This is the new website of the seminar series titled "New Directions in Group Theory and Triangulated Categories". This seminar series was started in November 2020. We usually meet every week with some occasional breaks. If you are interested in receiving updates and announcements regarding talks in this series, and also if you are interested in suggesting a speaker including yourself, please get in touch with me at rudradipbiswas@gmail.com.
Video recordings of talks from this series are posted on the seminar's YouTube channel which is called NDGTTC (Group Th, Triangulated Cat) Seminar Series. Note that most of the talks from the first season (Nov 2020 - March 2021) were not recorded.
-- Dr. Rudradip Biswas, organizer (for information about my research background, please visit my website)
125th Meeting
Date: June 5, 2025; Thursday
Time: 4 pm UK
Speaker: Shubhodip Mondal (University of British Columbia, Vancouver)
Title: Unipotent homotopy groups of Calabi-Yau varieties.
Abstract: In this talk, I will introduce the notion of unipotent homotopy group schemes and discuss its relation to Artin--Mazur formal groups. I will also explain why derived equivalent Calabi--Yau varieties over fields of characteristic p have isomorphic unipotent homotopy group schemes. This is joint work with Emanuel Reinecke.
124th Meeting
Date: May 29, 2025; Thursday
Time: 4 pm UK
Speaker: Logan Hyslop (University of California, Los Angeles)
Title: Towards the Nerves of Steel Conjecture.
Abstract: In arXiv:2001.00284, Balmer proposed the so-called “nerves of steel conjecture,” and proves that it is equivalent to an “exact-nilpotence condition” holding for local tensor triangulated categories. In this talk, we will explain how to improve this result to show an equivalence between the nerves of steel conjecture and a strong form of the exact-nilpotence condition for local tensor triangulated categories. We will then explain how to extend a number of cases where the exact-nilpotence condition holds. This talk is based on the preprint arXiv:2410.06320.
*No talks on 15 May and 22 May.
123rd Meeting
Date: May 8, 2025; Thursday
Time: 4 pm UK
Speaker: Dominik Kirstein (University of Bonn)
Title: A twisted Bass-Heller-Swan decomposition for localising invariants.
Abstract: Classically, the Bass-Heller-Swan decomposition relates algebraic K-theory of a Laurent polynomial ring to algebraic K-theory of its coefficient ring. In this talk, I will explain how to obtain a generalisation of this result in the setting of localising invariants of stable infinity-categories that come with an endomorphism. Furthermore, I’ll explain regularity conditions under which the relevant assembly map is an equivalence. As an application one gets a splitting result for Waldhausen’s A-theory of mapping tori, and more generally for the K-theory of certain tensor algebras. Based on joint work with Christian Kremer.
122nd Meeting
Date: May 1, 2025; Thursday
Time: 4 pm UK
Speaker: Sam Miller (University of California, Santa Cruz)
Title: On the tensor product property and the complete prime spectrum in noncommutative tensor-triangular geometry.
Abstract: A pertinent question for any support theory of a monoidal category is whether the tensor product property holds. In this talk, we'll discuss the complete prime Balmer spectrum, the universal support among support data satisfying the tensor product property. Analogous to the noncommutative algebraic geometry setting, the complete prime Balmer spectrum is functorial, but may also be empty. If time permits, we'll mention obstructions towards adapting Reyes's theorem, determining the universal functorial spectral theory for noncommutative rings, to the monoidal triangulated categorical setting.
*No talk on Thursday 24 April.
121st Meeting
Date: April 17, 2025; Thursday
Time: 4 pm UK
Speaker: Devarshi Mukherjee (Universität Münster)
Title: Rigid categories in functional analysis.
Abstract: In this talk, I will introduce a category of nuclear modules associated to a complete bornological algebra. In a sense that will be made precise, this category is a rigid symmetric monoidal category, and therefore serves as an input for continuous K-theory. I will place this category in a broader context of derived analytic geometry from the bornological perspective. All results are joint work with Jack Kelly.
120th Meeting
Date: April 3, 2025; Thursday
Time: 4 pm UK
Speaker: Lev Borisov (Rutgers University)
Title: No-phantom theorem for certain rational surfaces.
Abstract: Let E be a smooth cubic curve in the projective plane. We show that for n points on E in very general position, the corresponding blowup of the plane has no phantoms in its derived category. This is joint work with Kimoi Kemboi, arXiv:2405.01683.
119th Meeting
Date: March 27, 2025; Thursday
Time: 4 pm UK
Speaker: Mikhail Bondarko (St. Petersburg State University)
Title: On periodic objects and semi-orthogonal decompositions corresponding to exact transformations (joint work with Alexander Aizatov).
Abstract: For a family T of transformations between exact endofunctors on a triangulated category C we studied the subcategory Cone(T)^{\perp} of the corresponding periodic objects (that is, the orthogonal to the class {Cone(t(X)): t ∈ T, X ∈ C}) and the adjoints to the embedding Cone(T)^{\perp} (whenever they exist). Several important examples come from (motivic) stable homotopy theory and exact tensor actions. In particular, we took T_S = {X \mapsto s.id_X, s ∈ S \subset R}, where R is a commutative ring such that C is R-linear and X runs through C; Cone(T_S)^{\perp} consists of uniquely S-divisible objects. I will generalize this description to the case where T consists of transformations corresponding to tensoring (all X ∈ C) by some morphisms between projective R-modules (under the assumption that C is closed with respect to coproducts and products of the corresponding size).
In the case where C is well generated (in particular, if C is compactly generated) and T consists of transformations between endofunctors E^j and F^j that respect coproducts, we proved the existence of a semi-orthogonal decomposition s_T^{cl} whose left admissible component LA_T^{cl} equals Cone(T)^{\perp}. Respectively, there exists a left adjoint to the embedding Cone(T)^{\perp} \to C; in the case T=T_S, this functor is the R[1/S]-linearization one. This adjoint is weight-exact whenever E^j and F^j are weight-exact with respect to a smashing weight structure. The right admissible component RA_T^{cl} of s_T^{cl} is the triangulated subcategory of C generated by the aforementioned Cone(T) under coproducts; in particular, one may call RA_{T_S}^{cl} the subcategory of S-torsion objects of C. I will try to mention certain 2-functoriality of decompositions of this sort.
Lastly, if the right adjoints to all E^j and F^j respect coproducts, then Cone(T)^{\perp} is right admissible as well; respectively, it is the right admissible component of yet another semi-orthogonal decomposition.
118th Meeting
Date: March 20, 2025; Thursday
Time: 4 pm UK
Speaker: Nicholas Williams (University of Cambridge)
Title: Oriented matroids from cluster-tilting objects in type A cluster categories.
Abstract: Cluster categories are triangulated categories introduced by Buan, Marsh, Reineke, Reiten, and Todorov to categorify the cluster algebras of Fomin and Zelevinsky. In particular, reachable basic cluster-tilting objects in the cluster category are in bijection with clusters in the cluster algebra. In type A, one then has a bijection between cluster-tilting objects in the cluster category and triangulations of a convex polygon. A three-dimensional analogue of this bijection is that, for linearly oriented type A, equivalence classes of maximal green sequences are in bijection with triangulations of a three-dimensional cyclic polytope. It is natural to ask whether there exist similar polytopes for other orientations of the type A Dynkin diagram. In this talk, we will explain work in progress answering this question by showing how a choice of cluster-tilting object T in the type A cluster category naturally defines an oriented matroid, whose triangulations are conjecturally in bijection with equivalence classes of maximal green sequences of End(T).
117th Meeting
Date: March 13, 2025; Thursday
Time: 4 pm UK
Speaker: Martin Kalck (Universität Graz)
Title: Lengths spectra of triangulated categories.
Abstract: The thick subcategories of a triangulated category form a lattice. These lattices have been actively studied for about 40 years in fields ranging from algebraic topology and algebraic geometry to representation theory.
In many cases, it seems hard to give explicit and complete descriptions of these lattices. Instead of attempting to understand the whole structure, we look at a rather coarse invariant of these lattices and hence of the corresponding triangulated category. Namely, we study the lengths of maximal chains in this lattice. We collect all integers that arise as lengths of maximal chains for a fixed triangulated category T into a set called the "lengths spectrum of T".
We will illustrate this concept with examples from representation theory and algebraic geometry and (time permitting) also discuss open questions. This is based on joint work with Yuki Hirano and Genki Ouchi, arXiv:2404.07583.
116th Meeting
Date: March 6, 2025; Thursday
Time: 4 pm UK
Speaker: Juan Omar Gomez (Universität Bielefeld)
Title: Fiberwise building in tt-geometry and applications.
Abstract: In this talk, I will discuss certain conditions on a family of base-change functors between tt-categories with small coproducts that allow the descent of certain tt-geometric information. Specifically, we describe how the localizing ideal generated by an object can be understood in terms of such a family of functors - a property known as building. We also present some consequences of this property in the context of rigidly compactly generated tt-categories.
In particular, we consider the big derived category of permutation modules for a finite group over a Noetherian base, and study building in terms of the family of geometric functors induced by base change along the residue fields of the ground ring. As an application, we leverage Balmer and Gallauer’s stratification result in the case of fields to extend it to an arbitrary Noetherian base.
115th Meeting
Date: Februrary 20, 2025; Thursday
Time: 4 pm UK
Speaker: Jorge Vitória (Università degli Studi di Padova)
Title: Topological properties of cosilting mutation.
Abstract: Mutations are processes occurring in algebra that modify an object under study into another of the same kind by replacing part of it. They often enjoy good combinatorial properties. In this talk, we discuss mutation of bounded cosilting complexes in derived categories, as a generalisation of tilting and silting mutation. We will observe that cosilting mutation behaves well with respect to a canonical topology (defined on the set of indecomposable summands of a cosilting object) induced by the localisation theory of the corresponding hearts: the Gabriel topology. This is based on joint work with Sergio Pavon and Michal Hrbek.
114th Meeting
Date: Februrary 13, 2025; Thursday
Time: 4 pm UK
Speaker: Giovanna Le Gros (Charles University, Prague)
Title: Semi-Bousfield classes in some big tensor triangulated categories.
Abstract: We introduce semi-Bousfield classes in a rigidly-compactly generated tensor triangulated category with a fixed t-structure. In the particular case of the unbounded derived category of a commutative Noetherian ring with the standard t-structure, semi-Bousfield classes simultaneously include coaisles of compactly generated t-structures and localising subcategories. In this setting, if we restrict to the semi-Bousfield classes which are generated by bounded complexes of flat modules, these are classified by functions from the Zariski spectrum of the ring to the extended integers, in a way which is compatible with the classification of compactly generated t-structures by sp-filtrations due to Alonso-Jeremías-Saorín, and localising subcategories by subsets of the spectrum due to Neeman.
In this talk, we will discuss these results and some more recent results about semi-Bousfield classes in the setting of big tensor triangulated categories with some extra assumptions. This talk is based on current work-in-progress with Dolors Herbera and Michal Hrbek.
No talks on 30 January and 6 February
113th Meeting
Date: January 23, 2025; Thursday
Time: 4 pm UK
Speaker: Manuel Saorin (University of Murcia)
Title: On set-generation of hearts of t-structures.
