GAP XV — State College

Schedule

Long talks: 50 minutes + questions, Short talks: 25 minutes + questions.

Downloads: Schedule and Abstracts.

Mini-courses

Pavel Etingof (MIT):
D-modules on Poisson varieties, Poisson homology, and symplectic resolutions

In this minicourse I will explain how to use the theory of D-modules to study the zeroth Poisson homology (i.e., Poisson traces) of Poisson algebraic varieties. After reviewing the theory of D-modules, I will explain how to prove that the zeroth Poisson homology of varieties with finitely many symplectic leaves is finite, by defining a D-module whose solutions are invariants of Hamiltonian flows. More generally the technique applies to coinvariants of Lie algebra actions on varieties. In the case of varieties admitting symplectic resolutions, I will describe conjectures identifying the result with the cohomology of the resolution and their status. In the case of complete intersection surfaces with isolated singularities, I will describe the D-module, in which the Milnor numbers and genera of the singularities are encoded. The lectures will be based on my joint work with Travis Schedler, reviewed in the paper arXiv:1705.00423.

Marco Manetti (Rome Sapienza):
Homotopy theory of DG-Lie algebras and formality criteria

The aims of the lectures are:

1) to give an elementary description of the homotopy categories of differential graded Lie algebras over a field of characteristic 0, together with some applications to deformation theory;

2) to give some necessary and/or sufficient conditions on a DG-Lie algebra in order to be homotopy equivalent to its cohomology Lie algebra.

Boris Tsygan (Northwestern University):
Topics on noncommutative geometry and deformation quantization

I will discuss deformation quantization of symplectic manifolds and its enhanced version that allows for some nonlocal effects. I will do virtually everything on the examples of the plane, the sphere, and the two-torus. Time permitting, I will end with a discussion of noncommutative forms and their relation to Hochschild chains, as well as a notion of quantum Hamiltonian action and quantum Hamiltonian reduction for Hopf algebras.

Plenary talks

Vladimir Baranovsky (UC Irvine):
Quantization of vector bundles on coisotropic subvarieties

We consider the situation of a smooth algebraic variety with an algebraic symplectic form. Assume that an (untwisted) quantization of the structure sheaf exists, we discuss when a vector bundle on a coisotropic subvariety admits a deformation quantization to a module over the quantized functions.

Yuri Berest (Cornell):
Perverse sheaves and knot contact homology

Knot contact homology is a powerful geometric invariant of knots K in R3 obtained by Floer-theoretic counting of pseudoholomorphic disks associated to the (unit) conormal bundle ST^∗_KR^3 of K and depending on the isotopy type of ST^∗_KR^3 as a Legendrian submanifold of ST^∗R^3. In its simplest form, this invariant was introduced by L. Ng and has been extensively studied in recent years.

In this talk, we will give a new, purely algebraic construction of knot contact homology based on the homotopy theory of (small) DG categories. For a link L in R^3, we define a differential graded k-category A_L with a distinguished object, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of the distinguished object of A_L coincides with a geometric DG algebra model of the knot contact homology of L constructed by Ekholm, Etnyre, Ng and Sullivan (2013). The input of our construction is a natural action of the Artin braid group B_n on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson and Vilonen (1996). As an application, we show that the category of finite-dimensional representations of the homology category H_0(A_L) is equivalent to the category of perverse sheaves on R^3 that are singular along the link L.

Martin Bojowald (Penn State):
States in non-associative quantum mechanics

This talk introduces the mathematical notions relevant for an algebraic formulation of quantum mechanics. Open questions in an extension to non-associative algebras of observables are then discussed.

Vasily A Dolgushev (Temple University):
A Recursive Construction of Formality Quasi-isomorphisms for Algebraic Operads

Formality for differential graded (dg) operads (and other algebraic structures) is a subtle phenomenon. Currently, there are no effective tools for determining whether a given dg operad is formal or not. Moreover, in various interesting examples, all known proofs of formality require transcendental tools. In my talk, I will describe a class of formal dg operads which admit a recursive construction of formality quasi-isomorphisms. I will also present a software package for computing a formality quasi-isomorphism for the dg operad BRACES over rationals. My talk is based on joint paper arXiv:1610.04879 with Geoffrey Schneider.

