Abstracts
Giovanni Cerulli Irelli (Sapienza University of Rome)
Title: Specialization map for quiver Grassmannians
Abstract: We define a specialization map between cohomology algebras of quiver Grassmannians of Dynkin type and we prove that it is surjective in type A, generalizing a beautiful result of Lanini and Strickland. This is a joint work with Francesco Esposito, Xin Fang and Ghislain Fourier. If time permits, I will shortly report on a second project in collaboration with Martina Lanini, Francesco Esposito and Rui Xiong in which we describe the kernel of the specialization map and thus the cohomology of quiver Grassmannians of type A in terms of Schubert calculus.
Karin Baur (Ruhr University Bochum)
Title: Frieze patterns and cluster theory
Abstract: Cluster categories and cluster algebras can be described via triangulations of surfaces or via certain diagrams. In type A, such triangulations lead to frieze patterns or SL_2-friezes in the sense of Conway and Coxeter. We explain how infinite frieze patterns arise and discuss their growth behaviour. In particular, we show that tame module categories yield friezes with linear growth.
Lars Göttgens (RWTH Aachen University)
Title: PBW deformations of U(g) smash products in OSCAR
Abstract: A filtered deformation $A_k$ of $A_0 = S(V)#U(g)$ for a semisimple Lie algebra $g$ is called PBW if its associated graded algebra is isomorphic to $A_0$. We present combinatorial and computer algebraic methods to compute PBW deformations for a given $A_0$ using diagrammatic categories, both on a theoretical level and as an implementation using OSCAR. Based on joint work with J. Flake.
Ulrich Krähmer (TU Dresden)
Title: The ring of differential operators on a monomial curve is a Hopf algebroid
Abstract: The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of D-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of D-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman.
Sven Möller (University of Hamburg)
Title: Vertex algebras associated with symplectic singularities
Abstract: Symplectic singularities (Beauville) appear prominently in (geometric) representation theory. In physics, they arise as the Higgs branch of 3d and 4d superconformal quantum field theories.
Vertex algebras are mathematical structures that axiomatise 2d conformal field theories. The associated Poisson variety, a certain geometric invariant defined by a vertex algebra, is often a symplectic singularity. In that case, the vertex algebra can be regarded as a chiral quantisation of the latter.
The 4d/2d correspondence in physics (Beem et al.) proposes vertex algebras as invariants of 4d superconformal theories. Crucially, it is conjectured that the Higgs branch of the 4d theory can be recovered as the associated variety of the vertex algebra. Therefore, all vertex algebras arising from 4d theories are supposed to be chiral quantisations of singular symplectic varieties. In this talk, I will report on work related to vertex algebras for one family of symplectic singularities that can be realised as Nakajima quiver varieties
Arnaud Eteve (MPI Bonn)
Title: Jordan decomposition for representations of finite Lie groups
Abstract: Let $G$ be a reductive group over a finite field $k$, Jordan decomposition for the irreducible representations of the finite group $G(k)$ is a collection of two statements:
- there is a partition, into Lusztig series, of the set of irreducible representations of $G(k)$ indexed by conjugacy classes of semisimple elements in the Langlands dual group $G^*$. The part corresponding to $1 \in G^*$ is usually called the set of unipotent representations.
- there is a canonical bijection between the Lusztig series corresponding to a semisimple element $s \in G^*$ and the unipotent series for a smaller group.
In this talk, I will discuss a categorical form of this statement which remains valid for modular coefficients.
Istvan Heckenberger (Philipps University of Marburg)
Title: Braidings and Nichols algebras
Abstract: In the talk I will discuss braided Hopf algebras and their relationship with other mathematical areas, with particular emphasis on structural results on Nichols algebras related to reflection groups, combinatorics of words, and modular representation theory.