Abstract: Since the introduction of the concept of a t-structure in a triangulated category by Beilinson, Bernstein, and Deligne [1], it has been known that the heart of a t-structure is an abelian category. A natural vague question asks when such an abelian category is nice. When the ambient triangulated category has products and coproducts, the heart of any t-structure is Ab3 (=cocomplete) and Ab3* (=complete). In such a situation, the activity of researchers has mainly concentrated on deciding when the heart is a Grothendieck category, i.e. when it is Ab5 and has a set of generators. The Ab5 and the generators parts of the problem were of a different nature, and the second part was tackled mainly with ad hoc arguments which depended on the existence of a model for the triangulated category. In joint work with Jan Stovicek [2] we proved that the heart of any compactly generated (not necessarily with a model) is a locally finitely presented Grothendieck category, giving a precise set of finitely presented generators for that heart. We recently extended this result [3] by proving that if the ambient category is \kappa-compactly generated, for some regular cardinal \kappa, and the t-structure is generated by \kappa-compact objects then the heart is a (not necessarily Ab5) \kappa-presented abelian category. In this talk I will quickly recall the arguments for the result in [2] and will present the arguments for the result in [3], showing the (de)similarities.
[1] A. A. Beilinson, J. Bernstein, P. Deligne: "Faisceaux pervers". Astérisque 100, Soc. Math. France, Paris (1982), 5-171.
[2] M. Saorín, J. Stovicek: "t-Structures with Grothendieck hearts via functor categories". Selecta Math. 29 (2023), 73 pages.
[3] M. Saorín: "Hearts of set-generated t-structures have a set of generators". arXiv:2408.01378
112th Meeting
Date: January 16, 2025; Thursday
Time: 4 pm UK
Speaker: Anna Barbieri (University of Verona)
Title: A smooth compactification of the stability space for the A_n-quiver.
Abstract: The space of Bridgeland stability conditions is a non-compact complex manifold attached to a triangulated category D, parametrizing some t-structures of the category. In this talk, I will propose a notion of multi-scale stability conditions that gives a smooth compactification of an appropriate quotient of the stability manifold of the Ginzburg category of type A_n and a partial compactification for other Ginzburg categories attached to quivers with potential from triangulated marked Riemann surfaces.
111th Meeting
Date: January 9, 2025; Thursday
Time: 4 pm UK
Speaker: Alexander Vishik (University of Nottingham)
Title: On the Balmer spectrum of Voevodsky motives.
Abstract: Algebraic Geometry is substantially more complex than Topology, which is manifested by the abundance of different types of atomic objects (points) in the former, and is reflected in the Balmer spectrum of the category of geometric Voevodsky motives. While the Balmer spectrum of topological motivic category is just the Zariski spectrum of integers, its algebro-geometric counterpart is pretty wild. For example, over real numbers, it has the cardinality equal to that of the set of subsets of the category. The vast majority of known points there are given by isotropic realisations. These are parametrised by a choice of a prime p and a p-equivalence class of an extension of the base field. For p=2, these equivalence classes are detected by pure symbols in Milnor’s K-theory modulo 2 over flexible closure. The ideals are distinguished by the test spaces: Rost motives and reduced Rost motives. This allows to put the etale (=topological) and isotropic realisations into the common context.
110th Meeting
Date: December 13, 2024; Friday
Time: 4 pm UK
Speaker: Alexander Perry (University of Michigan)
Title: Flops, derived categories, and liftability of threefolds.
Abstract: Bridgeland famously proved that birational Calabi-Yau threefolds are derived equivalent, as a consequence of his description of threefold flops as moduli spaces of perverse coherent sheaves. I will explain joint work with Riku Kurama extending this theory to a relative setting, which we apply to prove that liftability to characteristic 0 is birationally invariant among Calabi-Yau threefolds in positive characteristic.
109th Meeting
Date: December 5, 2024; Thursday
Time: 4 pm UK
Speaker: Kabeer Manali Rahul (Australian National University)
Title: Metrics, approximability, and some new Brown representability theorems for triangulated categories.
Abstract: Neeman has recently initiated the use of certain analytic techniques to study triangulated categories. The central notions are those of a metric on a triangulated category, and that of an approximable triangulated category. In this talk, we will begin by going over the motivation and some intuition behind these ideas. We will then move on to a more general definition of approximability, which allows us to prove new Brown representability type theorems.
108th Meeting
Date: November 28, 2024; Thursday
Time: 4 pm UK
Speaker: Jon Woolf (University of Liverpool)
Title: Heart fans as universal phase diagrams.
Abstract: The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a `universal phase diagram' for Bridgeland stability conditions with the given heart. This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog (arXiv:2310.02844v2).
107th Meeting
Date: November 21, 2024; Thursday
Time: 4 pm UK
Speaker: Vladimir Sosnilo (Universität Regensburg)
Title: Resolutions of c-categories.
Abstract: (joint with Ishan Levy) Many results about schemes can be generalized to the noncommutative setting of stable infinity-categories. For bounded weighted categories, things are even better: one can formulate and prove a natural analogue of the theorem of Dundas-Goodwillie-McCarthy which is one of the fundamental tools in studying algebraic K-theory. However, in algebraic geometry bounded weighted categories do not show up very often: for instance, the infinity-category of perfect complexes over a scheme X only admits a reasonable weight structure when X is affine. We introduce a new notion of a c-category which is designed to cover a diverse class of geometric examples, including all quasi-compact quasi-separated schemes, yet allowing for all the weighted arguments to work out in this setting. Our main result shows that a c-category can be resolved in finitely many steps by categories of perfect complexes over connective ring spectra. This allows us to prove an analogue of the DGM theorem for c-categories, as well as the vanishing of their Hochschild homology below a certain degree. We also show that either of the following admits a structure of a c-category:
1. the derived category of any exact (infinity-)category that has finite Ext-dimension;
2. the subcategory of compact objects in any weakly approximable stable infinity-category in the sense of Neeman. In particular, using the computation of the Hochschild homology for c-categories mentioned before we obtain that the category of module-spectra over the ring C*(S^2) of cochains over the 2-sphere is not weakly approximable.
106th Meeting
Date: November 14, 2024; Thursday
Time: 4 pm UK
Speaker: Emanuele Macrì (Université Paris-Saclay)
Title: Deformation of t-structures.
Abstract: Bridgeland stability conditions were introduced about 20 years ago, with motivations both from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.
105th Meeting
Date: November 7, 2024; Thursday
Time: 4 pm UK
Speaker: David Jorgensen (University of Texas, Arlington)
Title: Asymptotic vanishing of cohomology in triangulated categories.
Abstract: Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given. This is joint work with Petter Bergh and Peder Thompson.
104th Meeting
Date: October 31, 2024; Thursday
Time: 4 pm UK
Speaker: Changhan Zou (University of California, Santa Cruz)
Title: Homological and Balmer--Favi--Sanders support.
Abstract: Various support theories have been used as key tools to establish tensor triangular classifications. The goal of this talk is to motivate the notion of Balmer--Favi--Sanders support and compare it with the homological support. A consequence of the comparison is that Balmer spectra of tt-stratified categories must be weakly Noetherian.
103rd Meeting
Date: October 17, 2024; Thursday
Time: 4 pm UK
Speaker: Martin Frankland (University of Regina)
Title: Toda brackets in n-angulated categories
Abstract: Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n=3). We will look at different constructions of the brackets, their properties, some examples, and some applications.
102nd Meeting
Date: October 10, 2024; Thursday
Time: 4 pm UK
Speaker: Lukas Bonfert (MPI Bonn)
Title: Derived projective covers and Koszul duality of simple-minded and silting collections
Abstract: Weight structures (also known as co-t-structures) and t-structures on triangulated categories are closely related by orthogonality. In terms of simple-minded collections and silting collections, the characterization of orthogonality is very similar to the relation of the indecomposable projectives to the simple objects in an abelian category. I will explain how this observation can be formalized using derived projective covers (an analog of projective covers for triangulated categories), and how this is related to the notion of enough derived projectives introduced by Genovese-Lowen-van den Bergh and Coelho Simoes-Pauksztello-Ploog. I will also discuss a result about Koszul duality between simple-minded and silting collections, which is a first step towards understanding the (conjectural) Koszul duality between weight structures and t-structures. The talk is based on arXiv:2309.00554v2.
101st Meeting
Date: July 25, 2024; Thursday
Time: 4 pm UK
Speaker: Janina Letz (Universität Bielefeld, UCLA)
Title: Generation time for biexact functors and Koszul objects in triangulated categories.
Abstract: One way to study triangulated categories is through finite building. An object X finitely builds an object Y, if Y can be obtained from X by taking cones, suspensions and retracts. The X- level measures the number of cones required in this process; this can be thought of as the generation time. I will explain the behavior of level with respect to tensor products and other biexact functors for enhanced triangulated categories. I will further present applications to the level of Koszul objects. This is joint work with Marc Stephan.
100th Meeting
Date: July 4, 2024; Thursday
Time: 4 pm UK
Speaker: Bertrand Toën (University of Toulouse)
Title: Geometric quantization for shifted symplectic structures.
Abstract: The purpose of this talk is to present an ongoing work (joint with Vezzosi) on geometric quantization in the setting of shifted symplectic structures. I will start by recalling the various notions involved as well as the results previously obtained by James Wallbridge, who constructed the prequantized (higher) categories of a given integral shifted symplectic structure. I will then explain our main result so far: the construction of the shifted analogues of the Kostant–Souriau prequantum operators, which will be realized as a "Poisson module over a Poisson category" (a categorification of the notion of a Poisson module over a Poisson algebra). This will be obtained by means of deformation theory arguments for categories of sheaves in the setting of (derived) differential geometry. If time permits, I will discuss further aspects associated to the notion of polarizations of shifted symplectic structures.
99th Meeting
Date: June 20, 2024; Thursday
Time: 4 pm UK
Speaker: Sondre Kvamme (NTNU Trondheim)
Title: Higher torsion classes and silting complexes.
Abstract: Higher Auslander-Reiten theory was introduced by Iyama in 2007 as a generalization of classical Auslander-Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of module categories. It turns out that many notions in algebra and representation theory have generalizations to higher Auslander-Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.
In this talk, I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type A_n.
This is based on joint work with Jenny August, Johanne Haugland, Karin M. Jacobsen, Yann Palu, and Hipolito Treffinger.
98th Meeting
Date: June 13, 2024; Thursday
Time: 4 pm UK
Speaker: Marc Stephan (Bielefeld University)
Title: An equivariant BGG correspondence and applications to free A_4 - actions on products of spheres
Abstract: A classical question in the theory of transformation groups asks which finite groups can act freely on a product of spheres. For instance, Oliver showed that the alternating group A_4 can not act freely on any product of two equidimensional spheres.
I will report on joint projects with Henrik Rüping and Ergün Yalcin and explain that for "most" dimensions m and n, there is no free A_4-action on S^m \times S^n and whenever there exists such a free action, then the corresponding cochain complex with mod 2 coefficients is rigid: its equivariant homotopy type only depends on m and n.
This involves an equivariant extension of Carlsson’s BGG correspondence in order to classify perfect complexes over F_2[A_4] with four-dimensional total homology.
