Last updated: July 20, 2019, 16:12 EEST

Conference Titles, Abstracts, and Slides

Mohamed Barakat, Chevalley’s theorem on constructible images made constructive. Barakat

Abstract: Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasi-affine. In this talk I will present a Gröbner-basis-based algorithm which computes the constructible image of a morphism of affine spaces, along with some applications.

Manuel Cortés-Izurdiaga, Tor-pairs closed under products and Mittag-Leffler modules.

Abstract: In a previous paper, the author has studied rings for which products of flat modules have finite flat dimension. In this talk we shall extend this theory to study when the left hand class of a Tor-pair is closed under products. The main property characterizing this Tor-pairs is about the relative Mittag-Leffler dimension of modules in the right hand class of the Tor-pair. Then, we shall discuss some applications.

Sergio Estrada, Infinite powers of ghost ideals.

Abstract: Ghost ideals were firstly introduced by Christensen in the stable homotopy category. We will present an abelian version of the ghost ideal associated to a class of objects. Then we study powers of the ghost ideal and show that, under mild assumptions, any (finite or infinite) power of a ghost ideal is a complete ideal cotorsion pair, and in fact it agrees with a nice complete cotorsion pair of objects for some infinite power. Applications will be given in the category of unbounded chain complexes of modules. The talk is based on a work in progress with Xianhui Fu, Ivo Herzog and Sinem Odabasi.

Alberto Facchini, Pretorsion theories in general categories. Facchini slides

Abstract: We will present a notion of (pre)torsion theory in general categories and two interesting examples of such pretorsion theories. Torsion theories in arbitrary categories have been studied by Grandis, Janelidze, M\'arki and several others. Our main examples will be in the category of preordered sets and the category of finite algebras with one operation, unary, and no axioms (i.e., the category of all mappings f : X --> X, where X is a finite set. The results presented appear in joint papers with Carmelo Antonio Finocchiaro, Marino Gran and Leila Heidari Zadeh.

Pedro Guil Asensio, Partial morphisms in additive exact categories. Guil Asensio slides

Abstract: We develop a general theory of partial morphisms over an exact category, extending the notion introduced by Ziegler for the special case of pure-exact sequences in the category of modules over a ring. We relate these partial morphisms with injective approximations and phantom maps and study the existence of such approximations in an exact category.

Dolors Herbera, The stable category and big pure projective modules. Herbera slides

Abstract: Let M be a finitely generated module over a ring R. In this talk we will explain that the structure of the  stable category generated by a direct summand of a number of copies of M gives information on, possibly infinitely generated, summands of copies of the module M. This relation is particularly interesting in the case R is a local commutative noetherian rings, because it allows to determine  the whole class of modules that are direct summands of copies of M. The results presented are joint work with Pavel Prihoda and Roger Wiegand.

Alina Iacob, Relative Gorenstein flat modules and model structures.

Abstract: A recent result by J. Saroch and J. Stovicek shows that there is a unique abelian model structure on the category of left R-modules, for any associative ring R with identity, whose cofibrant objects are the Gorenstein flat modules and whose fibrant objects are the cotorsion modules. We generalize this result. We introduce a relative version of Gorenstein flat modules, which we call Gorenstein B-flat modules, where B is a class of right R-modules. We give sufficient conditions on the class B so that the class of Gorenstein B-flat modules is closed under extensions. We also obtain a relative version of the model structure described above. This is joint work with Sergio Estrada and Marco Perez.

Alex Martsinkovsky, Deconstructing Auslander's formulas. Martsinkovsky slides

Abstract: Among the many contributions of Maurice Auslander to mathematics it is difficult to find anything more ubiquitous than the transpose of a finitely presented module. This innocuously looking and elementary construction is fundamental for Auslander-Reiten theory and also appears in a variety of formulas. Its basic purpose is to provide a duality on the category of finitely presented modules (modulo projectives), which is based on the well-known duality for finitely generated projectives. The lack of such duality for infinite projectives makes the transpose of little use for infinite modules. In this talk, I will discuss new results providing a homological formalism that completely bypasses the transpose and produces formulas which are valid for arbitrary modules over arbitrary rings and, for finitely presented modules, specialize to Auslander's formulas.

