Workshop on
Quiver Grassmannians and their Applications

University of Wuppertal, Germany

Tuesday 21st March - Friday 24th March 2017

All lectures take place in Building K at Campus Grifflenberg, Room K.11.20 (Seminarraum K5).

---> Plan of Campus Grifflenberg

To get there take a bus with number 615 (to Schulzentrum Süd) or 645 (to Remscheid Friedrich-Ebert-Platz) from bus station "Ohligsmühle" (closest to Hotel Astor) or bus station "Historische Stadthalle" (closest to Wuppertal Hauptbahnhof) to bus station "Universität".

From the train station Wuppertal Hauptbahnhof to Hotel Astor it is five minutes walk: leave the train station and follow the signs into direction city center, after crossing the river Wupper you reach the Schwebebahnstation "Haubtbahnhof" which is directly next to the Hotel Astor at Schlossbleiche 4-6.

---> Here you can find some information about Wuppertal

The idea of the workshop is to bring together experts working on the geometry of quiver Grassmannians and/or applying quiver Grassmannians to representation theory and cluster algebra theory. But we also want to encourage mathematicians who want to learn more about this subject to participate. There will be introductory talks by Giovanni Cerulli Irelli and Dylan Rupel. The number of talks will be kept low to allow for ample time for discussions.

---> Preliminary schedule

Speakers include:

Giovanni Cerulli Irelli

Frédéric Chapoton

Evgeny Feigin

Hans Franzen

Lutz Hille

Andrew Hubery

Igor Makhlin

David Pescod

Dylan Rupel

Julia Sauter

Organisers: Oliver Lorscheid (oliver<at>impa.br), Markus Reineke (markus.reineke<at>ruhr-uni-bochum.de), Thorsten Weist (weist<at>uni-wuppertal.de)

If you like to participate, please send an email to one of the organisers.

Talks and abstracts

Giovanni Cerulli Irelli: Cellular decompositions of quiver Grassmannians

I will report on an ongoing project with Markus Reineke, Hans Franzen and Francesco Esposito. We show that every quiver Grassmannian associated with a rigid representations of an acyclic quiver has property (S), which means no homology in odd degrees, no torsion in even degrees, and that the cycle map is an isomorphism. This improves a result of Nakajima who showed that such varieties have no odd cohomology. In case the quiver is affine, we improve this result by showing that for representations of such quivers every quiver Grassmannian admits a decomposition into affine spaces.

Frédéric Chapoton: Tree-shaped quivers and their cluster varieties

I will explain my work on the cluster varieties that can be attached to quivers that are trees, in particular about their number of points over finite fields and their cohomology.

Evgeny Feigin: From flag varieties to toric varieties via quiver Grassmannians

I will review the connection between the PBW degeneration of the flag varieties and the theory of quiver Grassmannians.

I will also explain how the degenerate flag varieties can be further degenerated to the toric varieties.

Hans Franzen: Cohomological properties of quiver Grassmannians

Lutz Hille: Moduli Spaces of Quiver Representations, Inverse Limits, and Quiver Grassmannian

Dylan Rupel:Categorification of Rank 2 Generalized Cluster Algebras

University of Wuppertal, Germany

Tuesday 21st March - Friday 24th March 2017

All lectures take place in Building K at Campus Grifflenberg, Room K.11.20 (Seminarraum K5).

---> Plan of Campus Grifflenberg

To get there take a bus with number 615 (to Schulzentrum Süd) or 645 (to Remscheid Friedrich-Ebert-Platz) from bus station "Ohligsmühle" (closest to Hotel Astor) or bus station "Historische Stadthalle" (closest to Wuppertal Hauptbahnhof) to bus station "Universität".

From the train station Wuppertal Hauptbahnhof to Hotel Astor it is five minutes walk: leave the train station and follow the signs into direction city center, after crossing the river Wupper you reach the Schwebebahnstation "Haubtbahnhof" which is directly next to the Hotel Astor at Schlossbleiche 4-6.

