We will continue online on Thursday, November 19th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Maru Sarazola (Cornell University)

The stable homotopy hypothesis

Abstract:
The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.

In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof. This is based on joint work with Moser, Ozornova, Paoli and Verdugo.

We will continue online on Thursday, November 12th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Marcos Mazari-Armida (Carnegie Mellon University)

Model-theoretic stability and superstability in classes of modules

Abstract:
Dividing lines in complete first-order theories were introduced by Shelah in the early seventies. A dividing line is a property such that the classes satisfying such a property have some nice behaviour while those not satisfying it have a bad one. Two of the best understood dividing lines are those of stability and superstability.
In this talk, I will study the notion of stability and superstability in abstract elementary classes of modules with respect to pure embeddings, i.e., classes of the form (K,≤p) where K is a class of R-modules for a fixed ring R and ≤p is the pure submodule relation. In particular, using that the class of p-groups with pure embed- dings is a stable AEC, I will present a solution to Problem 5.1 in page 181 of Abelian Groups by Laszlo Fuchs. Moreover, I will show how the notion of superstability can be used to give new characterizations of noetherian rings, pure-semisimple rings, and perfect rings.

We will continue online on Thursday, November 5th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Ivan Di Liberti

Enriched Locally Generated Categories

Abstract:
We introduce the notion of M-locally generated category for a factorization system (E,M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.  This is a report on joint work with Jiří Rosický.

The seminar on differential geometry continued with public habilitation lecture:

November 2, 11am, online on ZOOM.

Record of the lecture is available via this LINK.

Yaroslav Bazaikin (Hradec Králové):

On constructions of cohomogeneity one Spin(7)-holonomy Riemannian metrics

Abstract:

An intrinsic property of a curved Riemannian manifold is the a priori non-commutativity of directional derivatives and, as a consequence, the dependence of the parallel translation operation on the chosen path on the Riemannian manifold. The holonomy group serves as a measure of this dependence and is a global object related to a Riemannian manifold that characterizes the deep properties of its geometry. In particular, in many important cases, the presence of a special holonomy group allows us to conclude that the Riemannian manifold is Einstein, which explains the importance of the concept of holonomy in applications to theoretical physics.
After getting familiar with the basic concepts of holonomy groups, the talk will give a survey of the author's results on constructions of examples of Spin(7)-holonomy Riemannian manifolds of cohomogeneity one, based on the geometry of 3-Sasakian manifolds.