Seminar
Friday, 09 June 2023
-
13:15, Room U2-113
Paolo Stellari (Milano): Deformations of stability conditions with applications to Hilbert schemes of points and very general abelian varieties
Abstract: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk I will illustrate a new and very effective technique based on deformations. A key ingredient is a general result about deformations of bounded t-structures (with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for some irreducible holomorphic symplectic manifolds, again in all possible dimensions. This is joint work with C. Li, E. Macri' and X. Zhao.
-
14:30, Room U2-113
Amnon Neeman (Canberra): On generalizations of an old theorem of Rickard's
Abstract: In 1989 Rickard published a couple of papers on Morita equivalence of derived categories. We will begin with a quick review of his results.
The interest in this old result was revived in 2018 by Krause, who asked for improvements. Krause's question almost immediately led to a couple of results we will briefly review.
And then we will talk about very recent work, joint with Canonaco and Stellari, which goes much further in improving Rickard's theorem. This is work still in progress, in the sense that we are trying to sharpen and generalize the theorems that we can already prove.
-
16:00, Room U2-113
Wilberd van der Kallen (Utrecht): Reductivity and finite generation
Abstract: Recall that the first fundamental theorem of invariant theory is about finite generation of the subalgebra of invariants, when a reductive group acts on an algebra. We replace the group with an affine group scheme G and also ask about finite generation of the G-cohomology of the algebra.
For a regular email announcement please contact birep.
Future Talks
Thursday, 15 June 2023
Friday, 16 June 2023
Wednesday, 21 June 2023
-
Sondre Kvamme (Trondheim): tba
-
Karin Erdmann (Oxford): tba
Friday, 30 June 2023
-
13:15, Room U2-113
Aran Tattar (Köln): tba
-
14:30, Room U2-113
Wassilij Gnedin (Paderborn): tba
Friday, 07 July 2023
-
13:15, Room U2-113
Niko Naumann (Regensburg): Separable commutative algebras and Galois theory in stable homotopy theories
Abstract: We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We illustrate this with the motivating example of the stable module category of a finite gorup.
Seminar Archive
Friday, 26 May 2023
-
13:15, U2-113
Andrea Solotar (Buenos Aires): Strong stratifying Morita contexts
Abstract: We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context (after H. Bass) is an algebra built on the data of two algebras, two bimodules and two morphisms. For a strong stratifying Morita context - or equivalently for a strong stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. This is a work in collaboration with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos.
Wednesday, 17 May 2023
Friday, 12 May 2023
-
13:15, Room U2-113
Markus Schmidmeier (Boca Raton): Invariant subspaces of nilpotent linear operators: Level, mean and colevel
Abstract: Let S(n) be the category of all pairs X=(U,V) where V is a finite dimensional vector space with a nilpotent linear operator T which satisfies T^n=0, and where U is a T-invariant subspace of V. Note that S(n) is just the category of Gorenstein-projective modules over the triangular matrix ring with coefficients in k[T]/T^n.
We consider three invariants for a non-zero pair X: If u, v and w denote the dimensions of U, V and V/U and b is the number of Jordan-blocks of V, then the level, mean and colevel of X are p=u/b, q=v/b and r=w/b.
The pr-vector of X is just the pair (p,r); together, the pr-vectors of the indecomposable objects in S(n) form the pr-picture.
In my talk I discuss properties of level, mean and colevel, and how the pr-picture evolves as n increases.
This is a report about recent work with Claus Michael Ringel.
Friday, 05 May 2023
-
13:15, Room U2-113
Johannes Krah (Bielefeld): Phantoms and exceptional collections on rational surfaces
Abstract: A smooth projective rational surface over an algebraically closed field admits a full exceptional collection. Building on work of Hille--Perling, Perling, and Vial, we study mutations of (numerically) exceptional collections by analyzing the lattice theoretic behavior of the numerical Grothendieck group. On the one hand, we show that some results, known for del Pezzo surfaces, can be extended to the blow-up of the projective plane in 9 points in very general position. On the other hand, on the blow-up of 10 points in general position we construct an exceptional collection of maximal length which is not full. This disproves a conjecture of Kuznetsov and a conjecture of Orlov. As an application in representation theory, our example shows that the derived equivalent finite dimensional algebra carries a presilting object which is not a direct summand of a silting object.
