Graz Algebra and Geometry Seminar

Graz Algebra & Geometry Seminar

In Winter term 24, we meet on Tuesdays 16:00 -- 17:00 in SR 11.32 (Heinrichstr. 36, 3rd floor).
IMPORTANT: For talks marked "ONLINE" we do not meet in person! We meet here: https://unimeet.uni-graz.at/b/mar-bte-gbo-lyr

Everybody is very welcome to attend!

26.08.2024

Polynomials induced from matroids, and their singularities, and why Feynman is interested

Abstract: We discuss a number of constructions that start with a matroid or a realization thereof, and lead to a hypersurface. Some of these are very old, some are rather new. We discuss when these polynomials are irreducible, and engage in a discussion on the "badness" of the singularity. This may involve the Frobenius morphism, or jet spaces. Part of the talk is devoted to (my elementary understanding of) Feynman diagrams and integrals, and what it has to do with matroid polynomials.

08.10.2024

Derived categories of singular varieties

Abstract: We give a short introduction to semiorthogonal decompositions of derived categories and discuss the state of the art of semiorthogonal decompositions of derived categories of singular varieties. More concretely, we explain obstructions for such decompositions, and we also give concrete examples with interesting decompositions and hint at a general construction.

15.10.2024 (ONLINE)

A geometric model for semilinear gentle algebras

Abstract: Semilinear gentle algebras are path algebras over a division ring, where the underlying quiver with relations is subject to restriction similar to those for a “classical” gentle algebra. They are a type of semilinear clannish algebras studied by Bennett-Tennenhaus and Crawley-Boevey. Classically, one can model gentle algebras using geometric means. In this talk, we explain how we can extend these models to the semilinear case. Along the way we will also discuss how semilinear gentle algebras are nodal, and demonstrate how the Zembyk decomposition for nodal algebras can be interpreted geometrically. This is based on joint work (arXiv 2402.04947) with Raphael Bennett-Tennenhaus, Karin Jacobsen, and Kayla Wright.

22.10.2024

Rings with extremal cohomology annihilator

Abstract: Cohomology annihilator of Noether algebras was defined by Iyengar and Takahashi in their work on strong generation in module category. For a commutative Noetherian local ring, it can be observed that the cohomology annihilator ideal is the whole ring if and only if the ring is regular. Motivated by this, I will consider the question: When is the cohomology annihilator ideal of a local ring the maximal ideal? I will discuss various ring theoretic and category theoretic conditions towards understanding this question. This will include results from two joint works, one with Hailong Dao and Monalisa Dutta, and another with Jian Liu.

05.11.2024

Silting reduction using generalised concentric twin cotorsion pairs

Abstract: Given a triangulated category T and a rigid subcategory R, Iyama--Yang put forward a mild technical condition that lets us compute the Verdier quotient T/thick(R). There are good reasons to extend Iyama--Yang's work to extriangulated categories, but one has to grapple with a more complicated theory of localisation. In this talk, we propose a generalisation of Iyama--Yang's work. More precisely, given an extriangulated category C and a rigid subcategory R giving rise to a generalised concentric twin cotorsion pair, we show that the Verdier quotient C/thick(R) can be expressed as an ideal quotient. Time permitting, we show that if C is 0-Auslander, in the sense of Gorsky--Nakaoka--Palu, it suffices that C admits Bongartz completions for our results to hold. Moreover, the Verdier quotient C/thick(R) then remains 0-Auslander. The talk will be based on Section 5 in [arXiv:2405.00593].

12.11.2024

Abstract representation theory via coherent Auslander-Reiten diagrams

Abstract: In this talk we explain a method to study representations of quivers over arbitrary stable infinity categories in terms of Auslander-Reiten diagrams.

Our techniques allow us to internally visualize (a significant piece of) the Auslander-Reiten quiver of the derived category of a hereditary finite-dimensional algebra inside much more general (infinity) categories of representations, such as representations over arbitrary rings, schemes, dg algebras, or ring spectra. This is provided by an equivalence with a certain 'mesh subcategory' of representations of the repetitive quiver, which we build inductively using abstract reflection functors.

As an application we obtain that the automorphism group of the non-regular components of the Auslander-Reiten quiver acts on the infinity category of representations over any stable infinity category. Moreover, this action allows us to shed some light on the Picard group of the quiver with coefficients in importante examples (a field, the integers, spectra).