Alexander Gorokhovsky (University of Colorado):
Pairings for pseudodifferential symbols

I will describe a comparison theorem for different constructions of cyclic cocycles on the algebra of complete symbols of pseudodifferential operators. This comparison result leads to index-theoretic consequences and a construction of invariants of the algebraic K-theory of the algebra of pseudodifferential symbols. This is a joint work with H. Moscovici.

Donatella Iacono (Bari):
Deformation of pairs via DG-Lie algebras

In this seminar, we focus our attention on deformation theory via Differential Graded Lie Algebras (DG-Lie Algebras). In particular, we study infinitesimal deformations of pairs (X, F), where F is a coherent sheaf on a smooth projective variety X, describing an explicit DG-Lie algebras that controls this deformation problem. We also analyse a trace map and its relation with deformations.

Ajay Ramadoss (Indiana University Bloomington):
Representation homology of spaces

We discuss representation homology of topological spaces, which is a higher homological extension of the representation varieties of fundamental groups. We give a natural interpretation of representation homology as functor homology and relate it to other homology theories associated with spaces (such as Pontryagin algebras and S^1-equivariant homology of free loop spaces). One of our main results, which we call the Comparison Theorem, computes the representation homology of any simply connected space of finite rational homotopy type in terms of its Quillen and Sullivan models. We also compute representation homology for some interesting examples, such as spheres, Riemann surfaces, complex projective spaces and link complements in R^3. While the representation homology of spheres and complex projective spaces is related to the strong Macdonald conjecture of Feigin and Hanlon, the representation homology of link complements is a new homological link invariant similar to knot contact homology. This is joint work with Yuri Berest and Wai-Kit Yeung.

Travis Schedler (Imperial College London):
Holonomic Poisson manifolds and deformations of elliptic algebras

I will introduce the notion of holonomic Poisson manifolds, which can be thought of as a refinement of the log symplectic condition, and is closely related to the flow of the modular vector field (which measures the failure of Hamiltonian flow to preserve volume). These manifolds are characterized by having finite-dimensional spaces of local deformations and quantizations. As an application, I will prove that the the first families of Feigin–Odesski elliptic algebras quantizing P^{2n} (and the corresponding Poisson structures) are universal deformations.

Jim Stasheff (University of Pennsylvania):
Foliations from a homotopy point of view

Foliations, even if regular, can have a topologically very bad quotient space of leaves. This suggests looking at a homotopy quotient, called “space of leaves”, instead and appropriate versions of multi-vector fields and differential forms on the “space of leaves”. Historically, this first occurred with the work of Batalin–Fradkin–Vilkovisky on the foliation of a constraint surface in a symplectic manifold by a set of first-class constraints. A key ingredient of the BFV construction is a resolution of A/I for the ‘constraint’ ideal I in the commutative algebra A together with a Chevalley–Eilenberg complex for I/I^2 as a Lie algebra. This led to the strong homotopy version of Lie–Rinehart algebras, introduced by Kjeseth. He constructed such an sh-Lie–Rinehart algebra structure on these two pieces. Much later, such algebras were used to study general regular foliations by Huebschmann (2003) and then Vitagliano (2012). Singular foliations were quite recently studied from a similar point of view by Lavau in his thesis. Our goal is to clarify the relation between the various models for such a “space of leaves”.

Alexander A. Voronov (University of Minnesota):
r_∞ -Matrices and quantum ∞-groups

The goal of the talk is to propose an ∞-version of Drinfeld's scheme of quantization. We introduce r∞-matrices, which are solutions to the Maurer–Cartan equation in the exterior algebra of a dg Lie algebra or, more generally, an L∞-algebra. This equation generalizes the classical Yang–Baxter equation [r,r]=0 in the exterior algebra of a Lie algebra. We show that such r∞-matrices give rise to triangular L∞-bialgebras and quantum ∞-groups. This is a joint work with Denis Bashkirov.