97th Meeting
Date: June 6, 2024; Thursday
Time: 4 pm UK
Speaker: Kent Vashaw (MIT)
Title: A Chinese remainder theorem and Carlson's theorem for monoidal triangulated categories.
Abstract: Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
96th Meeting
Date: May 30, 2024; Thursday
Time: 4 pm UK
Speaker: Aslak Bakke Buan (NTNU Trondheim)
Title: From exceptional to tau-exceptional sequences in module categories
Abstract: This is based on joint work with Eric Hanson and Bethany Marsh.
Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite dimensional algebras. This is the theory of tau-exceptional sequences, which was developed in joint work with Marsh, motivated by tau-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.
The interplay between theories for tau-rigid modules, torsion pairs, and wide subcategories is central to our discussions.
95th Meeting
Date: May 23, 2024; Thursday
Time: 4 pm UK
Speaker: Timothy Logvinenko (Cardiff University)
Title: The Heisenberg category of a category.
Abstract: In the 90's, Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Later, Krug lifted this to derived categories and generalised it to the symmetric quotient stacks of any smooth projective variety.
On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e. the one associated to the rank 1 lattice. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.
We show how to associate the Heisenberg 2-category to any smooth and proper DG category and then define its Fock space 2-representation. This construction unifies all the results above and extends them to what can be viewed as the generality of arbitrary noncommutative smooth and proper schemes.
94th Meeting
Date: May 9, 2024; Thursday
Time: 4 pm UK
Speaker: Paolo Stellari (Università degli Studi di Milano)
Title: Comparing the homotopy categories of dg categories and of A-infinity categories
Abstract: In this talk, we show that the homotopy category of (small) dg categories and the homotopy category of A_infty categories are equivalent (even from a higher categorical viewpoint). We will discuss several issues related to the various notions of unity and provide several applications. The main ones are about the uniqueness of enhancements for triangulated categories and a full proof of a claim by Kontsevich and Keller concerning a description of the category of internal Homs for dg categories. This is joint work with A. Canonaco and M. Ornaghi.
93rd Meeting
Date: May 2, 2024; Thursday
Time: 4 pm UK
Speaker: Eloísa Grifo (University of Nebraska - Lincoln)
Title: Searching for modules that are not virtually small
Abstract: Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander--Buchsbaum--Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.
This is joint work with Ben Briggs and Josh Pollitz.
92nd Meeting
Date: March 28, 2024; Thursday
Time: 4 pm UK
Speaker: Maxime Ramzi (University of Copenhagen)
Title: Categorifying spectra and the theorem of the heart
Abstract: The goal of this talk will be to present the results from my recent joint work with Vova Sosnilo amd Christoph Winges, where we prove that every spectrum is the (nonconnective) K-theory spectrum of a stable category. Our main application of this is the disproof of a conjecture by Antieau-Gepner-Heller about a nonconnective version of the theorem of the heart in the non-noetherian setting; but I will also try to mention other perspectives on this result.
91st Meeting
Date: March 21, 2024; Thursday
Time: 4 pm UK
Speaker: Carles Casacuberta (Universitat de Barcelona)
Title: Homotopy reflectivity is equivalent to the weak Vopenka principle.
Abstract: We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka's principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.
90th Meeting
Date: March 14, 2024; Thursday
Time: 4 pm UK
Speaker: Gregory Arone (Stockholm University)
Title: The tensor triangular geometry of functor categories
Abstract: We consider the (infinity) category of excisive (aka polynomial) functors from Spectra to Spectra. Understanding this category is a basic problem in functor calculus. We will approach it from the perspective of tensor triangular geometry. Day convolution equips the category of excisive functors with the structure of a rigid monoidal triangulated category. We describe completely the Balmer spectrum of this category, i.e., its spectrum of prime tensor ideals. This leads to a Thick Subcategory Theorem for excisive functors. A key ingredient in the proof is a blueshift theorem for the generalized Tate construction associated with the family of non-transitive subgroups of products of symmetric groups. If there is time, I will say something about work in progress to extend these results to more general functor categories. Joint with Tobias Barthel, Drew Heard, and Beren Sanders.
89th Meeting
Date: March 7, 2024; Thursday
Time: 4 pm UK
Speaker: Matt Booth (Lancaster University)
Title: Global Koszul duality
Abstract: Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of infinity-categories) between augmented dg algebras and conilpotent dg coalgebras. One should think of this as a noncommutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I'll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to noncommutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra. This is joint work with Andrey Lazarev, which appears in arXiv:2304.08409.
88th Meeting
Date: February 29, 2024; Thursday
Time: 4 pm UK
Speaker: Jay Shah
Title: Real topological Hochschild homology, C2-stable trace theories, and Poincaré cyclic graphs
Abstract: To study topological Hochschild homology as an invariant of stable ∞-categories and endow it with its universal property in this context, Nikolaus introduced the formalism of stable cyclic graphs and trace theories (after Kaledin). On the other hand, Poincaré ∞-categories are a C2-refinement of stable ∞-categories that provide an adequate formalism for studying real and hermitian algebraic K-theory, which should be then well-approximated by the real cyclotomic trace. In this talk, we explain how to systematically provide Poincaré refinements of all the components of Nikolaus' approach to stable trace theories. This is work-in-progress with Yonatan Harpaz and Thomas Nikolaus.
87th Meeting
Date: February 22, 2024; Thursday
Time: 4 pm UK
Speaker: Benjamin Briggs (University of Copenhagen)
Title: Koszul homomorphisms and resolutions in commutative algebra.
Abstract: Koszul duality has taken many forms across algebra, geometry, and topology.
This is a talk about the situation in commutative algebra. A homomorphism f: S -> R of commutative local rings has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) = Tor^S(R,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition and how it is satisfied by many many examples.
The main application is the construction of explicit free resolutions over R in the presence of a Koszul homomorphism. These tell you about the asymptotic homological algebra of R, and so the structure of the derived category of R. This construction simultaneously generalizes the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.
This is all joint with James Cameron, Janina Letz, and Josh Pollitz.
86th Meeting
Date: February 15, 2024; Thursday
Time: 4 pm UK
Speaker: Srikanth Iyengar (University of Utah)
Title: The stable module category of a finite group is locally regular.
Abstract: The goal is to explain the title of the talk, and some consequences that flow from that property of the stable module category, having to do with locally dualisable objects. This is based on an ongoing collaboration with Dave Benson, Henning Krause, and Julia Pevtsova.
85th Meeting
Date: February 8, 2024; Thursday
Time: 4 pm UK
Speaker: Leonid Positselski (Czech Academy of Sciences, Prague)
Title: Semi-infinite algebraic geometry of quasi-coherent torsion sheaves
Abstract: This talk is based on the book "Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes" (arXiv:2104.05517). I will start with some examples serving as special cases of the general theory, such as the tensor structure on the category of unbounded complexes of injective quasi-coherent sheaves on a Noetherian scheme with a dualizing complex. Then I will proceed to explain the setting of a flat affine morphism of ind-schemes into an ind-Noetherian ind-scheme with a dualizing complex, and the main ingredient concepts of quasi-coherent torsion sheaves, pro-quasi-coherent pro-sheaves, and the semiderived category. In the end, I will spell out the construction of the semi-tensor product operation on the semi-derived category of quasi-coherent torsion sheaves, making it a tensor triangulated category.
84th Meeting
Date: January 25, 2024; Thursday
Time: 4 pm UK
Speaker: Lars Winther Christensen (Texas Tech University)
Title: The derived category of a regular ring.
Abstract: Recall that a noetherian ring R is regular if every finitely generated R-module has finite projective dimension. In a paper from 2009, Iacob and Iyengar characterize the regularity of R in terms of properties of (unbounded) R-complexes. Their proofs build on results of Jorgensen, Krause, and Neeman on compact generation of the homotopy categories of complexes of projective/injective/flat modules.
In the commutative case, these results can be obtained with derived category methods in local algebra. I will illustrate how this is done by proving that the following conditions are equivalent for a commutative noetherian ring R:
1) R is regular.
2) Every complex of finitely generated projective R-modules is semi-projective.
3) Every complex of projective R-modules is semi-projective.
4) Every acyclic complex of projective R-modules is contractible.
The second condition is new, compared to the 2009 results, and relating it to the regularity of R is the novel part of the proof. This argument also plays a central role in the new proof of the corresponding results for complexes of injective modules and complexes of flat modules.
83rd Meeting
Date: January 18, 2024; Thursday
Time: 4 pm UK
Speaker: Shachar Carmeli (University of Copenhagen)
Title: Cyclotomic Redshift.
Abstract: I will discuss a joint work with Ben-Moshe, Schlank, and Yanovski, proving the compatibility of T(n+1)-local algebraic K-theory with the formation of homotopy limits with respect to p-local π-finite group actions on T(n)-local categories. This is a generalization of the results of Thomason for height 0 and Clausen, Mathew, Naumann, and Noel for actions of discrete p-groups in arbitrary chromatic height. I will then discuss the compatibility of K-theory with the chromatic cyclotomic extensions, chromatic Fourier transform, and higher Kummer theory from previous works with Barthel, Schlank, and Yanovski, phenomena we refer to as "cyclotomic redshift''. Finally, I will explain how cyclotomic redshift gives hyperdescent for K-theory along the cyclotomic tower after K(n+1)-localization.
82nd Meeting
Date: December 14, 2023; Thursday
Time: 4 pm UK
Speaker: Tom Bachmann (Johannes Gutenberg University of Mainz)
Title: p-adic homotopy theory and E-infinity coalgebras.
Abstract: I will report in joint work with Robert Burklund. We prove that the canonical functor from p-complete, nilpotent spaces to E-infinity coalgebras over \overline{F}_p is fully faithful. This generalizes a theorem of Mandell.
81st Meeting
Date: November 22, 2023; Wednesday
Time: 4 pm UK
Speaker: Arend Bayer (University of Edinburgh)
Title: Non-commutative abelian surfaces and generalized Kummer varieties.
Abstract: Polarised abelian surfaces vary in three-dimensional families. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations; moreover, based on general principles, one should expect to get "algebraic families" of their categories over four-dimensional bases. Generalized Kummer varieties (GKV) are Hyperkaehler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have four-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarieties.
I present a construction that addresses both issues. We construct four-dimensional families of categories that are deformations of D^b(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces.
This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.
80th Meeting
Date: November 16, 2023; Thursday
Time: 4 pm UK
Speaker: Michal Hrbek (Czech Academy of Sciences, Prague)
Title: Topological derived Morita theory and some applications.
Abstract: The now classical derived Morita theory developed by Rickard at the end of 1980's can be briefly stated as follows: Given two rings R and S, their derived module categories are triangle equivalent if and only if there is a (compact) tilting complex of R-modules whose endomorphism ring is isomorphic to S. Recently, Positselski and Šťovíček discovered a topological algebra description of the heart of the t-structure induced by a large (=not compact) tilting module T: It is equivalent to the category of contramodules over the endomorphism ring of T endowed with a suitable natural topology. Building on this idea, we develop a topological version of derived Morita theory, in which large tilting complexes with cotilting duals parametrize a natural class of derived equivalences between modules over a base ring R and the topologically flavored categories of contramodules and discrete modules over a topological ring. If time permits, we discuss some first applications of the theory to commutative algebra. The talk is based on arXiv preprints arXiv:2205.11105 and arXiv:2307.16722 with Lorenzo Martini.
