Contact

Institute of Mathematics of the AS CR
Žitná 25, CZ - 115 67 Praha 1, Czech Republic

Phone: +420 222 090 715 (office)
E-mail: zapletal at math.cas.cz

Current research

I work on the interface between forcing, analysis, Ramsey theory, and descriptive set theory. I outlined the theory of definable proper forcing in my book "Forcing Idealized", Cambridge Tracts in Mathematics 174, Cambridge University Press 2008. My current work elaborates on the problems left open in that book and the relationship with the proper forcing technologies of Saharon Shelah. A new research program appeared concerning the connection between definable proper forcing and sigma-ideals on one hand and the theory of Borel equivalence relations, resulting in a joint book with Marcin Sabok and Vladimir Kanovei, "Canonical Ramsey Theory on Polish Spaces".

Continuing the work of the book Forcing idealized , I wrote a paper on the applications of the ergodic iteration theorem, showing that the n-localization property is iterable. In a joint work with Marcin Sabok, we investigated the forcing properties of ideals generated by closed sets in a greater detail than in the book. I found a simple restatement of P-point preservation for definable forcing notions. I connected the descriptive set theoretic notion of overspill with a forcing preservation property, giving an independence result in harmonic analysis. An important Abominable list of typos and omissions in Forcing Idealized also appears in this category.

I started work on Ramsey theorems for Polish spaces. Applying the creature forcing technology, with Saharon Shelah we proved a partition theorem for infinite products of finite sets with submeasures, and also its version parametrized by a measure. The theorem can replace the classical Milliken Theorem in many contexts, and it also works in contexts where the Milliken Theorem cannot be applied. Connecting with the theory of Borel equivalences, I am in the process of writing a book Canonical Ramsey Theory on Polish Spaces, jointly with Marcin Sabok and Vladimir Kanovei. Reader discretion is advised as this is a rough preliminary version.

In other recent work, I resolved the pinned equivalence conjecture of Kechris. I found a variant of Matrai's theorem on homogeneous sets of countable sequences for the Lebesgue measure. I showed that the property "having a sigma-closed dense subset" may not be invariant under the forcing equivalence of partial orders. I found that several sigma-ideals traditionally used in mathematical analysis satisfy the Komjath-Laczkovich property. A joint paper with Simon Thomas on the Bergman and Steinhaus properties of products of finite groups and ultrafilters is under preparation. In the paper The sigma-ideal generated by H-sets I discovered an intimate connection between the descriptive set theoretic notion of overspill and certain forcing fusion arguments, which leads to preservation theorems for forcing that are based purely on complexity grounds.

Resume

Education and degrees

• DSc. in Mathematics, Czech Academy of Sciences 2007
• Ph. D. in Mathematics, The Pennsylvania State University 1995, supervisor T. Jech
• M. A. in Mathematics, Charles University, Prague 1994
• B. A. in Mathematics, Charles University, Prague 1990

Employment

• 2009-present Purkyně Fellow, Czech Academy of Sciences
• 2009-present full professor, University of Florida
• 2005-2009 associate professor, University of Florida
• 2000-2005 assistant professor, University of Florida
• 1998-2000 John Wesley Young instructor, Dartmouth College
• 1996-1998 Bateman research instructor, California Institute of Technology
• 1995-1996 postdoctoral fellow, Mathematical Sciences Research Institute Berkeley
• 1990-1995 teaching assistant, The Pennsylvania State University

Grants and fellowships

• 2010/2011 AIP project MEB051006, cooperation between CAS and University of Wroclaw, CZK132000
• 2009/2010 AIP project MEB060909, cooperation between CAS and Kurt Goedel Center in Vienna, CZK126000
• 2009-present Purkyně fellowship, Czech Academy of Sciences
• 2009-present grant IAA100190902 of Grant Agency of the Academy of Sciences of the Czech Republic
• 2008-present NSF grant DMS 0801114, $110000
• 2006-2007 NSF grant DMS 0532644 (PI) to organize special year in logic at UF, $138000
• 2003-2006 grant GA ČR 201-03-0933 of the Grant Agency of Czech Republic
• 2003-2006 NSF grant DMS 0335481 to organize an annual logic conference at UF, $15000
• 2003-2006 NSF grant DMS 0300201 $103827
• 2000-2003 NSF grant DMS 0071437, $61431
• 2000-2003 grant GA ČR 201-00-1466 of the Grant Agency of Czech Republic
• 1997-2000 grant GA ČR 201-97-0216 of the Grant Agency of Czech Republic