19.11.2024

An explicit derived McKay correspondence for some reflection groups

Abstract: This talk is about an extension of the derived McKay correspondence for finite subgroups of SU(2) to complex reflection groups of rank 2 generated by reflections of order 2: The classical McKay correspondence connects the representation theory of a finite group in SU(2) and the geometry of the exceptional divisor of the minimal resolution of the corresponding quotient singularity. Kapranov and Vasserot showed that this may be realized as a derived equivalence between the derived category of coherent sheaves on the minimal resolution, and the derived category of equivariant coherent sheaves on the two-dimensional vector space the group is acting on. On the other hand, for a complex reflection group of rank 2 generated by order 2 reflections, the quotient is smooth. Using that each such reflection group contains a distinct subgroup of SU(2) as a subgroup of index 2, we derive a semi-orthogonal decomposition of the category of equivariant coherent sheaves, where the derived category of the quotient is one piece, and the other pieces are coming from branch divisors and exceptional objects. In particular, the total number of pieces of this decomposition is equal to the number of irreducible representations of the reflection group and it can be related to the (conjectured) motivic semi-orthogonal decomposition of the derived categories of equivariant coherent sheaves for the reflection groups of Polishchuk and Van den Bergh. This is joint work with A. Bhaduri, Y. Davidov, K. Honigs, P. MacDonald, E. Overton-Walker, and D. Spence.

03.12.2024 (ONLINE)

Dimension theory of noncommutative curves

Abstract: We discuss three notions of dimensions of triangulated categories. We provide a large class of examples for each notion. Further, we show these dimensions for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van den Bergh.

07.01.2025

Infinite symmetric group and concatenations of bordism of triangulated surfaces

Abstract: We say that infinite symmetric group is the group finitely supported permutations of natural numbers $\N$. We consider product $G$ of three copies of infinite symmetric group. Denote by $K$ the diagonal subgroup, which also is an infinite symmetric group. Denote by $K(n)$ the subgroup of $K$ fixing the first $n$ elements of $\N$. We show that double cosets $K(n)\backslash G/K(n)$ has a natural structure of a semigroup, and moreover there is a natural multiplication $K(n)\backslash G/K(m)\times K(m)\backslash G/K(l)\to K(l)$. We get a category and this category acts in any unitary representation of the group $G$.

We assign to each morhism of this category a triangulated surfaces colored black and white in checker order and show that morphisms corresponds to concatenation of surfaces.

The talk does not require any preliminary knowledge of representation theory.

14.01.2025

Subvarieties of polynomial representations and their singularities

This talk deals with subvarieties of polynomial representations, which allow for two equivalent descriptions. First, we can think of them as functors assigning each finite-dimensional vector space a finite-dimensional variety. Second, taking the limit gives an infinite-dimensional scheme endowed with an action of the infinite general linear group. Our main result asserts that there exists a good notion of singular locus in both settings. Moreover, we construct a uniform equivariant resolution of singularities for a special class of such subvarieties. This talk covers joint work with Alessandro Danelon, Jan Draisma and Andrew Snowden.

21.01.2025

Polytopes, graded rings and the projective coinvariant algebra

The coinvariant algebra, the quotient of the polynomial ring in n variables by the ideal generated by positive degree symmetric polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group, equipping its regular representation with a graded algebra structure. Based on an idea from algebraic geometry, this talk will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra. This algebra gives a bigraded structure on the regular representation of S_n with interesting Frobenius character, generalising a classical result of Lusztig and Stanley. I will also explain connections to Ehrhart theory, the theory of counting lattice points in polytopes. The talk will partially be based on joint work with Praise Adeyemo, University of Ibadan, and my PhD student Fabian Levican.

28.01.2025 (ONLINE)

On the geometry of the Hilbert scheme, from the viewpoint of finite subgroups of SL2(C)

Abstract: The Hilbert scheme of points in the plane is naturally equipped with an action of GL2(C). We will begin by presenting the decomposition into irreducible components of the fixed- point locus of the Hilbert scheme under the action of a finite subgroup of SL2(C). The irre- ducible components will then be described as Nakajima quiver varieties over the McKay quiver. Along the way, we will provide a combinatorial description in terms of roots in the affine root lattice of the indexing set these irreducible components. We will then detail a result obtained with Gwyn Bellamy concerning the isotypic decompo- sition of the fibers of the restriction of the Procesi bundle to the fixed-point locus. This theorem generalizes the result of Bonnafé, Lehrer, and Michel on the coinvariant algebra of the symmetric group.

Organizers: E. Faber, M. Kalck