79th Meeting
Date: November 2, 2023; Thursday
Time: 4 pm UK
Speaker: Daniel Nakano (University of Georgia)
Title: Realizing Rings of Regular Functions via the Cohomology of Quantum Groups.
Abstract: Let G be a complex reductive group and be a parabolic subgroup of G. In this talk the presenter will address questions involving the realization of the G-module of the global sections of the (twisted) cotangent bundle over the flag variety G/P via the cohomology of the small quantum group.
Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar, and provides a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided which relate the multiplicities of simple G-modules in the global sections with the dimensions of extension groups over the large quantum group.
This talk represents joint work with Zongzhu Lin.
78th Meeting
Date: October 26, 2023; Thursday
Time: 4 pm UK
Speaker: Scott Balchin (Queen's University Belfast)
Title: A jaunt through the tensor-triangular geometry of rational G spectra for G profinite or compact Lie.
Abstract: In this talk, I will report on joint work with Barnes--Barthel (arXiv:2401.01878) and Barthel--Greenlees (arXiv:2311.18808) which analyses the category of rational G equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.
77th Meeting
Date: October 20, 2023; Friday
Time: 4 pm UK
Speaker: Leovigildo Alonso Tarrio (University of Santiago de Compostela)
Title: Derivators in additive context.
Abstract: By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes form closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.
76th Meeting
Date: October 13, 2023; Friday
Time: 4 pm UK
Speaker: Yann Palu (Université UPJV Amiens)
Title: 0-Auslander extriangulated categories.
Abstract: Categorification of cluster algebras has instilled the idea of mutation in representation theory. Nice theories of mutation, for some forms of rigid objects, have thus been developed in various settings. In a collaboration with Mikhail Gorsky and Hiroyuki Nakaoka, we axiomatized the similarities between most of those settings under the name of 0-Auslander extriangulated categories. The prototypical example of a 0-Auslander extriangulated category is the category of two-term complexes of projectives over a finite-dimensional algebra. In this talk, we will give several examples of 0-Auslander categories, and explain how they relate to two-term complexes. This is based on joint work with Xin Fang, Mikhail Gorsky, Pierre-Guy Plamondon, and Matthew Pressland and is related to works by Xiaofa Chen and by Dong Yang.
75th Meeting
Date: October 5, 2023; Thursday
Time: 4 pm UK
Speaker: Paul Balmer (UCLA)
Title: The geometry of permutation modules.
Abstract: In joint work with Martin Gallauer, we study the tensor-triangular geometry of the derived category of permutation modules over a finite group, and more generally over profinite groups. Martin and I have already spoken on this topic in various venues. So I’ll try to comment on aspects that were not highlighted so far, like the construction of the modular fixed points (or Brauer quotients), the Koszul objects and the reduction to elementary abelian groups. If time permits, I’ll say a few words about the profinite case, which is still partially work-in-progress.
74th Meeting
Date: July 18, 2023; Tuesday
Time: 4 pm UK
Speaker: Amalendu Krishna (Indian Institute of Science, Bangalore)
Title: Ramified class field theory of curves over local fields.
Abstract: In this talk, I will present some results on the class field theory of smooth projective curves over a local field where one allows arbitrary ramification along a proper closed subset. We shall derive these results using some new results on the class field theory of 2-local fields and a duality theorem. This is based on a joint work with Subhadip Majumder (arXiv:2307.15416).
73rd Meeting
Date: July 4, 2023; Tuesday
Time: 4 pm UK
Speaker: Jan Trlifaj (Charles University, Prague)
Title: Deconstructible classes, approximations, and AECs of modules.
Abstract: Deconstructible classes of modules are among the main sources of approximations in relative homological algebra. They also occur in connection with abstract elementary classes (AECs). The latter were introduced by Shelah as far-reaching generalizations of classic first-order structures [2]. A direct connection is provided by the "AECs of roots of Ext": these are the AECs of the form P = (A,≼) where A = { M in Mod-R such that Ext^i_R(M,N) = 0 for all i > 0 and all N in C} for a class of modules C, and ≼ is a partial order on A satisfying X ≼ Y, iff Y/X is in A. By [1], P is an AEC, iff A is a deconstructible class closed under arbitrary direct limits.
A major open problem concerning AECs is Shelah's Categoricity Conjecture (SCC). It claims that categoricity of an AEC is a large enough cardinal λ (= existence of a unique structure in A of cardinality λ up to isomorphism) is equivalent to its categoricity in a tail of cardinals.
After recalling the role of deconstructible classes of modules, we will prove SCC for the AECs of roots of Ext, and more in general, for all "deconstructible" AECs (D,≤), i.e., such that D is a deconstructible class of modules [4]. We will also consider the open problem of whether for all deconstructible AECs, the class D is necessarily closed under direct limits. We will show that it is consistent with ZFC that D is closed under countable direct limits provided that D is closed under direct summands and ≤ refines direct summands [3].
References:
[1] J.T.Baldwin, P.C.Eklof, J.Trlifaj. ⊥N as an abstract elementary class. Annals of Pure Appl. Logic 149(2007), 25-39.
[2] S.Shelah. Classification Theory for Abstract Elementary Classes, vols. 1 and 2, Studies in Logic, no. 18 & 20, College Publ., London 2009.
[3] J.Saroch, J.Trlifaj. Deconstructible abstract elementary classes of modules. manuscript.
[4] J.Trlifaj. Categoricity for transfinite extensions of modules. arXiv:2212.04433v1.
72nd Meeting
Date: June 21, 2023; Wednesday
Time: 4 pm UK
Speaker: Francesco Genovese (Milan)
Title: Deforming t-structures.
Abstract: A guiding principle of non-commutative algebraic geometry is that geometric objects (i.e. rings and schemes) are replaced by categories of modules/sheaves thereof. In order to keep track of the homological information, we actually take derived categories of such modules/sheaves.
From this point of view, we are now interested in understanding typical geometric concepts directly in this categorical framework. A key example is given by deformations.
In this talk, I will report on joint work with W. Lowen and M. Van den Bergh, where we attempt to define and study deformations categorically, in the framework of (enhanced) triangulated categories with a t-structure. This will also shed light on Hochschild cohomology.
71st Meeting
Date: June 6, 2023; Tuesday
Time: 4 pm UK
Speaker: Julia Sauter (Universität Bielefeld)
Title: Tilting theory in exact categories.
Abstract: We introduce tilting subcategories for arbitrary exact categories and discuss the question when one can get a bounded derived equivalence to a functor category over it.
70th Meeting
Date: May 30, 2023; Tuesday
Time: 4 pm UK
Speaker: Gustavo Jasso (Lund University)
Title: (Non-)uniqueness of strong enhancements.
Abstract: The Derived Auslander-Iyama Correspondence, a recent theorem of Muro and myself, guarantees the existence of unique (DG) enhancements for algebraic triangulated categories that satisfy mild finiteness conditions as well as a dZ-cluster tilting object. I will explain a by-product of our work that permits us to construct, to the best of our knowledge, the first examples of algebraic triangulated categories that admit a unique enhancement but not a unique strong enhancement in the sense of Lunts and Orlov. This talk is based on joint work with Fernando Muro (Sevilla).
Slides are available here.
69th Meeting
Date: May 23, 2023; Tuesday
Time: 4 pm UK
Speaker: Amnon Yekutieli (Ben-Gurion University)
Title: A DG Approach to the Cotangent Complex.
Abstract: Let B/A be a pair of commutative rings. We propose a DG (differential graded) approach to the cotangent complex L_{B/A}. Using a commutative semi-free DG ring resolution of B relative to A, we construct a complex of B-modules LCot_{B/A}. This construction works more generally for a pair B/A of commutative DG rings.
In the talk, we will explain all these concepts. Then we will discuss the important properties of the DG B-module LCot_{B/A}. If time permits, we'll outline some of the proofs.
It is conjectured that for a pair of rings B/A, our LCot_{B/A} coincides with the usual cotangent complex L_{B/A}, which is constructed by simplicial methods. We shall also relate LCot_{B/A} to modern homotopical versions of the cotangent complex.
The slides are available here.
68th Meeting
Date: May 16, 2023; Tuesday
Time: 4 pm UK
Speaker: Federico Binda (University of Milan)
Title: Motivic monodromy and p-adic cohomologies
Abstract: In this talk, I will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. Building on work of Ayoub, Bondarko, we provide an equivalence between the category of “unipotent” rigid analytic motives over a non-archimedean field and the category of “monodromy maps” M → M (−1) of algebraic motives over the residue field. This allows us to build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.
This is a joint work in progress with Alberto Vezzani and Martin Gallauer.
67th Meeting
Date: May 9, 2023; Tuesday
Time: 4 pm UK
Speaker: Patrick Lank (University of South Carolina)
Title: High Frobenius pushforwards generate the bounded derived category.
Abstract: This talk is concerned with generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic p when the Frobenius morphism is finite. It is shown that for any compact generator G of D(X), the e-th Frobenius pushforward of G classically generates the bounded derived category whenever p^e is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.
The corresponding paper is here: arXiv:2303.18085.
66th Meeting
Date: May 2, 2023; Tuesday
Time: 4 pm UK
Speaker: Marion Boucrot (Université Grenoble Alpes)
Title: The relation between A-infinity morphisms and pre-Calabi-Yau morphisms.
Abstract: Pre-Calabi-Yau algebras were introduced in the last decade by M. Kontsevich, A. Takeda and Y. Vlassopoulos using the necklace bracket. This notion is equivalent to a cyclic A-infinity algebra for the natural bilinear form in the finite dimensional case. Moreover, W-K. Yeung showed that double Poisson dg structures provide an example of pre-Calabi-Yau structures. In 2020, D. Fernandez and E. Herscovich proved that given a morphism of double Poisson dg algebras from A to B, one can produce a cyclic A-infinity algebra and A-infinity morphisms between the latter and the cyclic A-infinity algebras associated to A and B. I will explain how to generalize this result to pre-Calabi-Yau algebras by doing an explicit construction of a (cyclic) A-infinity algebra and A-infinity morphisms given a pre-Calabi-Yau morphism.
The corresponding paper is here: arXiv:2304.13661.
65th Meeting
Date: April 25, 2023; Tuesday
Time: 4 pm UK
Speaker: Alexey Elagin (University of Edinburgh)
Title: Dimension for triangulated categories.
Abstract: I will talk about two notions of dimension of a triangulated category. The first one is the classical Rouquier dimension, based on generation time with respect to a generator, while the second one is the more recent concept of Serre dimension, based on behavior of iterations of the Serre functor. I will propose "ideal" properties of dimension that one would like to have, and compare them to properties of Rouquier and Serre dimension, both known and conjectural. Various examples of categories where dimension is known will be given and discussed. This is based on a joint work with Valery Lunts.
64th Meeting
Date: April 18, 2023; Tuesday
Time: 4 pm UK
Speaker: Emma Brink (Ludwig-Maximilians-Universität München)
Title: Condensed Group Cohomology.
