Piotr Koszmider mail: P.Koszmider@impan.pl

"Mathematics is the science of the infinite, its goal, the symbolic comprehension of the infinite with human, that is finite, means. It is the great achievement of the Greeks to have made the contrast between the finite and the infinite fruitful for the cognition of reality""

    -- Hermann Weyl--   

  Research project financed National Research Council of Brazil

Project's title: Analytic topology and consistency in combinatorially induced structures

Special Visiting Researcher - 04.2014-03.2017
Coordinator: Christina Brech
Project's webpage

  Research grant from the National Science center, Poland

Project's title: Applications of infinitary combinatorics and analytic topology in Banach spaces and related structures

General project in Panel ST1: Supported by the National Science Center for the period 12.2011 - 11.2014

  Research project financed by a grant of the Ministry of Science and Education,   Poland

Project's title: Combinatorial and topological methods in the Banach spaces of continuous functions

Individual project - 06.2008-05.2011

The subject of the project is the application of new methods of infinitary combinatorics and set-theoretic topology in the topics concerning unsolved until now classical problems of Banach spaces of continuous functions. The aim of the project is solving or getting closer to solutions of some of these problems and also obtaining conclusions in the general geometry of Banach spaces.

New combinatorial methods mentioned above are the continuously developing method of forcing (proper forcing, iterated forcing, OCA, PFA etc), the method of elementary submodels and new Ramsey type principles. On the other hand, the topological methods concern constructions of compact spaces, quite often using the Stone duality and the above combinatorial methods.

The classical unsolved problems concerning the Banach spaces of continuous functions are for example, the injective space problem, the problem if every complemented subspace of a C(K) is isomorphic to some C(K'), etc. These problems were often left unsolved already in the sixties because of the difficulties of the combinatorial nature.

Since, the techniques mentioned above were developed in the last 30 years in a relative isolation from the geometry of Banach spaces, the aim of the project is to test them in the context of the above classical problems. The recent work of the applicant shows that the outcome of of such a test could be unexpectedly striking.

  Procect financed by the National Research Council of Brazil

Project's title: Applications of infinitary combinatorics to undecidable problems on Boolean algebras, compact spaces and Banach spaces.

Project's category: Produtividade em pesquisa (B2), Individual project 2001-2007

There is a classical chain of mathematical constructions: infinite families of sets which form Boolean algebras which induce, by the Stone duality compact topological spaces on which we can consider Banach spaces of continuous functions C(K) and Radon measures, the latter is the dual space of the former, we can can even get the operator algebra on the C(K).
This classical chain which was completely formed in the middle of 20th century is revisited with the technology of undecidability proofs which mainly affect the combinatorial structure of the infinite families of sets.
The drama begins when we start showing that the combinatorics and the logical undecidability appear in all the links of the chain.

   Project financed by the Foundation for Support of Research of the State of São    Paulo

Project's title: Spaces of continuous functions over compact spaces

In collaboration with Uniwersytet Warszawski 01.2006 - 05.2006

Spaces of real continuous functions over a compact space K are classical objects with both topological an analitic meaning. Mainly we are going to consider these spaces as a Banach space C(K) and as topological vector space Cp(K) i.e., with the topology of pointwise convergence in place of the norm topology; the weak topology determined by the Banach space dual being an intermediate tool. The problems to be investigated are concerned both with the geometry of Banach spaces and with the purely topological questions related to K as well as to Cp(K). But the methods are prinicpally combinatorial and topological with the possibility of applications of logic. The project is concentrated on the colaboration with various researchers in Poland, mainly with Prof. Witold Marciszewski of Warsaw University, but also inludes short visits to centers in Czech republic, England and Spain.

  Project financed by Fields Institute, Canada

Project's title: Long Term Senior Visitor at Fields Institute

Participati in Thematic Programme: Set Theory and Analysis at Fields Institute 08.2006 - 12.2002

From its very beginnings, set theory has enjoyed a relationship with analysis which, while at times close and at others distant, has always allowed for the possibility of symbiosis. During the Fall of 2002 the Fields Institute will hosts a thematic program devoted to fostering the interaction between these two areas. Internationally recognized experts from both disciplines will be on site from September 2002 through to December 2002.

   Project financed by the Foundation for Support of Research of the State of São    Paulo

During the period from September 2002 till February 2003 I was realizing o project:

Aplicaçoes de combinatória infinitária em análise

This project was supported by FAPESP (bolsa de pesquisa no exterior, manutenção).

In the period from September 2002 till November 2002 I was visiting the Fields Institute for research in Mathematical Sciences in Toronto in Canada as a long term senior visitor where I took partin the Thematic Programme Set Theory and Analysis. This part of the project was partially supported the Fields Institute.

In December 2002 I was visiting Prof. Marek Malcerzak and his group at Lodz Technical University in Poland. This part of the project was partially supported by Lodz technical University. During this part of I the project I also visited Polish Academy of Sciences in Warsaw

In January and February of 2003 I was visiting Prof. Joan Bagaria while being a visiting professor at the Centre de Recerca Matematica in Barcelona in Catalonia. This part was partially supported by the Centre de Recerca Matematica. During this part I also visited Departments of Mathematics at University of Murcia and University of Valencia.

   Project financed by the Université Paris VII oraz Foundation for Support of    Research of the State of São

Project's title: Infinitary combinatorics and undecidable problems.
Collaboration with Prof. Stevo Todorcevic 07.2001

The local support was provided by Université Paris VII were I had a one month research position: Mâitre de Conférences at Paris Logic group

   Project financed by the National Science Foundation, USA

Project's title: Certain set-theoretic principles and their applications.

Individual project: 1994 - 1998