## Departments

### Staff:

**Piotr Achinger**/ Assistant Professor / email

**Jarosław Buczyński**/ Associate Professor / email

**Przemysław Chojecki**/ Assistant Professor / email

**Christophe Eyral**/ Associate Professor / email

**Łucja Farnik**/ Assistant Professor / email

**Joachim Jelisiejew**/ Assistant Professor / email

**Michał Kapustka**/ Assistant Professor / email

**Wojciech Kucharz**/ Professor / email

**Henryk Michalewski**/ Assistant Professor / email

**Mateusz Michałek**/ Assistant Professor / email

**Karol Palka**/ Assistant Professor / email

**Tomasz Pełka**/ Graduate Assistant / email

**Piotr Pokora**/ Assistant Professor / email

**Masha Vlasenko**/ Assistant Professor / email

#### About the Department

The Department of Algebra and Algebraic Geometry was founded in 2000. It is, in some sense, a continuation of the IM PAN Department of Algebra in Toruń, directed at first by Jerzy Ło and then by Stanisław Balcerzyk. For the history of the latter, see the summary by S. Balcerzyk published in the booklet edited on the occasion of the 50th anniversary of IM PAN. We describe first

#### PERIOD 2000-2008

The following mathematicians worked in the Department of Algebra and Algebraic Geometry in the period 2000-2008: Janusz Adamus, Marcin Chałupnik, Sławomir Cynk, Piotr Hajac, Joanna Jaszuńska, Zbigniew Jelonek, Paweł Kasprzak, Jarosław Kędra, Oskar Kędzierski, Mariusz Koras, Urlich Kraehmer, Adrian Langer, Tomasz Maszczyk, Jerzy Płonka, Piotr Pragacz (head), Agata Smoktunowicz, Tomasz Szemberg, Marek Szyjewski, Halszka Tutaj-Gasińska, Bronisław Wajnryb, Andrzej Weber, and Bartosz Zieliński.

The spectrum of the interest of mathematicians in the department was marked by the following topics (displayed in alphabetical order):

- Affine algebraic geometry (Jelonek)
- Algebraic topology (Chałupnik, Weber)
- Classical algebra and combinatorics (Pragacz)
- Complex projective algebraic geometry (Cynk, Kedzierski, Langer, Pragacz, Tutaj-Gasińska, Szemberg)
- Enumerative theory of singularities (Pragacz, Weber)
- Geometry and topology of surfaces (Wajnryb)
- K-theory (Szyjewski)
- Moduli spaces (Langer)
- Noncommutative geometry and quantum groups (Hajac, Kasprzak, Kraehmer, Maszczyk, Zieliński)
- Noncommutative rings (Smoktunowicz)
- Symplectic geometry and topology (Kędra)
- Universal algebra (Płonka)

The "center" of mathematical life of the department was the seminar IMPANGA. Its leading subject was complex algebraic geometry. The seminar worked each two weeks (for two sessions with a break for discussions), and gathered algebraic geometers from all around of Poland. The speakers of this seminar included: F. Hirzebruch, H. Esnault, G. van der Geer, J. Kollar and A. Lascoux.

IMPANGA also organized at the Banach Center mini-schools, directed especially towards young researchers from all around of Europe. Let us mention here, e.g., the following mini-schools of IMPANGA : "Characteristic classes" (2002), "Schubert varieties" (2003), and "Moduli spaces" (2005). In 2003, IMPANGA organized jointly with Institutes of Mathematics of Bulgarian, Hungarian and Romanian Academy of Sciences, the Conference and Summer School "Algebraic Geometry, Algebra, and Applications" in Borovetz (Bulgaria). Two sessions, prepared by IMPANGA at the Banach Center for a wider audience (from history and philosophy of sciences), were devoted to Grothendieck (2004) and Hoene-Wroński (2007).

The outgrowth of seminars and schools of IMPANGA was published in two volumes:

*Topics in cohomological studies of algebraic varieties* (2005),

and

*Algebraic cycles, sheaves, shtukas, and moduli* (2007), edited by Birkhauser-Verlag,

and in:

*Hoene-Wroński: Życie, Matematyka i Filozofia* (2008), edited by IM PAN.