-
14:30, Room U2-113
Lutz Hille (Münster): Polynomial invariants for triangulated categories with full exceptional sequences
Abstract: For a full exceptional sequence of vector bundles on the projective plane there is a remarkable equation, the so-called Markov equation, in terms of the ranks of the three vector bundles. This equation, slightly modified, has been used in a joint work with Beineke and Brüstle for cluster mutations for quivers with three vertices.
The aim of this talk is to define the natural generalization for full exceptional sequences with n members. This leads to the notion of a polynomial invariant, that is a polynomial in indeterminants x(i,j) for i < j between 1 and n. This allows to evaluate such a polynomial at any full exceptional sequence. We define a polynomial invariant to be a polynomial whose value does not depend on the full exceptional sequence, it only depends on the underlying category.
In the talk we define polynomial invariants, present several examples and relate them to the natural braid group action. Eventually, we classify all polynomial invariants.
Wednesday, 03 May 2023
Friday, 28 April 2023
-
13:15, Room U2-113
Chiara Sava (Prague): ∞-Dold-Kan correspondence via representation theory
Abstract: Both Happel and Ladkani proved that, for commutative rings, the quiver An is derived equivalent to the diagram generated by An where any composition of two consecutive arrows vanishes. We give a purely derivator-theoretic reformulation and proof of this result showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory; in fact we will see how our result is a derivator-theoretic version of the ∞-Dold-Kan correspondence for bounded cochain complexes.
Wednesday, 26 April 2023
Friday, 21 April 2023
-
13:15, Room U2-113
Edmund Heng (Bures-sur-Yvette): Quiver representations in fusion categories; extending Gabriel’s theorem to ABCDEFG and HI!
Abstract: One of the most celebrated theorem in quiver representations is undoubtedly Gabriel’s theorem, which reveals a deep connection between quiver representations and root systems arising from Lie algebras. It shows that the finite-type quivers are classified by the ADE Dynkin diagrams and the indecomposable representations are in bijection with the underlying positive roots. Following the works of Dlab—Ringel, the classification can be generalised to include all the other Dynkin diagrams (BCFG) by considering the more general notion of valued quivers (K-species) representations instead.
However, from the point of view of Coxeter theory, which also have well-defined root systems extending those of ADE types, the (non-crystallographic) types H and I are missing from the classification. The aim of this talk is to introduce a new notion of representations for a class of (labelled) quivers known as Coxeter quivers, where their representations are built in certain fusion categories. The relevance to Gabriel’s theorem is then as follows: a Coxeter quiver has finitely many indecomposable representations if and only if its underlying graph is a Coxeter-Dynkin diagram — including types H and I. Moreover, the indecomposable representations of a Coxeter quiver are in bijection with extended the positive roots of the underlying root system of the Coxeter group.
Wednesday, 19 April 2023
Friday, 14 April 2023
-
13:15, Room U2-113
Alexander Pütz (Bochum): GKM Theory for Quiver Grassmannians
Abstract: Projective varieties have various realisations as quiver Grassmannians. Finding good realisations and nice torus actions on the corresponding quiver Grassmannians allows to compute torus equivariant cohomology using GKM theory. So far we have established nice torus actions on quiver Grassmannians for equioriented quivers of type A and nilpotent representations of the equioriented cycle. This leads to a combinatorial description of torus fixed points and one-dimensional orbits. We believe that a similar construction also works for other quivers.
Friday, 20 January 2023
-
13:15, Room V5-148
Markus Schmidmeier (Boca Raton): Hammocks to visualize the support of finitely presented functors
Abstract: Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra.
When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor.
Examples include quiver representations and invariant subspaces of nilpotent linear operators.
Friday, 16 December 2022
-
13:00, Room V5-148
Daniel Bissinger (Kiel): Kronecker representations and Steiner bundles
Abstract: Let d < r be natural numbers, K_r be the generalized Kronecker algebra with arrow space A_r and Gr_d(A_r) be the Grassmannian of d-planes.
Jardim and Prata have shown that the category of Steiner bundles on Gr_d(A_r) is equivalent to a full subcategory of mod K_r.
We identify the objects of this category as relative projective Kronecker representations and give a homological description of the subcategory.
Then we explain by means of examples how questions regarding bundles can be answered in mod K_r and vice versa.
This talk is based on joint work with Rolf Farnsteiner.
For information on earlier talks please check the complete seminar archive.