---> Here you can find some information about Wuppertal

The idea of the workshop is to bring together experts working on the geometry of quiver Grassmannians and/or applying quiver Grassmannians to representation theory and cluster algebra theory. But we also want to encourage mathematicians who want to learn more about this subject to participate. There will be introductory talks by Giovanni Cerulli Irelli and Dylan Rupel. The number of talks will be kept low to allow for ample time for discussions.

---> Preliminary schedule

Speakers include:

Giovanni Cerulli Irelli

Frédéric Chapoton

Evgeny Feigin

Hans Franzen

Lutz Hille

Andrew Hubery

Igor Makhlin

David Pescod

Dylan Rupel

Julia Sauter

Organisers: Oliver Lorscheid (oliver<at>impa.br), Markus Reineke (markus.reineke<at>ruhr-uni-bochum.de), Thorsten Weist (weist<at>uni-wuppertal.de)

If you like to participate, please send an email to one of the organisers.

Talks and abstracts

Giovanni Cerulli Irelli: Cellular decompositions of quiver Grassmannians

I will report on an ongoing project with Markus Reineke, Hans Franzen and Francesco Esposito. We show that every quiver Grassmannian associated with a rigid representations of an acyclic quiver has property (S), which means no homology in odd degrees, no torsion in even degrees, and that the cycle map is an isomorphism. This improves a result of Nakajima who showed that such varieties have no odd cohomology. In case the quiver is affine, we improve this result by showing that for representations of such quivers every quiver Grassmannian admits a decomposition into affine spaces.

Frédéric Chapoton: Tree-shaped quivers and their cluster varieties

I will explain my work on the cluster varieties that can be attached to quivers that are trees, in particular about their number of points over finite fields and their cohomology.

Evgeny Feigin: From flag varieties to toric varieties via quiver Grassmannians

I will review the connection between the PBW degeneration of the flag varieties and the theory of quiver Grassmannians.

I will also explain how the degenerate flag varieties can be further degenerated to the toric varieties.

Hans Franzen: Cohomological properties of quiver Grassmannians

Lutz Hille: Moduli Spaces of Quiver Representations, Inverse Limits, and Quiver Grassmannian

**s**

Andrew Hubery: Grassmannians of submodules for hereditary algebras

Using Ringel-Hall algebras, we show that the number of rational points over a finite field of a quiver Grassmannian for a rigid module is given by a polynomial in the size of the field. This, together with the fact that such quiver Grassmannians are smooth, implies that they have vanishing odd cohomology and positive Euler characteristic, given by evaluating this polynomial at one. In fact, these results extend to all finite dimensional hereditary algebras over finite fields.

Igor Makhlin: Recent results on FFLV bases and FFLV polytopesIgor Makhlin: Recent results on FFLV bases and FFLV polytopes

**Integer points in Feigin-Fourier-Littelmann-Vinberg polytopes enumerate certain monomial bases in irreducible representations of type A and type C Lie algebras. These structures are closely related to the so-called Abelian degenerations of flag varieties that, in turn, constitute quiver Grassmannians. I will describe some recent progress in the study of FFLV bases and FFLV polytopes as well as the possible implications for the geometry of degenerate flag varieties. (Based on papers arXiv:1604.08844 and arXiv:1610.07984.)**

David Pescod: A Multiplication Formula for the Modified Caldero-Chapoton Map

Dylan Rupel:Categorification of Rank 2 Generalized Cluster Algebras

Since their introduction in 2001, cluster algebras have found applications across mathematics. In particular, the categorification of cluster algebras has provided a bridge to the representation theory of quivers and the geometry of quiver Grassmannians. More recently, examples have emerged which require a more general setup: the binomial exchange relations involved in the mutation process for cluster algebra seeds should be replaced by polynomial exchange relations. In this talk I will describe progress towards the construction of a categorification of these generalized cluster algebras. This is a report on joint work with Kiyoshi Igusa and Gordana Todorov

**Julia Sauter:**Desingularizations of quiver Grassmannians**Desingularizations of quiver Grassmannians for representation-finite algebras and more generally for modules with suitable finiteness conditions. This summarizes joint work with W. Crawley-Boevey and current work with M. Pressland. It generalizes work of Cerulli-Irelli-Feigin and Reineke for Dynkin quivers.**