The seminar IMPANGA also stimulated research of mathematicians not working directly in the department. Let us mention here, e.g., the papers: "A cascade of determinantal Calabi-Yau threefolds" by M. and G. Kapustka from the Jagiellonian University (math.AG/08023669), and "On Thom polynomials for A4(-) via Schur functions" by O. Ozturk from METU in Ankara (Serdica Math. J. 33 (2007), 301-320).

Another very active seminar in the department, organized by P. Hajac and T. Maszczyk, was

"Noncommutative geometry and quantum groups". For the content of this seminar (among the speakers were Fields medalists: A. Connes and M. Kontsevich), and for many related activities on noncommutative geometry at IM PAN, including the Conference in honor of Paul Baum's 70th Birthday (2007), see http://ncg.mimuw.edu.pl.

##### SELECTED PUBLICATIONS in the period 2000-2008:

- J. Adamus, E. Bierstone and P.D. Milman, Uniform linear bound in Chevalley's lemma, to appear in Canad. J. Math. (2008),
- T. Bauer, T. Szemberg, Local positivity of principally polarized abelian threefolds, J. Reine Angew. Math. 531 (2001) 191-200,
- P.F. Baum, P.M. Hajac, R. Matthes and W. Szymański, The K-theory of Heegaard-type quantum 3-spheres, K-Theory 35 (2005), 159-186.
- P.F. Baum, P.M. Hajac, R. Matthes and W. Szymański, a chapter in the forthcoming volume "Quantum Symmetry in Noncommutative Geometry", European Mathematical Society Publishing House.
- B. Bojarski, A. Weber, Generalized Riemann-Hilbert transmission and boundary value problems, Fredholm pairs and bordisms, Bull. Polish Acad. Sci. Math. 50 (2002), 479-496.
- P. Cassou-Nogues, M. Koras and P. Russell, Smooth embeddings of C* into C2, part I., to appear in J. Algebra (2008)
- S. Cynk, Defect of a nodal hypersurface, Manuscripta Math. 104 (2001), 325-331.
- T. Hadfield, U. Kraehmer, Twisted Homology of Quantum SL(2), K-Theory 34 (2005), 327-360.
- P.M. Hajac, R. Matthes and W. Szymański, A locally trivial quantum Hopf fibration, Algebr. Represent. Theory 9 (2006), 121-146.
- S. Janeczko, Z. Jelonek, Linear automorphisms that are symplectomorphisms, J. of the London Math. Soc. 69 (2004), 503-517.
- Z. Jelonek, The Łojasiewicz exponent and effective Nullstellensatz, preprint (2003).
- Z. Jelonek, K. Kurdyka, On asymptotical critical values of a complex polynomial, J. Reine Angew. Math. 565 (2001), 1-11.
- J. Kędra, D. McDuff, Homotopy properties od Hamiltonian group actions, Geometry and Topology, vol.9 (2005), Paper no.3, pp. 121-162.
- A. Langer, Semistable principal G-bundles in positive characteristic, Duke Math. J. 128 (2005), 511-540.
- A. Langer, Moduli spaces and Castelnuovo-Mumford regularity of sheaves on surfaces, Amer. J. of Math. 128 (2006), 373-417.
- A. Langer, T. Gomez, A. Schmitt, I. Sols, Moduli spaces for principal bundles in arbitrary characteristic, preprint (2006).
- A. Lascoux, P. Pragacz, Double Sylvester sums for subresultants and multi-Schur functions, J. Symb. Comp. 35 (2003), 689-710.
- T.H. Lenagan, A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, Journal of the A.M.S. 20 (2007), 989-1001.
- T. Maszczyk, A pairing between super Lie-Rinehart and periodic cyclic homology, Comm. in Math. Physics, 263 (2006), 737-747.
- T. Maszczyk, One-dimensional infinitesimal-birational duality through differential operators, Fund. Math. 191 (2006), 23-43.
- A. Parusiński, P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom. 10 (2001), 63-79.
- J. Płonka, Subvarieties of the clone extension of the variety of distributive lattices, Algebra Universalis 55 (2006), 175-186.
- P. Pragacz, Multiplying Schubert classes, in "Topics in cohomological studies of algebraic varieties", Trends in Math., Birkhauser (2005), 163-174.
- P. Pragacz, Thom polynomials and Schur functions: the singularities I22(-), Annales de l'Institut Fourier 57 (2007), 1487-1508.
- P. Pragacz, V. Srinivas and V. Pati, Diagonal subschemes and vector bundles, math.AG/0609381 - to appear in the special volume of Quarterly J. of Pure and Appl. Math., dedicated to J-P. Serre on his 80th Birthday (S.T. Yau et al. eds.).
- P. Pragacz, A. Weber, Positivity of Schur function expansions of Thom polynomials, Fund. Math. 195 (2007), 85-95.
- A. Smoktunowicz, There are no graded domains with GK dimension strictly between 2 and 3, Inv. Math. 164 (2006), 635-640.
- A. Smoktunowicz, Some results in noncommutative ring theory, Proceedings of the International Congress of Mathematicians, Madrid, Spain, August 22-30, 2006, Vol 2: Invited talks, 259-269.
- A. Weber, Pure homology of algebraic varieties, Topology 43 (2004), 635-644.

