Program na podzimní semestr 2019/2020
8. 10. 2019
Michael Joseph Lieberman, Ph.D.
Model theory and accessible categories
We give a general introduction to the field of model theory—classically conceived, a field which reduces mathematical objects to syntactic descriptions, then uses the powerful tools of first order logic to analyse their structure. We notice that first order logic is not adequate for many mathematical purposes (e.g. it cannot adequately capture complex exponentiation), which forces a move first toward generalized (infinitary) logics, then to abstract model theory. The latter field is a unifying theory of classes of mathematical structures that better connects with (a) actual mathematical practice, and (b) accessible categories, a notion found in every corner of mathematics.
This connection between model theory and category theory (and, ultimately, set theory) is currently driving a great deal of research, including by myself, Rosický, and Vasey.
22. 10. 2019
Mgr. Ivan di Liberti (Masarykova univerzita)
Topos-theoretic approaches to abstract model theory
Model theory is the study of mathematical structures axiomatizable in classical first order logic. These structures are typically organized in elementary classes, which are the main object of investigation of model theorists. In the 70’s, Shelah introduced abstract elementary classes as a framework to apply model theoretic techniques to infinitary logics. It’s always a relevant challenge to provide syntactic axiomatizations of an abstract elementary class. In full generality, this problem leads to complications and the results are not completely satisfactory. We try to give a categorical approach to this problem via topos theory. In a nutshell, topos theory is where geometry meets logic. A topos is both a generalized theory and a generalized space. Given an AEC, say A, (and even more generally an accessible category with directed colimits) we introduce the Scott topos S(A) of A. Since topoi can be seen as semantic incarnations of theories, the Scott topos of an AEC is a candidate axiomatization of the AEC it lf. We show that this intuition is natural and fruitful.
12. 11. 2019
John Denis Bourke, Ph.D. (Masarykova univerzita)
Variations on injectivity
The notion of an injective module is a basic one in algebra. More generally, injectivity (in the more general sense of category theory) appears in many areas, and is a central basic concept in homotopy theory. In the last ten years, the notion of ``algebraic injectivity" appeared and has begun to play a more prominent role, in particular in settings where constructive methods are of interest. In this talk I will talk about this relatively recent variant of injectivity, the mathematical structures it can encode and some of the things it is good for.
3. 12. 2019
Raffael Stenzel, Ph.D. (Masarykova univerzita)
Homotopy Type Theory - What functional programming and homotopy theory have in common
Type theory as a logical system to formalize mathematical reasoning is one of the first such systems ever developed in modern terms. Although it lost the competition to be "the" default mathematical foundation early on to set theory, it underwent crucial transformations and extensions since its formulation by Russell and Whitehead in the beginning of the last century.
Typed lambda-calculus found application in the formalization of the concept of computability and therefore as a programming paradigm in theoretical computer science. After a slowly progressing integration of the system into mainstream mathematics, a major breakthrough occurred in the 2000's with Voevodsky's intuition derived from homotopy theory, establishing what is now called Homotopy Type Theory - a formal recursive theory that has both computational features and natural interpretations in the worlds of algebraic topology, differential geometry and higher category theory.
In this talk, we will motivate and sketch the calculus of Homotopy Type Theory, outline its significance for automated theorem provers and discuss Voevodsky's fundamental Univalence Axiom. In the end, if time permits, we will explain how univalence yields a formal understanding of universal fibre bundles in classical algebraic topology.
17. 12. 2019
Mgr. Jan Pavlík, Ph.D.
Vícegrupový monoid a jeho zobecnění
Množina nezáporných reálných čísel s násobením je monoid s nulovým prvkem. Sám tento prvek tvoří podpologrupu, která je grupou. To platí i její pro doplněk, tj. množinu nenulových prvků. Tato situace, kdy monoid můžeme rozložit na podpologrupy, které jsou grupy, pak nastává také na kartézské mocnině tohoto monoidu. Naším cílem bude zobecnění této konstrukce a zkoumání jejích vlastností.
Program bude upřesňován. Předpokládáme další dva nebo tři příspěvky z teorie kategorií.
Seminář probíhá obvykle v úterý od 10:00 v seminární místnosti A1/1842 Ústavu matematiky FSI VUT, Technická 2, Brno. Dle zájmu budou vkládány další přednášky do volných termínů. Předpokládaná délka semináře je 60-90 minut.