Titles & Abstracts

Talks

Ryan Aziz (Université Libre de Bruxelles)
Quantum differentials on cross product Hopf algebras
Abstract. In the study of noncommutative geometry by quantum groups approach developed by Beggs, Majid, and many other collaborators, we start with a given algebra (which generalizes the role of the algebra C∞(M )), and equip it with a noncommutative differential calculus. From this data, we can start to build basic elements of noncommutative Riemannian geometry such as metric, connection, curvature, etc., all algebraically. A problem arises in this approach even in the first step, which is to choose a differential calculus on the given algebra, in particular in the case of our algebra being an inhomogeneous quantum group, where it can be written as a cross product Hopf algebra.
In this talk, I will try to explain my join work with Majid where we contruct a method to construct a canonical strongly bicovariant exterior algebra structure on various types of cross product Hopf algebras as a ‘super’ extension of the same flavour, but to save time, I will focus on biproducts A ⇆ A ⧔ B which coacts differentiably on its factor B. As an example, we find a strongly bicovariant exterior algebra on ℂ_q [P ] ≃ ℂ_q [GL₂] ⧔ ℂ²_q, which is a quantum deformation of maximal parabolic P ⊂ SL₃ and isomorphic to a quotient of ℂ_q [SL₃]. Moreover, from Manin, we know that the structure of ℂ_q [GL₂] is largely determined from its coaction on the quantum plane ℂ²_q. By requiring that this coaction is differentiable, we find that the structure of 4D strongly bicovariant Ω(ℂ_q [GL₂]) is largely determined by its coaction on Ω(ℂ²_q).
If time permits, I will also try to talk about further directions of this result, namely to study noncommutative geometry on quantum Poincaré groups where now its differential structure is nicely given by our construction.

Tiago Cruz (University of Stuttgart)
Ringel self-duality via relative dominant dimension
Abstract. It is well known that the blocks of the BGG category O are Ringel self-dual. In this talk, we discuss an integral version of Soergel's Struktursatz and deformations of the blocks of the BGG category O based on the work of Gabber and Joseph.
By studying relative dominant dimension over these deformations together with cover theory in the sense of Rouquier, we reprove Ringel-self duality of the blocks of the BGG category O.

Marvin Dippell (Universität Würzburg)
Towards an HKR-Theorem for coisotropic reduction
Abstract. Deformation quantization aims to construct a quantum analog of a classical phase space given by a Poisson manifold, by deforming the commutative multiplication on the classical observable algebra into a non-commutative, so called, star product. In classical mechanics symmetry reduction plays an important role, and it can be formalized in geometric terms using coisotropic reduction. In this talk I want to introduce a subcomplex of the classical Hochschild complex, called constraint Hochschild complex, which controls deformations compatible with the reduction data. As a first step towards a HKR-Theorem for this subcomplex we will compute the second constraint Hochschild cohomology class in the local situation. In the end I will give a glimpse into a general framework which allows to treat geometric and algebraic objects with reduction data on equal footing and might provide tools for the proof of a HKR-Theorem for coisotropic reduction.

Jiaqi Fu (Institut de Mathématiques de Toulouse)
Formal moduli problems and partition Lie algebras
Abstract. A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and dg Lie algebras over char 0. Brantner and Mathew extended the correspondence to arbitrary fields. To achieve this, they introduce an invariant of dg Lie algebras in char p > 0, that is partition Lie algebra. Then a Koszul duality functor from augmented simplicial algebras to partition Lie algebras is constructed. Finally, a correspondence between formal moduli problems and partition Lie algebras is established. In char 0, the theorem of Lurie and Pridham is recovered.

Wouter Rienks (University of Amsterdam)
Deformations of Fourier–Mukai transforms between Calabi–Yau varieties
Abstract. Let X and Y be Calabi–Yau varieties over a field of characteristic 0. Let X' and Y' be deformations of X and Y over some Artinian ring A. Suppose F is a Fourier–Mukai equivalence D(X ) → D(Y ) with Mukai vector v = v (F ) in the de Rham cohomology H*(X' ×Y' ). If, for all even 0 ≤ p ≤ 2d, the constant lift of v (F )ₚ remains within the F ᵖ-part of the (twisted) Hodge-filtration on H²ᵖ(X' ×Y' ), then in some cases it is known that F lifts to a Fourier–Mukai equivalence D(X' ) → D(Y' ). For example, the case of X and Y  K3-surfaces is treated in a paper by Addington and Thomas.

Philipp Schmitt (Leibniz Universität Hannover)
Strict quantization of polynomial Poisson structures
Abstract. The quantization problem is the problem of associating a non-commutative quantum algebra to a classical Poisson algebra in such a way that the commutator is related to the Poisson bracket. In a formal setting, this problem was solved for any Poisson manifold by Kontsevich, and for polynomial Poisson structures on ℝᵈ another construction was proposed by S. Barmeier and Z. Wang.
In this talk, we discuss the strict quantization of several quadratic Poisson structures. Using the approach by Barmeier and Wang, we obtain concrete formulas for formal star products, which make their convergence and continuity properties very accessible and allow us to obtain strict star products on certain classes of analytic functions. Surprisingly, the strict deformations are defined for larger classes of functions than similar deformations of constant or linear Poisson structures.
This is joint work with S. Barmeier.

Karandeep Singh (KU Leuven)
Stability results in geometry and differential graded Lie algebras
Abstract. When deforming a structure Q, a natural question is when any structure near Q of the same type shares a property with Q. If the deformations of Q are parametrized by Maurer–Cartan elements of a differential graded Lie algebra 𝔤, this property can sometimes be encoded in a differential graded Lie subalgebra 𝔥 ⊂ 𝔤.
Therefore, if Q ∈ 𝔥, the question can be phrased as when any Maurer–Cartan element Q' ∈ 𝔤 near Q lies in 𝔥 up to gauge equivalence, or in other words, when the inclusion map 𝔥 → 𝔤 is locally surjective around Q on equivalence classes of Maurer–Cartan elements. A geometric example of this problem is when given a Poisson structure π, and a symplectic leaf S of π, when any Poisson structure π' near π has a symplectic leaf S'  diffeomorphic to S, as studied by M. Crainic and R. Fernandes.
Assume that the codimension of 𝔥 in 𝔤 is finite. We show that the vanishing of a (finite-dimensional) cohomology group implies that any Maurer–Cartan element Q' ∈ 𝔤 near Q ∈ 𝔥 is gauge equivalent to a Maurer–Cartan element in 𝔥. As applications, we recover the stability results of Crainic–Fernandes and Dufour–Wade for (higher order) singular points of Poisson structures, as well as give a stability criterion for singular points of Dirac structures in split Courant algebroids.