Seminar

No talks have been announced for this week.

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Future Talks

Friday, 18 July 2025

• 13:15, Room B2-278
  Martin Kalck (Graz): Path into transcendence
  Abstract: Algebraic numbers are complex numbers that are roots of polynomials with rational coefficients. All other complex numbers are called transcendental. It is typically a hard question to decide whether a given complex number is transcendental.
  A more general, classical question in 'transcendental number theory' (cf. e.g. works of Lindemann and Weierstraß, Gelfond and Schneider, Baker, Wüstholz) is the following: determine the dimension of the vectorspace generated by a (finite) set of complex numbers over the algebraic numbers. For example, the vectorspace generated by 1 and π is two-dimensional since π is transcendental by Lindemann's Theorem.
  For certain complex numbers called periods, we will try to explain how this transcendence question can (sometimes) be translated into determining dimensions of certain finite dimensional algebras – in other words, into counting (equivalence classes of) paths in 'modulated' quivers (with 'multiplicities').
  The dimension formulas obtained in this way improve and clarify earlier results of Huber & Wüstholz and recover a dimension estimate of Deligne & Goncharov.
  This is based on joint work with Annette Huber (Freiburg).
  

Friday, 17 October 2025

• 13:15
  Tilman Bauer (Stockholm): tba

Seminar Archive

Friday, 04 July 2025

• 13:00, Room B2-278
  Georgios Dalezios (Verona): Quasi-hereditary algebras arising from Reedy categories
  Abstract: Reedy categories form a generalization of the category of finite ordinals, with morphisms the weakly monotone functions between them. Given a Reedy category C and a Quillen model category M, there is always a model structure on the category of functors from C to M. In this talk, we study Reedy categories which are enriched over a field and we present two main results. The first is an analogue of the aforementioned result for complete cotorsion pairs and abelian model structures. In the second result, we focus on linear Reedy categories having finitely many objects, which leads us to the concept of Reedy finite-dimensional algebras. We prove that any Reedy finite-dimensional algebra is quasi-hereditary with an exact Borel subalgebra. This is joint work with Jan Stovicek.
• 14:15, Room B2-278
  Øyvind Solberg (Trondheim): Endomorphisms of modules as linear maps
  Abstract: All algebras and modules in this talk are finite dimensional over a field. A module M is non-trivially decomposable if and only there exists an endomorphism f of M such that the characteristic polynomial of f has at least two different irreducible factors. Hence, all endomorphisms of an indecomposable module has a characteristic polynomial a power of an irreducible polynomial. The talk will discuss these observations and which irreducible polynomials occur when over a finite prime field.
  This is work in progress and based on joint work with Fernando Yamauti.
  

Friday, 06 June 2025

• 13:15, Room B2-278
  Erlend Borve (Graz): tau-tilting theory and tau-tilting finiteness under scalar extension
  Abstract: Let L:k be a field extension and let A be a finite-dimensional k-algebra. The extension of scalars of A along L:k is the L-algebra A^L, obtained by tensoring A and L over k. In the early 1980s, Jensen and Lenzing showed that extension of scalars preserves many module-theoretic and homological properties, particularly when L:k is MacLane separable. In particular, representation-finiteness is preserved in this case. However, if A is tau-tilting finte, i.e. it admits only a finite number of support tau-tilting modules up to isomorphism, this need not be true for A^L. We explore some examples and counter-examples of when tau-tilting finiteness is preserved. Along the way, we explain how tau-tilting theory and related notions lift under extension of scalars.
  The talk is based on joint work in progress with Max Kaipel (Cologne).
• 14:30, Room B2-278
  Mohammad Hossein Keshavarz (Nantong): Characterizing higher Auslander(-Gorenstein) algebras
  Abstract: It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa's theorem in 1974.
  Let n be a positive integer. In this talk, by using torsion theoretic methods, we show that n-Auslander algebras can be characterized by the abelianness of the category of modules with projective dimension less than n and a certain additional property, extending the classical Auslander-Tachikawa theorem. By Auslander-Iyama correspondence a categorical characterization of the class of Artin algebras having n-cluster tilting modules is obtained.
  Since higher Auslander algebras are a special case of higher Auslander-Gorenstein algebras, the results are given in the general setting as extending previous results of Kong. Moreover, as an application of some results, we give categorical descriptions for the semisimplicity and selfinjectivity of an Artin algebra.
  Higher Auslander-Gorenstein algebras are also studied from the viewpoint of cotorsion pairs and, as application, we show that they satisfy in two nice equivalences.

Friday, 16 May 2025

• 13:15, Room B2-278
  Simone Virili (Barcelona): A study of Sylvester rank functions via functor categories
  Abstract: Given a ring R, a Sylvester rank function on the category of finitely presented right R-modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value 1 on R. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left R-modules to Abelian groups.
  This enlargement of the setting comes with many advantages, in fact, the richer structure of the functor category makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.