We now pass to

#### PERIOD 2009-2015

The following mathematicians have worked in the Department of Algebra and Algebraic Geometry in the period 2009-2015: Maciej Borodzik, Weronika Buczyńska, Jarosław Buczyński, Sławomir Cynk, Lionel Darondeau, Christophe Eyral, Marek Hałenda, Grzegorz Kapustka, Oskar Kędzierski, Adrian Langer, Mateusz Michałek, Karol Palka, Piotr Pragacz (head), Sanjay Kumar Singh, Tomasz Szemberg, Saurabh Triverdi, Halszka Tutaj-Gasińska,

Masha Vlasenko.

The leading topic in the department is still complex algebraic geometry

and the spectrum of the interests in the department (displayed in alphabetical

order) is:

- Affine algebraic geometry (Palka)
- Algebraic geometry in positive characteristic (Achinger, Langer)
- Arithmetic algebraic geometry (Achinger, Vlasenko)
- Calabi-Yau varieties (Cynk, Kapustka)
- Characteristic classes (Darondeau, Pragacz)
- Combinatorial methods in geometry (Michalek, Pragacz)
- Combinatorics (Michałek)
- Holomorphic and contact symplectic geometry (Buczyński, Kapustka)
- Hyperbolic varieties (Darondeau)
- Intersection theory and Schubert calculus (Darondeau, Pragacz)
- Linear systems (Tutaj-Gasińska, Szemberg)
- Moduli spaces (Langer)
- Number theory (Vlasenko)
- Secant varieties and ranks of polynomials and tensors (Buczyńska, Buczyński)
- Singularities (Borodzik, Eyra, Triverdil)
- Vector bundles (Langer, Pragacz)

The center of mathematical life of the department is again the seminar IMPANGA. The seminar meets every second Friday (for two sessions) and gathers algebraic

geometers from all around Poland (notably from Warsaw, Kraków, Poznań, Gdańsk and Szczecin). The speakers at the IMPANGA Seminar have included: J.P. Brasselet, S. Capell, L. Gruson, L. Katzarkov, V. Kiritchenko, V. Lazic, V. Mehta, M. Oka, T. Peternell, C. Ranestad, B. Totaro and J. Wlodarczyk.

IMPANGA has organized at the Banach Center the following mini-schools: "Thom polynomials and the Green-Griffith conjecture" (2011), "The ubiquity of Wrońskians" (2011), "Okounkov bodies and Nagata type conjectures" (2013), "The geometry of homogeneous varieties" (2013), "Abelian varieties" (2014).

The two largest events, organized by IMPANGA at the Banach Center in Będlewo, were: "Impanga summer school on algebraic geometry" (2010) and the Conference "IMPANGA 15"

(2015). The former event was devoted to Prym varieties and their moduli, moduli spaces of curves and abelian varieties, differential forms and applications to moduli,

K3 and Enriques surfaces, invariants of singularities in birational geometry, minimal model program, toric varieties and equivariant cohomology. The latter conference was mainly devoted to Chern class formulas for degeneracy loci, equivariant cohomology of flag varieties, moduli spaces of abelian varieties and surfaces, classes of singular varieties, Thom polynomials, tropical algebraic geometry and its applications, geometry in positive characteristic and filtrations of B-modules.