Abstract: The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of \underline{G}, \underline{M} (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of \pi_0(G) with coefficients in M.
Generalising results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological \delta-functor in the condensed setting for a large class of topological groups.
63rd Meeting
Date: April 4, 2023; Tuesday
Time: 4 pm UK
Speaker: Milen Yakimov (Northeastern University)
Title: On the spectrum and support theory of a finite tensor category.
Abstract: Finite tensor categories are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples. This is a joint with Daniel Nakano (Univ Georgia) and Kent Vashaw (MIT).
62nd Meeting
Date: March 29, 2023; Wednesday
Time: 4 pm UK
Speaker: Wilberd van der Kallen (Utrecht University)
Title: A Friedlander-Suslin theorem over a noetherian base ring.
Abstract: We show that the celebrated Friedlander-Suslin theorem - on finite generation of cohomology of a finite group scheme G over a field - remains valid for a finite flat group scheme G over a commutative noetherian ring (arXiv 2212.14600). In view of earlier work it suffices to put a uniform bound, depending on G only, on torsion in cohomology of G-modules.
Date: March 31, 2023; Friday
Time: 4 pm UK
Speaker: Martina Conte (HHU Düsseldorf)
Title: Definability of the rank and the dimension of p-adic analytic pro-p groups.
Abstract: Recently, Nies, Segal and Tent started an investigation of finite axiomatizability in the realm of profinite groups. Among the classes of profinite groups under their consideration is the class of p-adic analytic pro-p groups. In joint work with Benjamin Klopsch, we consider two key invariants of these groups, namely rank and dimension, and show that they can be characterized by a single first-order sentence. Before discussing these results I will introduce the relevant background. If time permits, I will also present some natural generalisations.
61st Meeting
Date: March 14, 2023; Tuesday
Time: 4 pm UK
Speaker: Giovanni Cerulli Irelli (Sapienza - Università di Roma)
Title: Motzkin combinatorics in linear degenerations of the flag variety.
Abstract: In 2021, Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti (IMRN 2023, arXiv 2112.02539) we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term, there is a very beautiful (as yet conjectural) formula for the coefficients.
60th Meeting
Date: March 7, 2023; Tuesday
Time: 4 pm UK
Speaker: Rhiannon Savage (University of Oxford)
Title: Algebra and Geometry in Monoidal Quasi-abelian Categories.
Abstract: A quasi-abelian category is an additive category with all kernels and cokernels, along with some additional conditions allowing us to extend notions from homological algebra to them. A key example is the category of complete bornological spaces which is derived equivalent to the category of inductive limits of Banach spaces. In this talk, we will introduce the key concepts in the theory of quasi-abelian categories and we will discuss their potential applications. In particular, we will see how we can extend ideas from Koszul duality to quasi-abelian categories, as well as their use more generally as a setting for a new theory of derived analytic geometry proposed by my supervisor Kobi Kremnizer and his collaborators.
59th Meeting
Date: February 28, 2023; Tuesday
Time: 4 pm UK
Speaker: Liran Shaul (Charles University, Prague)
Title: Finitistic dimension, generation of injectives, and dualizing complexes.
Abstract: In the 1960's, Grothendieck showed that a commutative noetherian ring which admits a dualizing complex has finite Krull dimension. In 2018, Rickard showed that a finite-dimensional algebra A for which the localizing subcategory generated by the injective modules is equal to D(A) satisfies the finitistic dimension conjecture.
In this talk we explain how to view both of these results as special cases of a single result which is valid for any noncommutative noetherian ring which admits a dualizing complex.
58th Meeting
Date: February 21, 2023; Tuesday
Time: 4 pm UK
Speaker: Francesca Fedele (University of Leeds)
Title: Ext-projectives in subcategories of triangulated categories.
Abstract: Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext^1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b—>c in C, then the triangle x—>b—>c—> extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C' of T and see how this operation is related to Iyama-Yoshino mutation of C with respect to a rigid subcategory. Time permitting, we will see the application of the result to cluster categories of type A.
57th Meeting
Date: February 7, 2023; Tuesday
Time: 4 pm UK
Speaker: Henning Krause (Universität Bielefeld)
Title: Central support for triangulated categories.
Abstract: Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.
56th Meeting
Date: January 31, 2023; Tuesday
Time: 4 pm UK
Speaker: Niko Naumann (University of Regensburg)
Title: Quillen stratification in equivariant homotopy theory.
Abstract: We prove a variant of Quillen's stratification theorem in equivariant homotopy theory for a finite group working with arbitrary commutative equivariant ring spectra as coefficients, and suitably categorifying it. We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing tensor-ideals of the category of equivariant modules over Lubin-Tate theory, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.
This is joint work with Tobias Barthel, Natalia Castellana, Drew Heard and Luca Pol, available as arXiv:2301.02212.
55th Meeting
Date: January 24, 2023; Tuesday
Time: 4 pm UK
Speaker: Henrik Holm (University of Copenhagen)
Title: The Q-shaped derived category of a ring.
Abstract: The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q. The talk is based on a joint paper with Peter Jørgensen, which has the same title as the talk.
54th Meeting
Date: January 17, 2023; Tuesday
Time: 4 pm UK
Speaker: Bernhard Keller (Paris City University)
Title: On Amiot's conjecture.
Abstract: In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with a cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained with Idun Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on recent progress in the general case obtained in joint work with Junyang Liu and based on Van den Bergh's structure theorem for complete Calabi-Yau algebras.
53rd Meeting
Date: January 10, 2023; Tuesday
Time: 4 pm UK
Speaker: Jan Stovicek (Charles University, Prague)
Title: Mutation, t-structures, and torsion pairs.
Abstract: The operation of mutation has a long history in representation theory and algebraic geometry, be it in the context of exceptional collections of sheaves or in the combinatorial study of tilting modules. The aim is to create a new object from an old one by changing a designated part of it and keeping the other part. Here I discuss a variant in the context of cosilting objects in compactly generating triangulated categories (which are also known as derived injective cogenerators for t-structures of Grothendieck type). In that case, the operation of mutation corresponds to certain nice tilts of t-structures with respect to torsion pairs. Time permitting, I will explain how this is related to the lattice of torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra, as studied by Demonet, Iyama, Reading, Reiten and Thomas.
This is joint work with Lidia Angeleri Hügel, Rosanna Laking and Jorge Vitória.
52nd Meeting
Date: December 13, 2022; Tuesday
Time: 4 pm UK
Speaker: Teresa Conde (University of Stuttgart)
Title: A functorial approach to rank functions
Abstract: Motivated by the work of Cohn and Schofield on Sylvester rank functions on rings, Chuang and Lazarev have recently introduced the notion of a rank function on a triangulated category. It turns out that a rank function on a category C can be recast as translation-invariant additive function on its abelianisation mod C. As a consequence, integral rank functions have a unique decomposition into irreducible ones, and they are related to a number of important concepts associated to the localisation theory of mod C. When C is the subcategory of compact objects of a compactly generated triangulated category T, these connections become particularly nice and provide a link between rank functions on C and smashing localisations of T. In particular, when C is the perfect derived category per(A) of a dg algebra A, this allows us to classify homological epimorphisms from A to B with per(B) locally finite via special rank functions, extending a result of Chuang and Lazarev. This talk is based on joint work with Mikhail Gorsky, Frederik Marks and Alexandra Zvonareva.
51st Meeting
Date: December 6, 2022; Tuesday
Time: 4 pm UK
Speaker: Petter Bergh (NTNU Trondheim)
Title: Support varieties for finite tensor categories.
Abstract: This is a report on recent and ongoing joint work with Julia Plavnik and Sarah Witherspoon, where we have developed a theory of cohomological support varieties for finite tensor categories. Under suitable finite generation conditions - conjectured to hold for all finite tensor categories - the varieties encode homological properties of the objects, as in the classical case for group algebras. For example, the dimension of a variety equals the complexity of the corresponding object, so that the objects having trivial support varieties are precisely the projective ones. Moreover, every potential variety is actually the support variety of some object, and the support variety of an indecomposable object is connected. I will also discuss the so-called tensor product property for varieties.
50th Meeting
Date: November 29, 2022; Tuesday
Time: 4 pm UK
Speaker: Laura Pertusi (Università degli Studi di Milano)
Title: Moduli spaces of stable objects in Enriques categories.
Abstract: Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.
In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category. This is a joint work in preparation with Alex Perry and Xiaolei Zhao.
49th Meeting
Date: November 8, 2022; Tuesday
Time: 4 pm UK
Speaker: Grigory Garkusha (Swansea)
Title: Homological Algebra for Enriched Grothendieck Categories.
Abstract: Enriched Grothendieck categories naturally occur in algebraic geometry, where associated abelian categories rarely have projectives but have plenty of information encoded by enriched category theory. In this talk general properties of derived categories for Grothendieck categories of enriched functors and various recollements of such categories will be presented. Applications are given for Voevodsky's triangulated categories of motives. This is a joint work with Darren Jones.
48th Meeting
Date: November 1, 2022; Tuesday
Time: 4 pm UK
Speaker: Mikhail Bondarko (St. Petersburg State University)
Title: From weight structures to (adjacent) t-structures
Abstract: I will speak about adjacent weight and t-structures (this means that either left or 'right hand side halves' of these 'structures' w and t coincide) in triangulated categories. In particular, for any compactly generated t there exists a weight structure w right adjacent to it. This yields injective cogenerators for the heart of t; it follows that the heart of t is Grothendieck abelian. To construct this w I proved that any perfect set of objects (in a smashing triangulated category) generates a weight structure w. Moreover, if a triangulated category satisfies the Brown representability property then t that is left adjacent to w exists if and only if w is smashing (i.e., coproducts respect weight decompositions).
47th Meeting
Date: August 9, 2022; Tuesday
Time: 4 pm UK
Speaker: Elizabeth Tatum (University of Illinois Urbana-Champaign)
Title: On a Spectrum-level Splitting of the BP⟨2⟩-Cooperations Algebra.
Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ⋀ bo and l ⋀ l. These splittings helped make it feasible to do computations using the bo- and l-based Adams spectral sequences. In this talk, we will discuss an analogous splitting for BP⟨2⟩ ⋀ BP⟨2⟩ at primes larger than 3.
46th Meeting
Date: July 26, 2022; Tuesday
Time: 4 pm UK
Speaker: Jonathan Kujawa (University of Oklahoma)
Title: Support varieties for Lie superalgebras
Abstract: Support varieties are a method which uses cohomology to bring commutative algebra and algebraic geometry to places where it doesn't obviously belong. Lie superalgebras are a graded analogue of Lie algebras and they seem well-adapted this technology. In this talk, I will give an overview of recent efforts to use support varieties to study the representation theory of Lie superalgebras. The talk will cover joint work with Boe, Nakano, and Drupieski.
45th Meeting
Date: July 19, 2022; Tuesday
Time: 4 pm UK
Speaker: Mark Behrens (University of Notre Dame)
Title: tmf-resolutions
Abstract: I'll talk about the tmf-based Adams spectral sequence, and how it detects most of the v2-periodic elements in the known range of the 2-primary stable stems. Parts of the material I will discuss are joint with Dominic Culver, Prasit Bhattacharya, JD Quigley, and Mark Mahowald.