Friday, 09 May 2025

• 13:15, Room B2-278
  Otto Sumray (Dresden): Quiver Laplacians and data
  Abstract: Analysing the topology or geometry of data is a common task in data analysis, using methods such as clustering, networks, and persistent homology. Often, however, we wish to embellish data with extra local information, which mathematically can take the form of a quiver representation or cellular sheaf. One example of this enriched data comes from analysis of single-cell epigenetics, and this situation can be further abstracted to a question about local versus global feature selection.
  Graph Laplacians have had great success in the analysis of networks. We introduce the quiver Laplacian as an analogue of the graph Laplacian for quiver representations, and formulate a pipeline for its application to local versus global feature selection.
  Applying it to our case study in single-cell epigenetics, we show that the eigenvalues of the quiver Laplacian can aid in feature selection to recover biologically meaningful results. To show stability of this method, we provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways.
  
• 14:30, Room B2-278
  Mikhail Gorsky (Hamburg): Counting in Calabi-Yau categories, with applications to Hall algebras
  Abstract: I will discuss a replacement of the notion of homotopy cardinality in the setting of even-dimensional Calabi--Yau categories and their relative generalizations. This includes cases where the usual definition does not apply, such as Z/2-graded dg categories. As an application of the definition in the relative case, we define a version of Hall algebras for odd-dimensional Calabi-Yau categories. I will briefly explain its relation to some previously known non-intrinsic constructions of Hall algebras. Whenever a 1CY category C is equivalent to Z/2-graded derived category of a hereditary abelian category A, our intrinsically defined Hall algebra of C realises the Drinfeld double of the twisted Hall algebra of A, thus resolving a long standing problem in this CY case. The talk is based on joint work with Fabian Haiden, arxiv:2409.10154.
  

Friday, 02 May 2025

• 13:15, Room B2-278
  Giulia Iezzi (Aachen): Quiver Grassmannians for the Bott-Samelson resolution of type A Schubert varieties
  Abstract: Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. In this talk, we construct a special quiver with relations and consider two classes of quiver Grassmannians for this quiver. For an appropriate choice of dimension vector for this quiver, we provide an isomorphism between the corresponding quiver Grassmannians and certain Bott-Samelson resolutions of type A Schubert varieties. Furthermore, for smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety.

Friday, 11 April 2025

• 13:15, Room B2-278
  Umesh V. Dubey (Prayagraj): The Balmer Spectrum of integral permutation modules
  Abstract: Balmer and Gallauer recently studied the derived category of permutation modules of finite groups over a commutative ring and computed its Balmer spectrum when the ground ring is a field.
  In our work, we compute the Balmer spectrum for the derived category of permutation modules of finite groups over a commutative Noetherian base ring. To achieve this, we employed a triangular fixed point functor recently introduced by J. Omar Gomez, which enables us to describe the underlying set of the Balmer spectrum.
  To describe the topology, we use the permutation resolution of Balmer and Gallauer to get an analog of the Koszul object that provides the required support conditions. Again following their strategies we gave a Dirac scheme structure on the Balmer spectrum of permutation modules over elementary abelian p-groups.
  In this talk, we will briefly discuss Balmer and Gallauer's construction of a Koszul object and the Dirac scheme structure on the Balmer spectrum for the elementary abelian case.
  This talk is based on an ongoing joint project with J. Omar Gomez.

Tuesday, 25 February 2025

• 11:15, Room V4-116
  Hideto Asashiba (Shizuoka): Relative Koszul coresolutions and relative Betti numbers
  Abstract: Let G be a finitely generated right A-module for a finite-dimensional algebra A over a fieled k, and I the additive closure of G. We will define an I-relative Koszul coresolution K^.(V ) of an indecomposable direct summand V of G, and show that for a finitely generated A-module M, the I-relative i-th Betti number of M at V is given as the k-dimension of the i-th homology of the I-relative Koszul complex K_V(M)_. := Hom_A(K^.(V), M) of M at V for all i ≥ 0. This is applied to investigate the minimal interval resolution/coresolution of a persistence module M, e.g., to check the interval decomposability of M, and to compute the interval replacement of M.

Friday, 24 January 2025

• 13:15, Room X-E0-228
  Luca Pol (Regensburg): The etale topology in equivariant homotopy theory
  Abstract: In this talk, I will introduce the concept of a separable commutative algebra in the setting of tt-geometry and discuss its connection to Mathew’s work on Galois theory. Building on this, I will present several classification results for separable algebras in derived commutative algebra and chromatic homotopy theory. In the final part of the talk I will focus attention to the problem of classifying separable algebras in the category of genuine G-spectra for a finite group G.