Bachuki Mesablishvili, A categorical approach to tilting theory. Mesablishvili slides

Abstract: Tilting modules play a prominent role in representation theory of finite dimensional algebras. Tilting theory is mainly described as torsion theory in module categories. We show that it can also be accessed through the (co)monad associated to the tilting module and that some constructions related with ”tilting” can be described at this level of generality.

Jason Morton, An approach to computational category theory.

Abstract: I'll discuss some foundational efforts to build a practical computer algebra system for computational (monoidal) category theory. Such a system should be able to manipulate abstract categorical quantities such as morphism terms in a REPL, compile code expressed categorically to an efficient implementation in a particular category (e.g. numerical linear algebra, (probabilistic) databases, quantum simulation, belief networks), and finally scale to be useful for practical computational problems in modeling uncertainty in data analysis and simulation.

Sebastian Posur, Kamal Saleh, and Fabian Zickgraf, A framework for constructive category theory in computer algebra. (N.B. This is a software presentation) Posur-Saleh-Zickgraf slides

Abstract: Categorical abstraction is a powerful organizing principle in computer algebra. In this talk, we explain the concept of constructive category theory and how we implement this concept in our software project CAP - Categories, algorithms, programming. In CAP it is possible to implement algorithms and data structures using basic categorical operations as primitives. As an example, we show how our categorical framework can be used for solving linear systems over the exterior algebra by exploiting an abstraction of homomorphism functors. Moreover, we discuss the question of how to render stable categories and homotopy categories constructive, a task that heavily relies on solving such linear systems.

Sebastian Posur, Applications of computable free abelian categories. Posur slides

Abstract: Computing with universal mathematical objects can be interpreted as theorem proving. In this talk, we discuss the constructiveness of free abelian categories. Using a very concrete description of free abelian categories by Murray Adelman, we demonstrate how to validate homological lemmata like the Snake lemma computationally.
Moreover, we raise the question of how to work computationally with certain Serre quotients of free abelian categories, a question that naturally leads us to the decision problem whether two objects have the same support with respect to the Ziegler spectrum.

Mike Prest, Finitely presented functors and formulas.

Abstract: I will describe the equivalence between finitely presented functors (on finitely presented modules) and pairs of pp (=regular) formulas, as well as the duality that one has between these for right and left modules. I will also say something about the broader context this sits within.

Job Rock, Continuous representations. Rock

Abstract: In 2015 Igusa and Todorov introduced continuous representations of the real line and the resulting continuous cluster categories. In this talk we examine a revised model that begins with representations as a functor category. We compare the usual constructions arising from representation categories of type A quivers to those arising from representations of the real line. In particular, we will examine the bounded derived category and continuous versions of: cluster categories, clusters, and mutations. Joint work with Kiyoshi Igusa and Gordana Todorov.

Philipp Rothmaler
, Positive primitive torsion. Rothmaler slides

Abstract: Martsinkovsky and Russell introduced a torsion radical -- called injective torsion -- as the injective stabilization of the tensor product with the module, evaluated at the ring. In joint work with Martsinkovsky I proved that this torsion coincides with what I earlier called elementary torsion for the case where the vanishing class of modules is that of flat modules (on the other side) and the pp formulas in question are those which vanish on that class (or simply, on the regular module).

