Seminar
No talks have been announced for this week.
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Seminar Archive
Friday, 16 July 2021
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13:15, Zoom
Joseph Chuang (London): Rank functions on triangulated categories
Abstract: A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category satisfying the triangle inequality on distinguished triangles. Rank functions on the perfect derived category of a ring are related to Sylvester rank functions on finitely presented modules, and therefore, via the work of Cohn and Schofield, to representations of the ring over skew fields. In my talk I will focus on examples. This is joint work with Andrey Lazarev.
Friday, 09 July 2021
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13:15, Zoom
Paolo Stellari (Milano): Uniqueness of enhancements: derived and geometric categories
Abstract: In this talk we address several open questions and generalize the existing results about the uniqueness of enhancements for triangulated categories which arise as derived categories of abelian categories or from geometric contexts. If time permits, we will also discuss applications to the description of exact equivalences. This is joint work with A. Canonaco and A. Neeman.
Friday, 25 June 2021
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13:15, Zoom
Martin Kalck: A surface and a threefold with equivalent singularity categories
Abstract: We start with an introduction to singularity categories and equivalences between them. In particular, we recall known results about singular equivalences between commutative rings, which go back to Knörrer, Yang, Kawamata and a joint work with Karmazyn. Then we explain a new singular equivalence between an affine surface and an affine threefold. This seems to be the first (non-trivial) example of a singular equivalence involving rings of even and odd Krull dimension.
Friday, 18 June 2021
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13:15, Zoom
Norihiro Hanihara (Nagoya): Yoneda algebras from additive generators
Abstract: Yoneda algebras form a class of algebras which have widely been studied in ring theory and representation theory. They are defined for a ring A and an A-module M as the direct sum of Ext^i_A(M,M) over all i endowed with the Yoneda product. We discuss these Yoneda algebras in the following setting: A is a finite dimensional algebra of finite representation type, and M is the additive generator for the category of A-modules. We will give some fundamental results on such Yoneda algebras, such as coherence, Gorenstein property, periodicity, and a description of the stable category of Cohen-Macaulay modules.
Friday, 11 June 2021
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13:15, Zoom
Job Rock (Boston): Composition series of arbitrary cardinality
Abstract: We discuss a generalization of the notion of a composition series in an abelian category to one of arbitrary cardinality. Then we discuss sufficient axioms that yield "Jordan—Hölder—Schreier like" theorems. Examples of these settings include pointwise finite-dimensional persistence modules and Prüfer modules. We will conclude with evidence that suggests the axioms are necessary for our "Jordan—Hölder—Schreier like" theorems. This is joint work with Eric J. Hanson.
Friday, 30 April 2021
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13:15, Zoom
Sondre Kvamme (Uppsala): Admissibly presented functors
Abstract: Wanting to extend the functorial approach of Auslander to exact categories, we introduce the category of admissibly presented functors mod_{adm}(E) for an exact category E. Using this category, we extend Auslanders formula from abelian to exact categories. Furthermore, we characterize exact categories equivalent to categories of the form mod_{adm}(E), and we show that they have properties similar to module categories of Auslander algebras. For a fixed idempotent complete category C, we use this construction to show that exact structures on C are in bijection with certain resolving subcategories of mod C, and we compare this with the bijection to certain Serre subcategories of mod C due to Enomoto. This is joint work with Ruben Henrard and Adam-Christiaan Van Roosmalen.
Friday, 23 April 2021
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13:15, Zoom
Rene Marczinzik (Stuttgart): Homological algebra and combinatorics
Abstract: We show that the incidence algebra of a finite lattice L is Auslander regular if and only if L is distributive. As an application we show that the order dimension of L coincides with the global dimension of its incidence algebra when L has at least two elements and we give a categorification of the rowmotion bijection for distributive lattices. At the end we discuss the Auslander regular property for other objects coming from combinatorics. This is joint work with Osamu Iyama.
We also report on recent joint work with Aaron Chan, Erik Darpö and Osamu Iyama on fractionally Calabi-Yau algebras and their trivial extension algebras with relations to combinatorics and lattices.
Friday, 16 April 2021
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14:15, Zoom
Xiao-Wu Chen (Hefei): Skew group categories, algebras associated to Cartan matrices and folding of root lattices
Abstract: The folding of root lattices is fundamental in Lie theory when getting from the simply-laced cases to the non-simply-laced cases. Following Gabriel and Geiss-Leclerc-Schroer, the relevant root lattices are categorified by certain module categories. We obtain a categorification of the folding projection, namely a certain functor between the module categories whose K_0-shadow is the folding projection. The main tools are skew group categories and finite EI categories of Cartan type. This is joint with Ren Wang at USTC.
