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Jan Šťovíček: Mutation, t-structures, and torsion pairs

10th January, 2023

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.

Teresa Conde: A functorial approach to rank functions

13th December, 2022

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.

Grigory Garkusha: Homological Algebra for Enriched Grothendieck Categories

8th November, 2022

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.

Mikhail Bondarko: From weight structures to (adjacent) t-structures 

1st November, 2022

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).

Jordan Williamson: Duality and definability in triangulated categories

17th May, 2022

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.

Martin Frankland: On good morphisms of exact triangles

3rd May, 2022

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. 

Antonio Lorenzin: Formality and strongly unique enhancements

22nd March, 2022

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.

Luca Pol: Finite covers and tt-rings

8th March, 2022

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. 

Fernando Muro: Uniqueness of enhancements for Hom-finite triangulated categories with an n-cluster tilting object

1st March, 2022

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'. 

James Cameron: Homological residue fields for tensor triangulated categories and cooperations

15th February, 2022

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. 

Jun Zhang: Triangulated persistence category

8th February, 2022

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.

Amnon Neeman: Finite approximations as a tool for studying triangulated categories

30th November, 2021

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.

Yu-Wei Fan: Shifting numbers of endofunctors of triangulated categories

17th November, 2021

One can consider endofunctors of triangulated categories as dynamical systems, and study their long-term behaviours 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.

Sergio Estrada: The singularity category of an exact category – applications

11th November, 2021

We consider (big) singularity categories and Gorenstein defect categories in the setting of exact categories, and especially in the presence of a complete hereditary cotorsion pair. As a main result, we show that the vanishing of this more general Gorenstein defect category characterizes finiteness of certain Gorenstein dimensions and provide an equivalence of a big singularity category with the stable category of Gorenstein objects. Applications include a viable non-affine analogue of the (big) singularity category of a ring and a perspective on the finitistic dimension conjecture.

Scott Balchin: The smashing spectrum of a tt-category

26th October, 2021

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.

John Greenlees: The torsion Adams spectral sequence for rational torus-equivariant spectra

19th October, 2021

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, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence

Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [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.

Daniel Nakano: Non-commutative Tensor Triangular Geometry and Applications

27th September, 2021

In this talk, I will show how to develop a general non-commutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MΔC). Insights from non-commutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MΔC, K, and then to associate to K a topological space: the Balmer spectrum Spc(K). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc(K) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an MΔC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc(K). Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.