Chelsea Walton (Temple):
Poisson Geometry of 3-dimensional Sklyanin algebras

We give the 3-dimensional Sklyanin algebras S that are module-finite over their center Z the structure of a Poisson Z-order in the sense of Brown–Gordon. We show that the induced Poisson bracket on Z is non-vanishing and is induced by an explicit potential. This is used to analyze the finite-dimensional quotients of S by central annihilators: there are three distinct isomorphism classes of such quotients in the case when n divisible by 3, and two in the case when n is coprime to 3, where n is order of the elliptic curve automorphism associated to S. The Azumaya locus of S is determined, extending results of the speaker for the case when n is coprime to 3. This is joint work with Xingting Wang and Milen Yakimov.

Contributed talks

Olivier Elchinger (University of Luxembourg):
A remark on the L∞ perturbation lemma

The Perturbation Lemma is a result in homological algebra stating that given a contraction between two chain complexes and a perturbation of one differential, we can obtain a new contraction and a new perturbed differential on the other complex. (This is also known as homotopy transfer theorem.) This result was extended more generally for L∞ algebra and on other operads. The goal of this talk is to show that the original formulas given modified morphisms of chain complex automatically extend in maps preserving the L∞ structures. Thus we have explicit formulas that can be used to compute transferred L∞ structures in some examples.

Chiara Esposito (University of Würzburg):
Universal Deformation Formula, Formality and Actions

In this talk we provide a quantization via formality of Poisson actions of a triangular Lie algebra (g,r) on a smooth manifold M. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization of M together with a quantum group U_ℏ(g) and a map of associated DGLA's. This motivates a definition of quantum action in terms of L∞-morphisms which generalizes the one given by Drinfeld. This is a joint project with N. de Kleijn.

Anthony Giaquinto (Loyola University Chicago):
Peter–Weyl bases, preferred deformations, and Schur–Weyl duality

We discuss the deformed function algebra O_h(G) of a simply connected reductive Lie group G over C using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation in that basis, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections with earlier work on preferred deforations that involved Schur–Weyl duality.

Hsuan-Yi Liao (PSU):
Formality theorem for g-manifolds

To a g-manifold, a smooth manifold with a Lie algebra action, one can associate two dglas which play roles similar to the spaces of polyvector fields and polydifferential operators. We establish the formality theorem for g-manifolds, i.e., an L∞ quasi-isomorphism from the dgla analogous to polyvector fields to the dgla analogous to polydifferential operators. As an application, we obtain a Kontsevich–Duflo type theorem for g-manifolds. The parallel theorems for other cases will also be mentioned briefly.

Yipeng Mi (The University of Hong Kong):
Quantization of Poisson CGLs

Quantum CGLs, named after Cauchon, Goodearl, and Letzter, are a special class of quantum algebras that are iterated Ore extensions of associative algebras with compatible torus actions. Examples of quantum CGLs include quantum Schubert cells and quantized coordinate rings of double Bruhat cells. They have recently been studied extensively in connection with quantum groups and quantum cluster algebras. Semi-classical limits of quantum CGLs are Poisson CGLs, which are Poisson polynomial rings with compatible torus actions for which the Poisson bracket is an iterated Poisson–Ore extension. In this talk, I will describe an explicit procedure for constructing a quantum CGL from a Poisson CGL and prove that such a quantization is unique in a proper sense. I will give examples coming from the standard Poisson structures on Bott–Samelson varieties.

Yanli Song (Dartmouth College):
Geometric quantization of b-symplectic manifolds

b-symplectic manifolds are manifolds whose symplectic forms have singularities. In this talk, I will discuss a notion of pre-quantization for b-symplectic manifolds. I will also show how to use the APS type index and deformation of Dolbeault operators to construct a geometric quantization of b-symplectic manifolds, which are shown to be finite dimensional. Our results provide a solution to a conjecture by Guillemin–Miranda–Weitsman. This is a joint work with Braverman and Li.

Wai-kit Yeung (Indiana University Bloomington):
Relative Calabi-Yau completions

In this talk, I will discuss a generalization of Keller's Calabi–Yau completion to the relative context. This construction is applied to give a conjectural description of the partially wrapped Fukaya category of the conormal bundle of an embedded submanifold in terms of homotopy data. As special cases, this construction recovers the multiplicative preprojective algebra studied by Crawley–Boevey and Shaw, as well as knot contact homology studied by Ng.