Alexander Slavik, Purity and flatness for quasicoherent sheaves. Slavik slides

Abstract: The category QCoh(X) of quasicoherent sheaves on a quasicompact quasiseparated scheme X has (at least) two natural notions of purity: the categorical one, defined using the finitely presented objects, and the geometric one, defined via the tensor product of sheaves. We investigate the relation between these two, focusing mainly on the associated pure-injective objects. We show that while the "geometric part" of the Ziegler spectrum is always compact, this might not be the case for the whole space. We also address the question of when QCoh(X) has a flat generator and its consequences for the "geometric dual" functor, i.e., the contravariant internal hom into the injective cogenerator. This is joint work with Mike Prest and Jan Šťovíček.

Alex Sorokin, Injective stabilization in categories. Sorokin slides

Abstract: To define Tate homology over arbitrary rings, Martsinkovsky and Russell introduced and studied the asymptotic stabilization of the tensor product of modules. We shall provide a similar construct for finitely complete and cocomplete closed symmetric monoidal categories with an adjoint cylinder-cocylinder pair. Our main examples will be categories of modules, pointed compactly-generated Hausdorff topological spaces, pointed (finite) sets, certain categories of rings, etc., equipped with various (co)cylinder functors.

Alberto Tonolo, Filtrations induced by tilting modules. Tonolo slides

Abstract: Let A be a ring. A n-tilting left module T naturally gives rise to n + 1 classes KE0, KE1, … , KEn of left A-modules , called Miyashita classes. If n=1, (KE0, KE1) is a torsion pair on the category of left A-modules. Therefore, each module admits a filtration with two factors belonging to KE0 and KE1. For n > 1, the Miyashita classes are too small in order to filter every left A-module. Working on the derived category D(A), we provide a decomposition of any module in terms of objects of suitable generalised Miyashita classes.

Jan Trlifaj, Transfinite extensions of simple artinian rings and the Dual Baer Criterion. Trlifaj

Abstract: Von Neumann regular semiartinian rings with primitive factors artinian (pfa for short) are particular transfinite extensions of simple artinian rings. These rings form a good testing ground for investigation of various variants of projectivity, such as R-projectivity, weak R-projectivity, and level projectivity, in the category Mod-R of all right R-modules. We will be particularly interested in small pfa's, that is, those of finite Loewy length, with each layer countably generated, and of cardinality at most the continuum.
    It is known that projectivity coincides with R-projectivity (i.e., the Dual Baer Criterion holds in Mod-R) for all right perfect rings R. Also, it is consistent with ZFC + GCH that for each non-right perfect ring R there is an R-projective module of projective dimension 1. We prove that when R is a small pfa then also the opposite consistency result holds true: in the extension of ZFC with the Axiom of Constructibility, the notions of projectivity, R-projectivity and weak R-projectivity coincide in Mod-R. (Based on my preprint arXiv:1901.01442v1.)

Peter Webb, Biset functors for categories. Webb

Abstract: In the context of group theory, biset functors have been useful in various ways: for example, computing the values of group cohomology, and providing fundamental constructions such as (the torsion free part of) the Dade group. Biset functors can also be done for categories in general, with similar goals in mind. To establish this theory we must first say what a biset is. Such things have been studied for a long time under the name of profunctors, or distributors. We form a biset category, in the same way as for groups, and functors defined on this category share many of the properties of biset functors for groups. An important role is played by the Burnside ring functor, which is now defined for categories, not just groups. It turns out the that Burnside ring of a category is a more delicate construction than for groups, and we examine its structure. A key part of the work is to establish conditions under which the cohomology of a category has the structure of a biset functor. For groups, the bisets considered must be free on one side. We introduce an extension of this condition that works for categories.

Dali Zangurashvili, Confluent term rewriting systems for the descent problem.

Abstract: One of the central notions of Grothendieck descent theory is the one of an effective descent morphism. The characterization of these morphisms in varieties of universal algebras is simple and well-known. However little is known on effective descent morphisms in categories dual to varieties. In this work the problem of characterizing effective descent morphisms in such categories is related to the notion of confluency, which is one of the central notions of the term rewriting systems theory. This gives an opportunity to employ the computational techniques of this theory to the considered problem.

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