The lecturers at the conferences, schools and workshops of IMPANGA have included: K. Altmann, D. Anderson, G. Berczi, A. Buch, P. Cascini, C. Ciliberto, I. Coskun, G. Farkas, G. van der Geer, B. Harbourne, J. Huh, J.M. Hwang, M. Kazarian, S. Kebekus, J. Keum, M. Lehn, R. Miranda, S. Mukai, M. Mustata, K. Ono, K. Ranestad, F. Russo, F.O. Schreyer,

V. Srinivas, H. Tamvakis and M. Vlasenko.

The outgrowth of seminars, schools and conferences of IMPANGA was published in: Contributions to algebraic geometry (EMS Publishing House, 2012). The next volume of IMPANGA Lecture Notes is in preparation.

For more on Department of Algebra and Algebraic Geometry, please consult:

http://www.impan.pl/~pragacz/impanga.htm

##### SELECTED RECENT PUBLICATIONS:

- P. Achinger, K($\pi,1$)-neighborhoods and comparison theorems, Compositio Math. 151 (2015), 1945-1964.
- P. Achinger, A characterization of toric varieties in characteristic p, International Mathematics Research Notices 16 (2015), 68796892.
- T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora, T. Szemberg, Bounded Negativity and Arrangements of Lines, Intern. Math. Research Notices 2015 (2015), 9456-9471
- I. Biswas, S. Singh, Diagonal property of the symmetric product of a smooth curve, Comptes Rendus Mathematique, 353 no. 5 (2015), 445-448
- M. Borodzik, A. Némethi, The Hodge spectrum of analytic germs on isolated surface singularities. J. Math. Pures Appl. (9) 103 (2015), no. 5, 11321156.
- M. Borodzik, A. Némethi, A. Ranicki, On the semicontinuity of the mod 2 spectrum of hypersurface singularities. J. Algebraic Geom. 24 (2015), no. 2, 379398.
- S. Boucksom, A. Kuronya, C. Maclean, T. Szemberg, Vanishing sequences and Okounkov bodies, Math. Ann. 361 (2015), 811-834
- J. Buczyński, G. Kapustka, M. Kapustka, Special lines on contact manifolds, arXiv:1405.7792
- W. Buczyńska, J. Buczyński, J. Kleppe, Z. Teitler, Apolarity and direct sum decomposability of polynomials, to appear in Michigan Math Journal, arXiv:1307.3314
- S. Cynk, S. Rams, Non-factorial nodal complete intersection threefolds. Commun. Contemp. Math. 15 (2013), no. 5, 1250064, 14 pp.
- S. Cynk, M. SchUtt, Non-liftable Calabi-Yau spaces. Ark. Mat. 50 (2012), no. 1, 2340.
- S. Cynk, D. van Straten, Calabi-Yau conifold expansions, Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, 499515, Fields Inst. Commun., 67, Springer, New York, 2013.
- L. Darondeau, Fiber integration on the Demailly tower, Annales de l'Institut Fourier to appear (2015)
- L. Darondeau, P. Pragacz, Universal push-forwards, arXiv: 1510.07852
- C. Eyral, M. Ruas, Deformations with constant Le numbers and multiplicity of nonisolated hypersurface singularities, Nagoya Math. J. 218 (2015), 29-50
- C. Eyral and M. Oka, On the fundamental groups of non-generic R-join-type curves, II
- J. Math. Soc. Japan (to appear)
- C. Eyral, Topics in equisingularity theory, IMPAN Lecture Notes, Polish Academy of Sciences, Institute of Mathematics, Warsaw (to appear)
- V. Golyshev, M. Vlasenko, Equations D3 and spectral elliptic curves, in Feynman Amplitudes, Periods and Motives, Contemporary Mathematics 648 (2015), 135-152
- A. Iliev, G. Kapustka, M. Kapustka, K. Ranestad, EPW cubes, arXiv:1505.02389
- T. Kahle, M. Michałek, Plethysm and Lattice Point Counting, Foundations of Computational Mathematics (to appear)
- S. Kaji, P. Pragacz, On diagonals of flag bundles, arXiv: 1503.03217
- G. Kapustka, On IHS fourfolds with b2 = 23, Mich. Math. J. vol. 65 (2016), 23 pages
- M. Koras, K. Palka, The Coolidge-Nagata conjecture, arXiv: 1502.07149
- A. Langer, Generic positivity and foliations in positive characteristic, Adv. Math. 277 (2015), 1-23
- A. Langer, Bogomolov's inequality for Higgs sheaves in positive characteristic, Inv. Math. 199 (2015), 889-920
- A. Mellit, M. Vlasenko, Dwork's congruences for the constant terms of powers of a Laurent polynomial, International Journal of Number Theory 2016
- M. Michałek, B. Sturmfels, C. Uhler, P. Zwiernik, Exponential Varieties, Proceedings of London Mathematical Society (to appear)
- M. Mikosz, P. Pragacz, A. Weber, Positivity of Legendrian Thom polynomials, Journal of Differential Geometry 89(1) (2011), 111-132
- N. Nguyen, S. Trivedi, and D. Trotman, A geometric proof of the existence of definable Whitney stratifications, Illinois J. Math. 58, no. 2 (2014), 381-389
- K. Palka, The Coolidge-Nagata conjecture, part I, Adv. Math. 267 (2014), 1-43
- K. Palka, Classification of singular Q-homology planes I. Structure and singularities, Israel J.M. 195 (2013), 37-69
- P. Pragacz, A Gysin formula for Hall-Littlewood polynomials, Proc. Amer. Math. Soc. 143 (2015) no.11, 4705-47011
- S. Triverdi, G. Valette, Flat currents on subanalytic pseudomanifolds, prepint 2015
- H. Tutaj-Gasińska, C. de Volder, Higher order embeddings of certain blow-ups of $P^2$, Proc. Amer. Math. Soc. 137 (2009) 4089-4097