44th Meeting
Date: June 28, 2022; Tuesday
Time: 4 pm UK
Speaker: David Barnes (Queen's University Belfast)
Title: Sheaf models for rational stable equivariant homotopy theory.
Abstract: Sheaves sit at an interface of algebra and geometry. Equivariant sheaves offer even more structure, allowing for different group actions at different stalks. We are interested in the case where both the base space and group of equivariance are profinite (that is, compact, Hausdorff and totally disconnected). This combination provides many useful consequences, such as a good notion of equivariant presheaves and an explicit construction of infinite products.
The 2019 PhD thesis of Sugrue used equivariant sheaves to give an algebraic model for rational G-equivariant stable homotopy theory, where G is profinite. In this talk I will explain the model and related results, such as the equivalence between equivariant sheaves and rational Mackey functors (for profinite G).
43rd Meeting
Date: June 21, 2022; Tuesday
Time: 4 pm UK
Speaker: Hans-Werner Henn (Université de Strasbourg)
Title: On the Brown Comenetz dual of the K(2)-local sphere at the prime 2
Abstract: The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n)-local version I_n of I. For a type n-complex X, this is given by the Pontryagin dual of the 0-th homotopy group of the K(n)-localization of X. By work of Hopkins and Gross, the homotopy type of the spectra I_n for a prime p is determined by its Morava module if p is sufficiently large. For small primes, the result of Hopkins and Gross determines I_n modulo an "error term". For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almos,t 30 years. For n>2 very little is known if the prime is small. For n=2 every prime bigger than 3 is sufficiently large. The case p=3 has been settled in joint work with Paul Goerss. This talk is a report on work in progress with Paul Goerss on the case p=2. The "error term" is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 2^9. We use chromatic splitting in order to get information on the error term.
42nd Meeting
Date: May 31, 2022; Tuesday
Time: 4 pm UK
Speaker: Serge Bouc (Université de Picardie - Jules Verne)
Title: Functorial equivalence of blocks
Abstract: In this talk, I will introduce the notion of functorial equivalence of blocks of finite groups, developed in recent joint work with Deniz Yilmaz.
For a commutative ring R, and a field k of characteristic p>0, we introduce the category of diagonal p-permutation functors over R. To a pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor F_{G,b}, and we say that two such pairs (G,b) and (H,c) are functorially equivalent over R if the functors F_{G,b} and F_{H,c} are isomorphic.
We show that the category of diagonal p-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a precise description of the simple functors, and explicit formulas for their multiplicities as summands of F_{G,b}. It follows that functorial equivalence preserves the defect groups of blocks, their number of simple modules, and their number of ordinary irreducible characters. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan's finiteness conjecture.
41st Meeting
Date: May 24, 2022; Tuesday
Time: 4 pm UK
Speaker: Jesper Grodal (University of Copenhagen)
Title: A guided tour to the Picard group of the stable module category.
Abstract: The Picard group T_k(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 70’s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate T_k(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.
40th Meeting
Date: May 17, 2022; Tuesday
Time: 4 pm UK
Speaker: Jordan Williamson (Charles University, Prague)
Title: Duality and definability in triangulated categories.
Abstract: In the category of modules over a ring, purity may be viewed as a weakening of splitting - a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. However, in general, it can be hard to check whether a class is closed under purity operations. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and illustrate this with a range of examples, often from the tensor-triangular perspective. I will also discuss an application to the study of definable subcategories of triangulated categories. This is joint work with Isaac Bird.
39th Meeting
Date: May 10, 2022; Tuesday
Time: 4 pm UK
Speaker: Inna Entova-Aizenbud (Ben Gurion University)
Title: Representation stability for GL_n(F_q).
Abstract: I will present some results from a work in progress joint with Thorsten Heidersdorf on the Deligne categories for the family of groups GL_n(F_q), for non-negative integers n. The Deligne categories interpolate the tensor categories of complex representations of GL_n(F_q), and have been previously constructed by F. Knop and E. Meir (for certain values of n). I will describe some properties of these categories as well as their relation to the category of algebraic representations of the infinite group GL_{\infty}(F_q).
38th Meeting
Date: May 3, 2022; Tuesday
Time: 4 pm UK
Speaker: Martin Frankland (University of Regina)
Title: On good morphisms of exact triangles
Abstract: When studying the Adams spectral sequence in triangulated categories, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles. The talk will include examples from stable module categories of group algebras.
37th Meeting
Date: April 19, 2022; Tuesday
Time: 4 pm UK
Speaker: Neil Strickland (University of Sheffield)
Title: Questions around the Chromatic Splitting Conjecture
Abstract: The chromatic splitting conjecture (CSC) is an important open problem in stable homotopy theory. Although Beaudry has shown that the strongest version fails when n=p=2, one can still hope that the conjecture is valid at large primes, or up to a filtration that is nearly split in some appropriate sense.
The CSC gives a description of L_{n-1}L_{K(n)}(S), but from that one can deduce similar descriptions of many other spectra, including those of the form
L_{K(n_1)}L_{K(n_2)}...L_{K(n_r)}(S) with n_1<...<n_r. The spectrum L_n(S) can be expressed as the homotopy inverse limit of a diagram of spectra of that form, and one can ask whether that, and various similar phenomena, are consistent with the CSC. We will explain a conjecture about how all this fits together, which involves some interesting algebra and combinatorics. We will also explain how Morava K-theory Euler characteristics can be used to do some basic consistency checks.
36th Meeting
Date: March 29, 2022; Tuesday
Time: 4 pm UK
Speaker: Andrew Baker (University of Glasgow)
Title: Locally Frobenius Hopf algebras and their modules.
Abstract: Motivated by work on the Steenrod algebra, Moore and Peterson introduced the notion of (graded) nearly Frobenius algebras; these were later renamed P-algebras by Margolis. This is a preliminary report on the development of an analogous theory for non-graded Hopf algebras which as far as I know is not in the algebra literature.
I will give a very brief overview of the graded theory, then explain one approach to emulating it based on filtered colimits of finite-dimensional Hopf algebras which are Frobenius extensions of each other.
35th Meeting
Date: March 22, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Antonio Lorenzin (Milano-Bicocca, Pavia)
Title: Formality and strongly unique enhancements.
Abstract: Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.
34th Meeting
Date: March 15, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Paolo Stellari (Università degli Studi di Milano)
Title: Stability conditions in the trivial canonical bundle case: Hilbert schemes of points.
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 review some results and techniques related to the latter setting. I will specifically concentrate on the case of Hilbert scheme of points on K3 surfaces and (as a work in progress) on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macri' and X. Zhao.
33rd Meeting
Date: March 8, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Luca Pol (University of Regensburg)
Title: Finite covers and tt-rings.
Abstract: Balmer initiated the study of separable commutative algebras (tt-rings in short) in tt-geometry: these are commutative algebras for which the multiplication map admits a bimodule section. Their importance has grown in recent years due to the fact that the category of modules over a tt-ring is again a tt-category, and that tt-rings allow to prove strong descent results. However, the classification of all tt-rings in a tt-category is an open problem in many cases of interest. In this talk, I will relate the notion of tt-ring to the notion of finite cover due to Mathew, and use this connection to provide classification results for tt-rings in some special cases of interest. This is joint work with Niko Naumann.
32nd Meeting
Date: March 1, 2022; Tuesday.
Time: 3 pm GMT
Speaker: Fernando Muro (Seville)
Title: Uniqueness of enhancements for Hom-finite triangulated categories with an n-cluster tilting object.
Abstract: In this talk, we will report on ongoing joint work with Gustavo Jasso. The goal is to show that algebraic triangulated categories satisfying the assumptions in the title have a unique DG-enhancement over a ground perfect field, up to Morita equivalence. This extends previous work on finite triangulated categories. The key step is the connection with Geiss-Keller-Oppermann's notion of n-angulated categories, which are like triangulated categories but with longer ‘triangles'.
31st Meeting
Date: February 22, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Beren Sanders (University of California, Santa Cruz )
Title: Stratification in tensor triangular geometry with applications to spectral Mackey functors.
Abstract: In a recent paper, joint with Tobias Barthel and Drew Heard, we develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. In this talk, I will provide an introduction to the problem of classifying thick and localizing tensor-ideals via theories of support, describe in broad strokes some of the highlights of our theory (which builds on the work of Balmer-Favi, Stevenson, and Benson-Iyengar-Krause) and, time-permitting, discuss our applications in equivariant homotopy theory. The starting point for these equivariant applications is a recent computation (joint with Irakli Patchkoria and Christian Wimmer) of the Balmer spectrum of the category of derived Mackey functors. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors, and harness our geometric theory of stratification to classify the localizing tensor-ideals of both categories.
30th Meeting
Date: February 15, 2022; Tuesday.
Time: 4 pm GMT
Speaker: James Cameron (University of Utah)
Title: Homological residue fields for tensor triangulated categories and cooperations.
Abstract: Many tensor triangulated categories admit 'residue field functors' that control their large-scale structure. The derived category of a ring is controlled by the residue fields of the ring, the structure of the stable homotopy category is controlled by the Morava K-theories, and in modular representation theory there are the pi-points. Unfortunately, it is not known if every tensor triangulated category has a notion of tensor triangulated residue fields. Homological residue fields were introduced by Balmer, Krause, and Stevenson as an abelian avatar of the putative tensor triangulated residue fields. They exist in complete generality, but they are hard to understand and compute with in general. I will discuss how to connect homological residue fields with the tensor triangulated residue fields that exist in examples. I will show that for the derived category of a ring, homological residue fields are closely related to usual residue fields, and in stable homotopy theory they are closely related to Morava K-theories. In fact, the homological residue fields have even more structure, and can be identified with comodules for a Tor coalgebra which in the case of the stable homotopy category is the coalgebra of coooperations for a Morava K-theory. I will introduce homological residue fields, give some examples, and mention some open problems. This is joint work with Paul Balmer and with Greg Stevenson.
29th Meeting
Date: February 8, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Jun Zhang (University of Montreal)
Title: Triangulated persistence category.
Abstract: In this talk, we will introduce a new algebraic structure called triangulated persistence category (TPC). A TPC combines the persistence module and the classical triangulated structure so that a meaningful measurement, via cone decomposition, can be defined on the set of objects. We will also elaborate on various examples of TPC that come from algebra, topology, and symplectic geometry. Finally, we will investigate the Grothendieck group of a TPC and explain several unexpected properties. This talk is based on joint work with Paul Biran and Octav Cornea.
28th Meeting
Date: February 1, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Niall Taggart (Utrecht University)
Title: Homological localizations of orthogonal calculus.
Abstract: Orthogonal calculus is a version of functor calculus that sits at the interface between geometry and homotopy theory; the calculus takes as input functors defined on Euclidean spaces and outputs a Taylor tower of functors reminiscent of a Taylor series of functions from differential calculus. The interplay between the geometric nature of the functors and the homotopical constructions produces a calculus in which computations are incredibly complex. These complexities ultimately result in orthogonal calculus being an underexplored variant of functor calculus.
