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Grants





Filters, Ultrafilters and Connections with Forcing (GF1733849L)
from 01/01/2017
to 31/12/2019 main investigator

Objectives:
The project falls within the scope of Set Theory & Foundations of Mathematics, specifically Combinatorial Set Theory and Forcing. We will investigate new combinatorial and forcing methods for constructing ultrafilters with special properties in different models of Set Theory.It will use these methods to answer independence questions about ultrafilters (no Ppoints with a large continuum or small character filters), structural questions about filters (which ultrafilters/filters contain towers) and questions about the related cardinal invariants (independence number, free sequence number). It is known that current methods cannot answer some of these questions so the project will have to come up with novel ideas. The methods used will include forcing iterations, diamondlike constructions, preservation theorems and methods from descriptive set theory.



The continuum, forcing and large cardinals (GF1534700L)
from 01/01/2015
to 31/12/2017 main investigator

Objectives:
The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics:
(1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections.
(2) Second subtopic is called"Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology.
(3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on nondefinable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.



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 main 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.



Set theory and its applications (MEB060909)
from 01/01/2009
to 31/12/2010 investigator

Programme type: MOBILITY  Austria
Objectives:
Exploring the interaction between set theory on one hand, and abstract analysis and combinatorics on the the other.