### Staff:

**Tomasz Adamowicz**/ Associate Professor / email

**Jan Burczak**/ Assistant Professor / email

**Tomasz Cieślak**/ Associate Professor / email

**Michael Eastwood**/ prof. wizyt. / email

**Michał Gaczkowski**/ Assistant Professor / email

**Piotr Gwiazda**/ Professor / email

**Stanisław Janeczko**/ Professor / email

**Antoni Kijowski**/ Graduate Assistant / email

**Karolina Kropielnicka**/ Assistant Professor / email

**Wojciech Kryński**/ Assistant Professor / email

**Marcin Małogrosz**/ Assistant Professor / email

**Jarosław Mederski**/ Assistant Professor / email

**Katharina Neusser**/ prof. wizyt. / email

**Bernard Nowakowski**/ Assistant Professor / email

**Jan Peszek**/ Assistant Professor / email

**Joanna Rencławowicz**/ Associate Professor / email

**Iwona Skrzypczak**/ Assistant Professor / email

**Guillaume Valette**/ Associate Professor / email

**Aneta Wróblewska-Kamińska**/ Assistant Professor / email

**Wojciech Zajączkowski**/ Professor / email

#### About the Department

The Section of Differential Equations was guided, at the early period of its existence, by Tadeusz Ważewski and then by Andrzej Pliś and Bogdan Ziemian. Ważewski, in addition to his well known results in ordinary differential equations, had fundamental contributions to creation and the initial development of the theory of differential inclusions. The basic achievement of Pliś was the discovery of the phenomenon of nonuniqueness of solutions for smooth linear partial differential equations and, in particular, proving existence of smooth solutions with compact support for certain smooth elliptic equations. Bodgan Ziemian, who passed away prematurely, used integral transforms in order to build a theory of generalized analytic functions suitable for analysis of singular partial differential equations. He obtained new integral representations of fundamental solutions with integration over subsets of the complex characteristic variety.

The present members of the Section present themselves below.

### Roman Dwilewicz

Areas of interest: geometric analysis, complex analysis, partial differential equations, some aspects of algebraic geometry and analytic number theory.