On the other hand, homological localizations are ubiquitous in homotopy theory. They are employed to split ‘integral' information into ‘prime' pieces, typically simplifying both computation and theory.
In this talk, I will describe a 'local' version of orthogonal calculus for homological localizations, and survey several immediate applications.
27th Meeting
Date: January 18, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Doug Ravenel (University of Rochester)
Title: String cobordism at the prime 3.
Abstract: String cobordism refers to the Thom spectrum for the 7-connected cover of BO, the classifying space for real vector bundles. I will describe progress toward a description of its 3-primary homotopy type in joint work with Vitaly Lorman and Carl McTague. It supports a map to tmf (the spectrum associated with topological modular forms) which is surjective in homotopy groups.
26th Meeting
Date: January 11, 2022; Tuesday.
Time: 4 pm GMT
Speaker: Jon Carlson (University of Georgia)
Title: Idempotent modules and endomorphisms.
Abstract: We work in the group algebra kG of a finite group scheme defined over a field of characteristic p greater than 0. The stable category stmod(kG) of finitely generated kG-modules is tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory C is a distinguished triangle → E → k → F → where E and F are idempotent modules in the stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to C. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture, we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.
25th Meeting
Date: December 14, 2021; Tuesday.
Time: 4 pm GMT
Speaker: Martin Gallauer (Max Planck Institute for Mathematics, Bonn)
Title: The derived category of permutation modules.
Abstract: To a field k and a finite group G, one associates the derived category of kG-modules, an important invariant that is difficult to understand in general. At least, its tensor-triangulated structure admits a familiar description in terms of the support variety.
We propose to study a refinement, the derived category of G-permutation modules over k. It has interesting interpretations in algebraic geometry, representation theory, and equivariant homotopy theory. We will say a few things we know about its tensor-triangulated structure. This is based on joint work, mostly in progress, with Paul Balmer.
24th Meeting
Date: December 7, 2021; Tuesday.
Time: 4 pm GMT
Speaker: Jérôme Scherer (École Polytechnique Fédérale de Lausanne)
Title: Floyd’s manifold is a conjugation space.
Abstract: This is joint work with Wolfgang Pitsch. We illustrate how equivariant stable homotopy methods can help us recognize the structure of a conjugation space, as introduced by Hausmann, Holm, and Puppe. We first explain their definition and present a characterization in terms of purity (obtained in previous joint work with Nicolas Ricka). We then perform equivariantly Floyd's construction from the 1970’s of a pair of 5- and 10-dimensional manifolds with four cells, relying on Lück and Uribe’s work on equivariant bundles. The 10-dimensional one is a conjugation space.
23rd Meeting
Date: November 30, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Amnon Neeman (The Australian National University, Canberra)
Title: Finite approximations as a tool for studying triangulated categories.
Abstract: A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll begin with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.
And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.
And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a major generalization of a theorem of Rouquier's, and a short, sweet proof of Serre's GAGA theorem.
22nd Meeting
Date: November 23, 2021; Tuesday.
Time: 4 pm GMT
Speaker: Fosco Loregian (Tallinn Technical University)
Title: Towards a formal category theory of derivators.
Abstract: Derivator theory, initiated by Grothendieck and Heller in the '90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (\infty,1)-category theory, and its applications to algebraic geometry/topology.
For a 2-category theorist, a (pre)derivator is a familiar object --(a suitably co/complete) prestack on the category cat of small categories--, and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.
After briefly outlining the essentials of "formal category theory'' (2-categories can be used to organize the theory of "categories with structure" just as category theory organizes the theory of "sets with structure"), I will report on a conjecture regarding the possibility to provide a "yoneda structure" or a "proarrow equipment" to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and _, this allows to speak about "locally presentable" and "accessible" objects (showing that Adamek-Rosický and Renaudin's definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).
21st Meeting
Date: November 17, 2021; Wednesday.
Time: 2 pm GMT
Speaker: Yu-Wei Fan (University of California, Berkeley)
Title: Shifting numbers of endofunctors of triangulated categories.
Abstract: One can consider endofunctors of triangulated categories as dynamical systems, and study their long-term behaviors under large iterations. There are (at least) three natural invariants that one can associate to endofunctors from this dynamical perspective: categorical entropy, and upper/lower shifting numbers. We will recall some background on categorical dynamical systems and categorical entropy, and introduce the notion of shifting numbers, which measure the asymptotic amount by which an endofunctor of a triangulated category translates inside the category. The shifting numbers are analogous to Poincare translation numbers. We additionally establish that in some examples the shifting numbers provide a quasi-isomorphism on the group of autoequivalences. Joint work with Simion Filip.
20th Meeting
Date: October 26, 2021; Tuesday.
Time: 4 pm UK time
Speaker: Scott Balchin (Max Planck Institute for Mathematics, Bonn)
Title: The smashing spectrum of a tt-category.
Abstract: In joint work with Greg Stevenson, we prove that the frame of smashing tensor ideals of a big tt-category is always spatial. As such, by Stone duality, we are afforded a space: the smashing spectrum. In this talk, I will report on the construction of this new invariant via lattice theoretic techniques, and its relation to the Balmer spectrum. In particular, we will see that there is a surjective comparison map which detects the failure of the telescope conjecture.
19th Meeting
Date: October 19, 2021; Tuesday.
Time: 2 pm UK time
Speaker: John Greenlees (University of Warwick)
Title: The torsion Adams spectral sequence for rational torus-equivariant spectra.
Abstract: We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) \Longrightarrow [X,Y]^G_* for rational G-spectra X and Y.
This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.
18th Meeting
Date: October 7, 2021; Tuesday.
Time: 4.30 pm UK time
Speaker: Clover May (NTNU Trondheim)
Title: Classifying perfect complexes of Z/2-Mackey functors.
Abstract: Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=Z/2. Our decomposition leads to a computation of the Balmer spectrum of the derived category. We extend these results to classify all finite modules over the equivariant Eilenberg--MacLane spectrum HZ/2.
17th Meeting
Date: March 23, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Paul Balmer (University of California, Los Angeles)
Title: Mackey 2-functors.
Abstract: This talk is based on ongoing joint work with Ivo Dell'Ambrogio. I will give an introduction to the notion of Mackey 2-functors, as a categorization of ordinary Mackey functors. I will discuss examples beyond linear representation theory and present some applications, including Green equivalences. Finally, I will say a word about our most recent work on cohomological Mackey 2-functors.
16th Meeting
Date: March 16, 2021; Tuesday.
Time: 4 pm GMT
Speaker: Julia Pevtsova (University of Washington, Seattle)
Title: Support and tensor product property for integrable finite-dimensional Hopf algebras.
Abstract: For a finite dimensional Hopf algebra A the cohomological support on the stable category Stab A can be defined via the Benson-Iyengar-Krause theory of local cohomology functors, with no reference to the tensor structure.
Yet, for various finite tensor categories the cohomological support turns out to respect that structure via the "tensor product property": supp(M \otimes N) = supp M \cap supp N. When the property holds, it often appears to be intimately connected with some kind of alternative description of the cohomological support, "a rank variety". I will describe such an alternative construction, the hypersurface support, which goes back to the work of Eisenbud, Avramov, Buchweitz and Iyengar in commutative algebra, in the case of a finite-dimensional integrable Hopf algebra. Applications include (only some) small quantum groups, quantum linear spaces, Drinfeld doubles of finite group schemes, rings of functions on finite group schemes and elementary finite supergroup schemes. Joint work with C. Negron and D. Benson, S. Iyengar, H. Krause.
15th Meeting
Date: March 9, 2021; Tuesday.
Time: 2 pm GMT
Speaker: Robert Kropholler (University of Munster)
Title: Homological filling functions and the word problem for almost finitely presented groups.
Abstract: The Dehn function of a finitely presented group is a classical invariant to study the complexity of the word problem. One can study the homological analogue of this function, namely, the homological filling function. This function is defined on the larger class of almost finitely presented groups. In recent joint work with Noel Brady and Ignat Soroko we study these functions and give surprising examples of almost finitely presented groups with quartic homological filling function but unsolvable word problem.
14th Meeting
Date: February 23, 2021; Tuesday.
Time: 2 pm GMT
Speaker: Pallavi Dani (Lousiana State University)
Title: Right-angled Coxeter groups commensurable to right-angled Artin groups.
Abstract: A well-known result of Davis-Januszkiewicz states that every right-angled Artin group (RAAG) is commensurable to some right-angled Coxeter group (RACG). I will talk about joint work with Ivan Levcovitz, in which we explore the converse statement. We establish criteria for constructing finite-index RAAG subgroups of RACGs. As an application, we prove that a 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG.
13th Meeting
Date: February 16, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Mima Stanojkovski (MPI, Leipzig)
Title: Intense automorphisms of finite groups.
Abstract: Let G be a finite group and let Int(G) be the subgroup of Aut(G) consisting of those automorphisms (called "intense") that send each subgroup of G to a conjugate. Intense automorphisms arise naturally as solutions to a problem coming from Galois cohomology, still they give rise to a greatly entertaining theory on its own. We will discuss the case of groups of prime power order and we will see that, if G has prime power order but Int(G) does not, then the structure of G is (surprisingly!) almost completely determined by its nilpotency class.
The results I will present are part of my Ph.D. thesis, which was supervised by Hendrik Lenstra.
12th Meeting
Date: February 9, 2021; Tuesday.
Time: 2 pm GMT
Speaker: Marco Pellegrini (Universita Cattolica del Sacro Cuore)
Title: A brief survey of the (2,3)-generation of the finite simple groups.
Abstract: A finite group is said to be (2,3)-generated if it can be generated by an involution and an element of order 3. Lots of results have been obtained on the (2,3)-generation of the nonabelian finite simple groups. In particular, I will review some of them and I will describe some recent developments obtained in collaboration with M. Chiara Tamburini. Thanks to these results, the classification of the (2,3)-generated finite simple groups is now complete, except for the finite orthogonal groups.
11th Meeting
Date: February 2, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Dawid Kielak (University of Oxford)
Title: The Friedl-Tillmann polytope.
Abstract: I will introduce the Friedl-Tillmann polytope for one-relator groups, and then discuss how it can be generalized to the L^2-torsion polytope of the Friedl-Lück polytope, how it connects to the Thurston polytope, and how we can view it as a convenient source of intuition and ideas.
Time: 2.15 pm GMT
Speaker: Jens Harlander (Boise State University)
Title: Wirtinger presentations of deficiency one.
Abstract: Wirtinger presentations of deficiency one arise in a variety of topological settings such as knots, virtual knots, and higher dimensional ribbon knots. Adding a generator to the list of relators gives a balanced presentation of the trivial group. Thus the Wirtinger complex, the standard 2-complex built from the presentation, embeds in a contractible 2-complex, and hence, as conjectured by J. H. C. Whitehead (1941), should be aspherical. This has been shown to be the case only in the classical knot setting. Wirtinger presentations of deficiency one are an important testing ground for the validity or failure of Whitehead’s asphericity conjecture.
In my talk, I will survey results concerning the geometry and topology of Wirtinger complexes and their groups. Special attention will be paid to Wirtinger complexes of Coxeter type.
