Gene Abrams

Title: Sandpile models and Leavitt path algebras  (joint work with Roozbeh Hazrat)

Abstract: The notion of a ‟sandpile model” has been studied by physicists since the 1980s, in part to model forest fires, traffic jams, and various electromagnetic systems.  These models posit a grid, in which particles move according to a specified set of rules. Associated to any sandpile model $G$ is its ‟sandpile monoid” $SP(G)$.      We show how these sandpile monoids arise naturally as the $\mathcal{V}$-monoid of weighted Leavitt path algebras.  We use as our approach to (weighted) Leavitt path algebras the Ara/Moreno/Pardo realization of these algebras as the Bergman algebra corresponding to a specified monoid given by specific generators and relations.
The talk is intended for a general mathematical audience; no prior knowledge of Leavitt path algebras (nor of sandpile models) will be assumed!

Lidia Angeleri Hügel

Title: Wide coreflective subcategories and torsion pairs

Abstract: A subcategory X of the module category Mod A over a ring A is said to be
reflective, respectively coreflective, if the inclusion functor X ↪Mod A admits a left, respectively right, adjoint. A result of Gabriel and de la Peña characterizes the subcategories which are both reflective and coreflective as those which arise as module categories X = Mod B from some ring epimorphism A → B. Much less is known when only one of the two conditions is satisfied, even when restricting to wide, i.e. exact abelian, subcategories of Mod A.
In my talk I will review a construction going back to work of Ingalls and Thomas which assigns to a torsion pair two wide subcategories in Mod A. These subcategories are often coreflective, and I will address the question of which wide coreflective subcategories can be obtained in this way. When A is the Kronecker algebra, this leads us to an open problem of Henning Krause and Greg Stevenson concerning the classification of localizing subcategories in the derived category of quasi-coherent sheaves on the projective line: are there more localizing subcategories beyond the ones constructed from our understanding of the compact objects?
The talk will be based on joint work with Francesco Sentieri

Dikran Dikranjan

Title: Tampering with topologicsal modules and closure operators — my collaboration with Alberto

Abstract: in this talk I will review the work of Alberto Tonolo in the field of topological groups and modules, as well as in a special field of category theory, namely categorcal closure operators.

Rosanna Laking

Title: Critical and special objects for cotilting modules

Abstract: In joint work with Lidia Angeleri Hügel and Ivo Herzog we calculate the simple objects and their injective envelopes in the heart of the t-structure associated to a 1-cotilting module.  In this talk I will describe how we can identify them in the module category as certain modules, called critical and special, that are neg-isolated the cotilting class.

Manuel Saorín

Title: (Quasi)tilting objects and (semi)special preenveloping torsion classes in abelian categories

Abstract: We will give a definition of (quasi)tilting object in an arbitrary abelian category that restricts to the classical ones appeared in the literature, both in the context of 'big' abelian (e.g. module categories or categories of quasi-coherent sheaves on schemes) and 'small' abelian categories (e.g. Hom-finite categories overs a field). Emulating the situation of module categories, where there are bijections (both in the big and small context) between equivalence classes of tilting objects and special preenveloping torsion classes, we shall try to understand to what extent these bijections hold in our fully general context.  If time permits, we will study in particular (quasi)tilting objects in the category of finitely presented modules over a coherent ring, a study that rises questions of interest in classical module theory.
 

Jan Trlifaj

Title: Relative Mittag-Leffler modules, approximations, and Zariski locality

Abstract: The interest in flat Mittag-Leffler modules goes back to the 1971 proof by Raynaud and Gruson of Grothendieck’s conjecture on Zariski locality of
the notion of an infinite dimensional vector bundle. Much later, Drinfeld suggested to study generalized vector bundles as the quasi-coherent sheaves whose modules of sections are flat Mittag-Leffler. However, possibilities of computing cohomology using generalized vector bundles/flat Mittag-Leffler modules turned out to be rather limited: ˇSaroch confirmed Enochs’ conjecture for those modules by proving in 2018 that they form a precovering class only if the underlying ring is perfect. In the meantime, relative Mittag-Leffler modules were introduced by Rothmaler, and studied in connection with infinite dimensional tilting theory by Angeleri Hügel and Herbera.
We will show that the classes of relative flat Mittag-Leffler modules share many properties with the class of all absolute ones: their structure is locally determined by their countably presented modules, the Enochs Conjecture holds for them, and for the particular case of relative flat Mittag-Leffler modules induced by definable classes of finite type, Zariski locality holds for the corresponding notions of quasi-coherent sheaves (based on joint work
with Asmae Ben Yassine, arXiv:2110.06792v2 and arXiv:2208.00869v1).