads of little discs.
Poisson geometry of Moduli stack of complexes
The moduli stack of bounded complexes of vector bundles on a projective Calabi-Yau d-fold X defines a Lagrangian correspondence on moduli stack of perfect complexes on X, therefore admits a natural (1-d)-shifted Poisson structure. When X has dimension 1, components of this moduli stack include many classical Poisson varieties arising in integrable system and representation theory. Two notable examples are the Feigin-Odesskii manifolds and the Hilbert scheme of points on Fano surfaces. In this talk, I will introduce a construction called “Bosonization” of the moduli stack of complexes. It provides a powerful tool to analyse the Poisson structure. This is a joint work with Sasha Polishchuk.
L-infinity liftings of semiregularity maps
We introduce the notion of Chern–Simons classes for curved DG-pairs and we prove that a particular case of this general construction provides canonical L-infinity liftings of Buchweitz-Flenner semiregularity maps for coherent sheaves on complex projective manifolds. By general results this implies that semiregularity maps annihilates all the obstructions to deformations. Joint work with R. Bandiera and E. Lepri.
A Theory of Gopakumar-Vafa Invariants for Calabi-Yau Threefolds with an Involution
The Gopakumar-Vafa (GV) invariants are the “ideal” curve-counting invariants which conjecturally underlie the Gromov-Witten or Pandharipande-Thomas theories. A geometrical definition was recently proposed by Maulik-Toda. In this talk, I will describe a version of the GV invariants for Calabi-Yau threefolds with an involution. The quantities are interpreted as a virtual count of invariant curves. In addition to tracking the genus of the invariant curve, we track the genus of the quotient. The example I will focus on is a local Abelian or Nikulin K3 surface, where these new invariants are encoded into formulas analogous to the Katz-Klemm-Vafa formula.
Three homotopy theories on simplicial cocommutative coalgebras
I will discuss the construction and significance of three different homotopy theories on the category of simplicial cocommutative coalgebras corresponding to the following notions of weak equivalence:
Notion (1) was used by Goerss to provide a fully-faithful model for spaces up to F-homology equivalence, for a F an algebraically closed field. Notion (2) corresponds to a linearized version of the notion of categorical equivalence between simplicial sets studied by Joyal and Lurie. I will explain how notion (3) leads to a fully-faithful model for the homotopy theory of spaces up to maps that induce isomorphisms on fundamental groups and on the F-homology of the universal covers, for F an algebraically closed field. This is joint work with G. Raptis.
Formal exponential maps and the Atiyah class of dg manifolds
Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré–Birkhoff–Witt isomorphism and the complete symbols of differential operators.
We investigate the question on how to extend these maps to dg manifolds. As an application, we show there is an L-infinity structure on the space of vector fields in connection with the Atiyah class of a dg manifold.
In particular, for the dg manifold arising from a foliation, we induce an L-infinity structure on the deRham complex associated with the foliation. As a special case, it is related to Kapranov’s L-infinity structure on the Dolbeault complex of a Kähler manifold.
This is a joint work with Mathieu Stiénon and Ping Xu.
Categorical wall-crossing formula of Donaldson-Thomas theory and applications
Donaldson-Thomas invariants count stable sheaves on Calabi-Yau 3-folds and they satisfy wall-crossing formula under change of stability conditions. It is expected that there exist dg-categories which categorifies DT invariants, and categorical wall-crossing formula as semiorthogonal decomposition. In this talk, I will explain a construction of categorical DT invariants on local surfaces and some examples of their categorical wall-crossing formula. I will also explain that an idea of categorical wall-crossing provides several interesting semiorthogonal decompositions of derived categories of classical moduli spaces, e.g. symmetric products of curves, Brill-Noether loci, Hilbert schemes of points, Quot schemes, etc.
Involutions of Topological Azumaya Algebras
A topological Azumaya algebra is a bundle of algebras that is locally isomorphic to the algebra of square matrices. The structure groups of such algebras are the projective general linear groups. An involution of an Azumaya algebra is an additive self-map of order 2 that reverses the order of multiplication—the prototypical examples being transposition and hermitian conjugation of matrices. I will discuss the problem of classifying of involutions of Azumaya algebras, especially the problems that arise when the base space is not left fixed by the involution. I will also discuss the use of equivariant bundle theory to produce counterexamples in algebra.