We will continue on Thursday, October 3, in M5 at 1pm by the talk

P. Arndt

Ranges of functors and geometric elementary classes

Abstract:
Given first order theories S,T and a functor F:Mod(S)-->Mod(T) between their categories of models, one can ask whether objects in the image of F satisfy first order sentences other than those of T, or whether the essential image of F can be described as Mod(T') for an extension T' of T. If Mod(S), Mod(T) are k-accessible and F is a strongly k-accessible functor for some cardinal k, we can give criteria for this in the realm of Espíndola's k-geometric first order theories.
To this end we consider k-classifying toposes associated to S and T. The hypotheses ensure that the functor F is induced by a k-geometric essential morphism between them. The criteria are then obtained by factorizing this geometric morphism appropriately. We will explain the involved notions and give examples and applications.

The seminar on differential geometry will continue with this lecture:

September 30, 10am, lecture room M5.

Omid Makhmali:

Causal structures and related geometries

Abstract:

A  causal structure is defined by a field of tangentially nondegenerate projective hypersurfaces over a manifold, which is an extension of conformal pseudo-Riemannian structures. Using Cartan's method of equivalence, we will solve the local equivalence problem for causal structures, realize them as parabolic geometries and give a geometric interpretation of their fundamental invariants. We will focus on four dimensional causal structures and extend several twistorial constructions that arise in conformal geometry. This work is partly joint with W. Kry'nski.

We will continue on Thursday, September 26, in M5 at 1pm by the talk

R. Stenzel

From Univalence to descent via "split indexed quasi-categories"

Univalence is a type theoretical notion at the heart of Voevodsky’s Univalent Foundations Program, and Descent is a property of presentable (infinity,1)-categories introduced by Rezk as a slick way to define higher toposes. In recent years it has been understood that univalence (i.e. the existence of certain univalent maps) and descent are two sides of the same coin when the (infinity,1)-category is presentable.

In the talk I will explain this correspondence in the world of model categories, but instead via local classes in the sense of Gepner and Kock, we will take an excursion to complete Segal objects and their associated "indexed
quasi-categories". We will see that in this way a third property arises naturally - call it P for short - which relates univalence and descent directly to two properties crucial for the model theory of HoTT: the fibration extension property and the weak equivalence extension property.

In the first half of the talk I will briefly introduce all notions referred to above and draw a big diagram relating them.
In the second half I will introduce the property P and elaborate on its relation to descent both in the presentable and non-presentable case, with a view towards the space between logoi in the sense of Anel and elementary higher toposes in the sense of Rasekh.

Module Theory in Sup and its applications

Ulrich Höhle

The lecture series will take place at the Department of Mathematics and Statics, Faculty of Science, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic. Each talk will be approx. 90 minutes.

1. Talk: Module Theory in Sup and Enriched Order Theory. (Thursday, October 3, 10.00, Seminar room)

2. Talk: Topological Representation of Semi-Unital Quantales with Applications to C*-algebras. (Thursday, October 3, 14.30, Seminar room)

3. Talk: Part I: Three-Valuedness --- The First Step Towards Many-Valuedness (Tutorial).
Part II: Applications of Module Theory in Sup to Linear Stochastic Programming. (Friday, October 4, 13.00, Seminar room)

4. Talk: Preservation of Projective Right Modules in Sup under Duality. (Monday, October 7, 14.00, lecture room M3)

Abstract

1. Since many people are educated in module theory over abelian groups and  NOT over complete lattices, as a first step I will give an introduction into the general principles of module theory in Sup. Among other things I will refer to A. Joyal and M. Tierney 1984.  One interesting thing is of course how we derive the related Q-preorder from a given right action over Q. Since the tensor product in Sup plays here an important role, I will also say something about its construction.
   
   
2. Applications of module theory in Sup. I proceed now from pure to applied mathematics.
   
   (a) Topological  representation of semi-unital quantales with applications to C*-algebras.
   (b) Three-valuedness ---- the first step towards many-valuedness.
   (c) Applications of module theory in Sup to linear stochastic programming.
   (d) Projective right modules  and their dual right modules over  involutive and unital quantales: The problem of the preservation of their projectivity.