Publications

• Canonical Ramsey theory on Polish spaces, with Marcin Sabok and Vladimir Kanovei, a book submitted to Cambridge University Press
• Pinned equivalence relations, accepted to Mathematical Research Letters
• More ideals with the Komjath-Laczkovich property, accepted to Topology and Applications
• Forcing properties of ideals of closed sets, with Marcin Sabok, accepted to J. Symbolic Logic
• Ramsey theorem for product of finite sets with submeasures, with Saharon Shelah, accepted to Combinatorica
• On the existence of a sigma-closed dense subset, Comment.Math.Univ.Carolin. 51,3 (2010) 513-517
• Applications of the ergodic iteration theorem. Math. Log. Q. 56 (2010), no. 2, 116–125
• Regular embeddings of the stationary tower and Woodin's Sigma Two Two maximality theorem, with Richard Ketchersid and Paul Larson, J. Symbolic Logic 75 (2010), no. 2, 711–727
• Preserving $P$P-points in definable forcing. Fund. Math. 204 (2009), no. 2, 145–154
• Canonization of equivalence relations and proper forcing, with Vladimir Kanovei, submitted to Fundamenta Mathematicae
• Increasing delta one two by a Namba-style forcing , with Richard Ketchersid and Paul Larson, J. Symbolic Logic 72 (2007), 1372--1378
• On the structure of stationary sets, with Qi Feng and Thomas Jech, Sci. China Ser. A 50 (2007) 615-627
• Forcing with quotients, with Michael Hrušák, Archive Math. Logic 47 (2008), 719-739
• Forcing idealized, Cambridge Tracts in Mathematics, Cambridge University Press 2008, ISBN 9780521874267
• Proper forcing and rectangular Ramsey theorems, Israel J. Math. 152 (2006), 29--47
• Between Maharam's and von Neumann's problem, with Ilijas Farah, Math. Research Letters 11 (2004), 673--684
• Four and more, Ann. Pure Appl. Logic, with Ilijas Farah, Ann. Pure Appl. Logic 140 (2006), 3--39
• Descriptive set theory and definable forcing, Memoirs Amer. Math. Soc. 793 (2004)
• Games with creatures, with S. Shelah, Comm. Math. Univ. Carolinae 44 (2003), 9--23
• Duality and the PCF theory, with S. Shelah, Math. Research Letters 9 (2002), 585--595
• Forcing with ideals of closed sets, Comm. Math. Univ. Carolinae 43,1 (2002), 181--188
• Isolating cardinal invariants, J. Math. Logic, 2003, 143-162
• Terminal notions in set theory, Ann. Pure Appl. Logic 109 (2001), 89--116
• Transfinite open games, Topology and Its Applications 111 (2001), 289--297
• Killing ideals and adding reals, J. Symbolic Logic 65 (2000), 747--755
• The nonstationary ideal and the other sigma ideals on omega one, Trans. Amer. Math. Soc. 352 (2000), 3981--3993
• Terminal notions, Bull. Symbolic Logic 5 (1999), 470--484
• On the Alaoglu-Birkhoff equivalence of posets, with S. Todorcevic, Illinois J. Math. 43 (1999), 281--292
• Canonical models for aleph one combinatorics, with S. Shelah, Ann. Pure Appl. Logic 98 (1999), 217--259
• Proper forcing and absoluteness in L(R), with I. Neeman, Comm. Math. Univ. Carolinae 39 (1998), 281--301
• A dichotomy for forcing notions, Math. Res. Lett. 5 (1998) 213--226
• Preserving sigma-ideals, J. Symbolic Logic 63 (1998), 1437--1441
• Keeping additivity of the null ideal small, Proc. Amer. Math. Soc. 125 (1997), 2443--2451
• Embeddings of Cohen algebras, with S. Shelah, Adv. Math. 126 (1997), 93--119
• Semi-Cohen boolean algebras, with B. Balcar and T. Jech, Ann. Pure Appl. Logic 87 (1997), 187--208
• Strongly almost disjoint functions, Israel J. Math. 97 (1997), 101--111
• Small forcings and Cohen reals, J. Symbolic Logic 62 (1997), 280--284
• Splitting number at uncountable cardinals, J. Symbolic Logic 62 (1997), 35--42
• A classification of definable partial orders on omega one, Fund. Math. 153 (1997), 141-144
• Characterization of the club forcing, in Papers on General Topology and Applications, S. Andima, R. Flagg, G. Itzkowitz, Y. Kong, R. Kopperman amd P. Misra, eds., Annals of the New York Academy of Sciences 806 (1996), 476--484
• A new proof of Kunen inconsistency, Proc. Amer. Math. Soc. 124 (1996), 2203-2205
• More on the cut and choose game, Ann. Pure Appl. Logic 76 (1995), 291--301

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