Friday, 10 January 2025

• 13:15, Room X-E0-228
  Florian Tecklenburg (Bonn): Generalising Keller's counterexample on the telescope conjecture
  Abstract: Building on results of Bazzoni–Šťovíček, we give an explicit construction of an infinite family of commutative rings which generalise the first counterexample on the generalised telescope conjecture of Bernhard Keller in 1994. In particular, the telescope conjecture for their derived categories does not hold. Generalising the construction of these rings further, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.
  The talk is based on joint work with Scott Balchin, see https://arxiv.org/abs/2407.11791.

Friday, 13 December 2024

• 13:15, Room X-E0-228
  Claudius Heyer (Paderborn): A 6-functor formalism for smooth mod p representations
  Abstract: The formalism of six operations was introduced by Grothendieck to show that many phenomena in the étale cohomology of schemes can be formally deduced from a small set of axioms. Since then these six operations have been constructed in many of other contexts like D-modules, motives and rigid-analytic geometry. But only recently has there been a formal definition of a 6-functor formalism, mainly due to Liu–Zheng and then further simplified by Mann in his PhD thesis. Also in Fargue–Scholze's geometrization of the local Langlands correspondence the six operations are a guiding theme.
  In this talk I will report on joint work with Lucas Mann where we construct a full 6-functor formalism in the setting of smooth representations of p-adic Lie groups with mod p coefficients. As an application we use the formalism to construct a canonical anti-involution on derived Hecke algebras generalizing earlier work by Schneider–Sorensen.
  

Friday, 06 December 2024

• 13:15, Room X-E0-228
  Mahrud Sayrafi (Leipzig): Splitting of Vector Bundles on Toric Varieties
  Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.
• 14:30, Room X-E0-228
  Markus Schmidmeier (Boca Raton): Wild categories: screwy or adaptable?
  Abstract: For n a natural number, the invariant subspace category S(n) consists of all systems (V,T,U) where V is a finite dimensional vector space, T a nilpotent linear operator acting on V with nilpotency index at most n, and U a T-invariant subspace of V. For increasing n, the S(n) form a chain of categories with increasing complexity; in particular, S(n) has wild representation type for n>6.
  A dimension vector is called Brauer-Thrall if each positive multiple can be realized by a parametrized family of pairwise nonisomorphic indecomposable objects. We discuss the question in the title with regards to regions of BTh-vectors for S(n).
  This is a report about joint work with Claus Michael Ringel, see arxiv.org/abs/2405.18592.

Friday, 29 November 2024

• 13:15, X-E0-228
  Markus Kirschmer (Bielefeld): Chow groups of one-dimensional noetherian domains
  Abstract: We discuss various connections between ideal classes, divisors, Picard and Chow groups of one-dimensional noetherian domains. As a result of these, we give a method to compute Chow groups of orders in global fields and show that there are infinitely many number fields which contain orders with trivial Chow groups. This is joint work with J. Klüners.

Friday, 22 November 2024

• 13:15, Room X-E0-228
  Jan Schröer (Bonn): Projective presentations of maximal rank
  Abstract: First, I will discuss the connection between projective presentations of maximal rank and generically tau-regular components of module varieties. Then I will present some classification results for generically tau-regular components. This is joint work with Grzegorz Bobinski.

Friday, 15 November 2024

• Colloquium in Honour of Lutz Hille

Friday, 25 October 2024

• 13:15, Room X-E0-228
  Peter Schneider (Münster): What is smooth representation theory about?
  Abstract: This talk will be a survey talk first on how number theory leads to the introduction of smooth representation theory of p-adic groups in characteristic zero. Then I will report on the recent development of the characteristic p coefficient case, which is radically different and where little is known yet.

Friday, 18 October 2024

• 13:15, Room X-E0-228
  David Nkansah (Aarhus): Rank Functions in the Framework of Higher Homological Algebra
  Abstract: Chuang and Lazarev introduced the concept of rank functions on triangulated categories as a generalisation of classical work by Cohn and Schofield on Sylvester rank functions. In this talk, we propose a generalisation of this notion to the broader framework of higher homological algebra.

Friday, 11 October 2024

• 13:15, Room X-E0-228
  Alexandre Minets (Bonn): Hecke symmetries of sheaves on surfaces
  Abstract: For a smooth algebraic surface S, I will explain how to construct an action of a quantum toroidal-type algebra on the cohomology of moduli of coherent sheaves on S. I will recall how this unifies several known results in geometric representation theory, and sketch the role of this action in our proof of P=W conjecture of de Cataldo-Hausel-Migliorini.
  

For information on earlier talks please check the complete seminar archive.