
Abstract convergence schemes and their complexities (GX2031529X)
from 01/01/2020
to 31/12/2024 investigator

Objectives:
Abstract convergence schemes are basic categorytheoretic structures which serve as universes for studying infinite evolutionlike processes and their limiting behavior. Convergence schemes endowed with extra structures provide an applicable framework for studying both discrete and continuous processes as well as their random variants.
The main goal of the project is unifying and extending several concepts from model theory, algebra, topology and analysis, related to generic structures. We propose studying selected topics within the framework of abstract convergence schemes, addressing questions on their complexity and classification. One of our inspirations is the theory of universal homogeneous models, where convergence of finite structures is involved. Another motivation is settheoretic forcing, where a convergence scheme is simply a partially ordered set of approximations of some ''unreachable" objects, living outside of the universe of set theory.



Groups and their actions, operator algebras, and descriptive set theory (GJ1905271Y)
from 01/01/2019
to 31/12/2021 main investigator

Objectives:
The goal of the project is to to find new connections and prove some interesting conjectures on the boundaries of three, currently very attractive mathematical disciplines  geometric group theory, operator algebras, and descriptive set theory.



Logic and Topology in Banach spaces (GF1634860L)
from 01/01/2016
to 31/12/2018 investigator

Objectives:
The project is devoted to the study of topological and geometric properties of Banach spaces and their duals, aiming at a better understanding of their structure. Properties of the weak topology often imply important geometric properties of the Banach space in question. On the other hand, geometric properties of the Banach space often give information about its weak topology. Similar statements are true for duals of Banach spaces with the weakstar topology. We are going to explore this interplay in detail. The main project goals are:
1. Developing new tools for constructing and studying Banach spaces, using techniques from set theory and category theory.
2. Exploring different types of networks and related concepts in weak topologies, determining connections with renorming theory.
Results of Goal 1 will lead to new examples, settling some of the problems concerning interplay between geometric and topological properties of nonseparable Banach spaces. Goal 2 will lead to a better understanding of the weak topology and its relations to the geometric structure of a Banach space.



Topics in set theory: Traces of large cardinals, variants of Hechler's theorem, and ultrafilters on countable sets (7AMB15AT035)
from 01/01/2015
to 31/12/2016 investigator

Objectives:
The goal of the project is to expand the collaboration between specialists working in different areas of Set Theory. We shall, in particular, try to exploit the complementary expertise of the Czech and Austrian teams while solving problems on the borders of traditional areas of research (large cardinals, forcing, infinitary combinatorics, topology) and encourage mutual crossfertilization and inspiration sharing. The main goal is to encourage new members of both Czech and Austrian research teams to interact in join international research, to explore new possibilities for scientific collaboration, and to introduce international contacts enabling applications for funding with more ambitious join projects in future years.



Mathematical logic, complexity, and algorithms (IAA100190902)
from 01/01/2009
to 31/12/2013 investigator

Objectives:
Project of basic research in mathematical logic and theoretical computer science. We focus on bounded arithmetic and proof complexity, set theory, computational complexity theory, and the theory of algorithms. The topics range from foundational areas of mathematics to algorithmic problems motivated by applied research. The results of the project will be published in high quality international scientific journals and in the proceedings of selective conferences.