10th Meeting
Date: January 26, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Peter Webb (University of Minnesota)
Title: Mackey functors, biset functors and their uses.
Abstract: From their introduction around 1970, Mackey functors, and subsequently biset functors, have provided a way to formalize operations of induction, restriction, and conjugation. Typical examples of structures with such operations are representation rings of groups, the Burnside ring, group cohomology, and K-groups of group rings. Viewing things in this light has led to methods of computation of group cohomology and K-groups, for example, as well as other applications. At the same time, the study of the algebraic structure of Mackey functors has revealed a language in which to reformulate conjectures from local representation theory, such as Alperin's weight conjecture. I will review these topics, indicating applications, and will go on to describe recent work extending the theory of biset functors to a situation where they are defined on categories, not just groups.
Time: 2.15 pm GMT
Speaker: Akhil Mathew (University of Chicago)
Title: Induction theorems for ring spectra.
Abstract: The classical Artin induction theorem states that the representation ring of a finite group (considered as a Green functor), after tensoring with Q, is induced from the family of cyclic groups. A generalization due to Swan implies that this is true when one considers representations with integral coefficients. I will describe some induction theorems for ring spectra, where the relevant family of subgroups varies, and some applications to algebraic $K$-theory. Joint with Dustin Clausen, Niko Naumann, and Justin Noel.
9th Meeting
Date: January 19, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Henning Krause (Bielefeld)
Title: Local versus global representations of finite groups.
Abstract: We consider the stable category of modular representations of a finite group. There are local cohomology functors which are parametrized by the prime ideals of the cohomology ring. These functors provide categories of local representations that are supported at a single prime. The talk will focus on these local categories. They are tensor triangulated and it turns out that compact and dualizing objects do not coincide, in contrast to the category of global representations. The talk presents recent progress from an ongoing collaboration with Dave Benson, Srikanth Iyengar, and Julia Pevtsova.
Time: 2.15 pm GMT
Speaker: Janina Letz (Bielefeld)
Title: Generation in derived categories and applications to commutative algebra.
Abstract: If an object X is contained in the thick subcategory generated by G, then X can be obtained from G by taking finitely many summands and cones. The number of cones required is the generation time of X with respect to G. I will give a survey on what is known about this invariant in the derived category of modules over a ring, and how the structure, as a triangulated category, of the bounded derived category of a commutative noetherian ring can be used to characterize properties of a ring or a ring map.
8th Meeting
Date: January 12, 2021; Tuesday.
Time: 1 pm GMT
Speaker: Timothee Marquis (Universite Catholique de Louvain)
Title: Structure of conjugacy classes in Coxeter groups.
Abstract: In this talk, I will present a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. After motivating the problem and reviewing its history, I will explain the key idea, of geometric nature, behind the proof of its solution.
Time: 2.15 pm GMT
Speaker: Amit Kuber (Indian Institute of Technology, Kanpur)
Title: Combinatorics of the bridge quiver and the stable rank of a string algebra.
Abstract: String algebras are a class of tame representation type finite-dimensional algebras whose Auslander-Reiten (AR) quivers, i.e., the classification of their finite-dimensional representations and maps between them is completely known, thanks to Gelfand-Ponomarev and Butler-Ringel, in terms of certain walks on the quivers known as "strings" and "bands".
The bridge quiver associated with a string algebra comprises of bands as vertices and some special strings called bridges as arrows and encodes useful information about certain algebraic invariants. Domestic string algebras are characterized by their acyclic bridge quivers and are well-studied whereas there is a lot to explore on the non-domestic side.
After explaining the basics of string algebras in the talk I will explain a new combinatorial technique of "terms" that helps to capture the geometry of the AR quiver. This technique allows us to connect graph-theoretic properties of the bridge quivers of non-domestic string algebras with the study of their category-theoretic radical.
This is joint work with E. Gupta and S. Sardar.
7th Meeting
Date: January 6, 2021; Wednesday.
Time: 1.30 pm GMT
Speaker: Srikanth Iyengar (University of Utah)
Title: Rank varieties for elementary supergroup schemes.
Abstract: This talk will be about the modular representations of finite supergroup schemes and based on ongoing joint work with Benson, Krause, and Pevtsova, partly reported here. The cohomology and representation theory of a finite group is controlled to a large extent by its family of elementary abelian subgroups. And for elementary abelian groups, Jon Carlson’s theory of rank varieties has proved to be an efficacious tool. A similar picture has emerged also for finite supergroups, except that there is a larger family of ``elementary” supergroups that one has to contend with. Nevertheless, there is an analogue of the theory of rank varieties, and that forms the foundation for a theory of pi-points for general supergroups, akin to the one for finite group schemes developed by Friedlander and Pevtsova. The plan for my talk is to present the construction and some noteworthy features of these new rank varieties. I will not assume familiarity with the theory of supergroups. In fact, elementary supergroup schemes can be treated entirely from the point of view of commutative algebra.
6th Meeting
Date: December 15, 2020; Tuesday.
Time: 1 pm GMT
Speaker: Anastasia Hadjievangelou (University of Bath)
Title: Left 3-Engel elements in locally finite p-groups.
Abstract: Engel Theory is of significant interest in group theory as there is an unmistakable correlation between Engel and Burnside problems. In this talk we first introduce Engel elements and Engel groups and in particular we expand our knowledge on locally finite Engel groups. It is important to know that M. Newell proved that if a is a right 3-Engel element in a group G then a lies in HP(G) (Hirsch-Plotkin radical) and in fact he proved the stronger result that the normal closure of a is nilpotent of class at most 3. The natural question arises whether the analogous result holds for left 3-Engel elements. We will give various examples of locally finite p-groups G containing a left 3-Engel element x whose normal closure is NOT nilpotent.
5th Meeting
Date: December 8, 2020; Tuesday.
Time: 1 pm GMT
Speaker: John Nicholson (University College London)
Title: Projective modules and exotic group presentations.
Abstract: Two presentations for a group G that have the same deficiency are called exotic if the corresponding presentation complexes are not homotopy equivalent. Despite early interest by Cockroft-Swan and Dyer-Sieradski, it was not until 1976 that the first examples of exotic presentations were found by Dunwoody (for the trefoil group) and Metzler (for finite abelian groups). In recent years, applications to Wall’s D2 problem and the classification of 4-manifolds have sparked renewed interest in this problem. In this talk, we will discuss exotic presentations for groups G with 4-periodic (group-) cohomology and their relation to the classification of projective ZG-modules. We will also briefly explain how to construct a 4-manifold from a group presentation and show how exotic presentations lead to interesting examples of 4-manifolds.
4th Meeting
Date: December 2, 2020; Wednesday.
Time: 1 pm GMT
Speaker: Lleonard Rubio y Degrassi (Padova)
Title: On the (restricted) Lie algebra structure of Hochschild cohomology.
Abstract: Let A be a finite-dimensional associative algebra and let k be an algebraically closed field. Every algebra A decomposes into indecomposable algebras called blocks. This implies that a classification and an explicit description of the blocks allow to address fundamental problems in representation theory such as the study of the indecomposable A-modules. The study of blocks depending on their number of indecomposable modules leads to a trichotomy: finite (in which indecomposable are finite), tame (in which the indecomposable can be parameterized), or wild (in which the indecomposable modules are 'unclassifiable').
Hochschild cohomology is an important invariant that allows us to understand blocks up to three categorical equivalences: Morita, derived, and stable equivalence of Morita type. The Hochschild cohomology of an algebra A possesses a great deal of structure, perhaps most famously, HH*(A) is a Gerstenhaber algebra, which endows HH^1(A) with a Lie algebra structure.
In the first part of this talk, we will focus on the Lie structure of HH^1(B) in the case when B is a block of a group algebra of a finite group G and the characteristic of k divides the order of G. Apart from a few exceptions, we will show that if B is a block with cyclic defect or tame, then HH^1(B) is a solvable Lie algebra.
In the second part, we will study another operation defined on Hochschild cohomology that arises over a field of positive characteristic, making HH^{>0}(A) into a restricted graded Lie algebra. We will show that if A and C are self-injective algebras that are stably equivalent of Morita type, then the induced transfer map on Hochschild cohomology gives an isomorphism of restricted graded Lie algebras between HH^{>0}(A) and HH^{>0}(C). As a consequence, all of the below are invariants under stable equivalences of Morita type:
the restricted Lie algebra HH^1(A);
the induced restricted Lie algebra on the centre Z(HH^1(A));
the restricted universal enveloping algebras of HH^1(A);
This is joint work with Briggs, Linckelmann, Schroll, and Solotar.
3rd Meeting
Date: November 25, 2020; Wednesday.
Time: 1 pm GMT
Speaker: Sam Hughes (University of Southampton)
Title: The Unstable Gromov-Lawson-Rosenberg Conjecture for certain S-arithmetic groups.
Abstract: In this talk, we will introduce the Unstable Gromov-Lawson-Rosenberg Conjecture relating a K-theoretic invariant of compact spin manifolds to the non-existence of metrics of positive scalar curvature. If time permits, we will discuss a proof of the conjecture for manifolds with fundamental group isomorphic to certain low dimensional S-arithmetic lattices.
Time: 2.30 pm GMT
Speaker: Aparna Upadhyay (SUNY Buffalo)
Title: The non-projective part of the tensor powers of modules of the symmetric group.
Abstract: In a recent paper, Dave Benson and Peter Symonds defined a new invariant \gamma_G(M) for a finite-dimensional module M of a finite group G. This invariant quantifies the asymptotic behaviour of the non-projective part of the tensor powers of a finite-dimensional modular representation of a finite group. In this talk, we will see some interesting properties of the invariant \gamma_G(M) and determine this invariant for a class of modules of the symmetric group.
2nd Meeting
Date: November 17, 2020; Tuesday.
Time: 1 pm GMT
Speaker: Oihana Garaialde Ocana (University of the Basque Country)
Title: On the cohomology of groups.
Abstract: How much information does the cohomology ring of a group give us about the group itself? More precisely, which type of properties of a group does its cohomology ring encode? Or, how hard is it to compute such cohomology rings? In this talk, we will discuss several applications of cohomology on the theory of finite groups.
1st Meeting
Date: November 10, 2020; Tuesday.
Time: 1 pm GMT
Speaker: Marialaura Noce (University of Gottingen)
Title: Engel conditions in groups.
Abstract: The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this talk, we survey on Engel elements and Engel groups, and we focus on their development during the last two decades, presenting new results and some open problems.
Time: 2 pm GMT
Speaker: Anitha Thillaisundaram (University of Lincoln)
Title: Amit's conjecture for words in finite nilpotent groups.
Abstract: Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that N_w(1) \geq |G|^{k-1}, where N_w(1) is the number of k-tuples (g_1,...,g_k) of elements in G such that w(g_1,...,g_k)=1. This conjecture is known to be true for finite groups of nilpotency class 2. In this talk, we consider a generalised version of Amit's conjecture and discuss known results. This is joint work with Rachel Camina and Ainhoa Iniguez.
Time: 3.15 pm GMT
Speaker: Rudradip Biswas (University of Manchester)
Title: 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 consequences of these results. And finally, using those results and some other results from the literature, 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.