Friday, 12 February 2021
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15:15, Zoom
Martin Gallauer (Oxford): Cohomological singularity
Abstract: Let M be a representation of a finite group with coefficients in a ring. I want to discuss the following slogan: M is controlled by permutation modules if and only if its cohomology is non-singular. This is joint work with Paul Balmer whose talk at ICRA2020 I intend to complement with this discussion.
Saturday, 30 January 2021
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16:15, Zoom
Joshua Hunt: Decompositions of the stable module ∞-category
Abstract: The Picard group of the stable module category of a finite group G has been studied by representation theorists as the group of "endotrivial modules". In this talk, I will outline an approach to studying endotrivial modules via descent, using the fact that we can decompose the stable module category as a limit of ∞-categories. This is joint work with Tobias Barthel and Jesper Grodal.
Friday, 22 January 2021
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15:15, Zoom
Gustavo Jasso (Bonn): Universal properties of derived categories, after Lurie
Abstract: Let G be a Grothendieck category. The derived category D(G) of G and the homotopy category K(Inj G) of complexes of injective objects in G play important roles in representation theory and algebraic geometry. In this talk I will explain–following Lurie–the strong universal properties enjoyed by the infty-categorical refinements of D(G) and K(Inj G) as well as how these relate to the construction of realisation functors and derived equivalences.
Knowledge of infinity-category theory will not be assumed.
Friday, 15 January 2021
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15:15, Zoom
Teresa Conde (Stuttgart): Quasihereditary algebras with exact Borel subalgebras
Abstract: Exact Borel subalgebras of quasihereditary algebras emulate the role of "classic" Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra A has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to A that has a (regular) exact Borel subalgebra. Despite that, an explicit characterisation of such "special" Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko's work. In this talk, I shall present a numerical criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and I will provide a method to compute all Morita representatives of A that have a regular exact Borel subalgebra. We shall also see that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra A only depends on the composition factors of the standard and costandard A-modules and on the dimension of the Hom-spaces between standard A-modules. I will conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.
Friday, 18 December 2020
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17:15, Zoom
Benjamin Briggs (Salt Lake City): Stable invariance of structures on Hochschild Cohomology
Abstract: If two self-injective finite dimensional algebras are connected by a stable equivalence of Morita type then their Hochschild Cohomology algebras are isomorphic in positive degrees.
In characteristic p this positive part of Hochschild cohomology is actually a restricted graded Lie algebra. Since the restricted structure is nonlinear it can be difficult to handle functorially, and so Linckelmann asked whether this too passes across a stable equivalence of Morita type. I'll talk about sone work with Rubio y Degrassi where we answer this question and give a few applications.
If there’s time I’ll talk about ongoing work with Rubio y Degrassi and Saorín which connects this with the fundamental group of a finite dimensional algebra.
Friday, 11 December 2020
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14:30, Zoom
Rudradip Biswas (Manchester): Generation of derived and stable categories for groups in Kropholler's hierarchy
Abstract: We will look at the generation of a range of derived categories of modules over groups in Peter Kropholler's hierarchy. For this, we will be mostly using the language of generation in triangulated categories with localizing and colocalizing subcategories. We'll then look at a range of interesting applications and questions related to these results. And finally, we will prove a couple of generation properties of the stable module category of groups in these hierarchies that admit complete resolutions. Such stable module categories (note that the groups here need not be finite) have only recently been defined and studied by Mazza and Symonds.
Friday, 04 December 2020
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14:30, Lecture Hall H12
Greg Stevenson (Glasgow): Differential graded algebras with finite dimensional cohomology
Abstract: It is not necessarily the case that a dg algebra with finite dimensional cohomology is quasi-isomorphic to one which is honestly finite dimensional. However, in the case that the cohomology is concentrated in negative cohomological degrees there is always a finite dimensional model. I'll explain how to prove this, give some consequences, and (probably) present some open questions.
Friday, 27 November 2020
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14:15, Lecture Hall H12
Eike Lau (Bielefeld): Balmer spectra of certain Deligne-Mumford stacks
Abstract: Let G be a finite group and k a field. It is classically known that prime tensor ideals in the bounded derived category of finite kG-modules correspond to homogeneous prime ideals in the cohomology ring of G with coefficients in k. Assume that A is a commutative ring on which G acts. We consider the corresponding question for representations of G on finite projective A-modules, or equivalently for perfect complexes on the stack quotient of Spec A by the action of G.
Friday, 30 October 2020
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14:15, Zoom
Ulrich Thiel (Kaiserslautern): Introduction to Soergel Bimodules
Abstract: Motivated by a recent publication, I'll give a gentle overview of the theory of Soergel bimodules, only assuming a little background and ignoring many details.
For information on earlier talks please check the complete seminar archive.