In the last years I have been working on problems related to geometry of real submanifolds in complex manifolds and the corresponding theory of functions. A part of the above is the Cauchy-Riemann (CR) theory. I proved, also with my collaborators, various theorems on (local and global) extension of CR functions and properties of CR manifolds.

Recently I am also involved in algebraic aspects of the above problems, and, independently, in some properties of the Riemann zeta function.

### Bronisław Jakubczyk

Research field: geometry and singularities in differential equations and control theory.

I have worked on the realization problem in control theory (the problem of finding a system on a differential manifold which realizes a given nonlinear causal operator), finding existence criteria and giving a geometric construction of the realization (SIAM J. Control Optim. 1980, 1986 and C. R. Acad. Sci. Paris 1986).

Together with W. Respondek, we have proved that control systems linearizable by feedback transformations are characterized by integrability of certain distributions (Bull. Acad. Polon. Sci. 1980). Together with E. Sontag we have shown that geometric methods, including infinitesimal notions, can be used for analysing discrete-time systems (SIAM J. Control Optim. 1991).

I have been studying local geometry of distributions on manifolds (subbundles of the tangent bundle). In particular, together with M. Zhitomirskii I have characterized local invariants of singular contact structures (C. R. Acad. Sci. Paris 1997). My recent research concentrates on local invariants of distributions and systems of ordinary differential equations (especially control systems) using techniques of singularity theory.

### Grzegorz Łysik

My field of interest is the analysis of generalized analytic functions, i.e. holomorphic functions with branch singularities. Such functions near an isolated singular point (say 0) have a "continuous" Taylor type expansion into powers xα, α in R, with some density which is a generalized function. They are closely related to resurgent functions of J. Ecalle. They behave well under algebraic and differential operations and appear as solutions to singular differential equations, both linear and non-linear. In my research I mainly use methods of complex analysis and the theory of ultradistributions and hyperfunctions. As my main achievement I consider a modification of the Mellin transformation in a way suitable for the study of generalized analytic functions with exponential growth at zero. Another one is the derivation of new versions of the quasi-analyticity principle for functions holomorphic in a half plane.

#### Selected publications:

- Generalized Taylor expansions of singular functions, Studia Math. 99 (1991), 235-251.
- The Taylor transformation of analytic functionals with non-bounded carrier, Studia Math. 108 (1994), 159-176. Laplace ultradistributions on a half line and a strong quasi-analyticity principle, Ann. Polon. Math. 63 (1996), 13-33.
- A Phragmén-Lindelöf type quasi-analyticity principle, Studia Math. 123 (1997), 217-234.
- The Mellin transformation of strongly increasing functions, J. Math. Sci. Univ. Tokyo 6 (1999).

### Czesław Olech

Research field: ordinary differential equations and optimal control theory.

My earlier results include:

(1960) Proving that the Markus-Yamabe problem "If the characteristic roots of the Jacobian matrix f '(x) have the real parts negative, for any x, then the system x'=f '(x) is globally asymptotically stable" is, in dimension 2, equivalent to the problem "the same hypothesis implies that the map x→f(x) is injective as a map of the plane". For further developments concerning this result see my review in: Featured Reviews in Mathematical Reviews 1995-1996, Amer. Math. Soc., 1998, E18-E20.

(1964) Elucidation of the essence of the bang-bang principle for linear control systems and, in consequence, a new proof of Lyapunov's theorem on convexity of the set of values of a vector-valued measure (and of the integral of a multivalued function). See: C. Olech, The Lyapunov theorem: its extensions and applications, in: Methods of Nonconvex Analysis, A. Cellina (ed.), Lecture Notes in Math. 1446, Springer, 1990, 84-103.

Organizational achievement: As the chairman of the Organization Committee of the International Congress of Mathematicians in Warsaw 1983 I managed to reach the main goal so that the Congress took place despite unfavorable political situation in and around Poland. See: Olli Lehto, Mathematics without Borders, A History of the International Mathematical Union, Springer, 1988, 219-237.

### Wojciech Zajączkowski

Research field: partial differential equations describing the motion of fluids.

For about 10 years I have been working on problems concerning viscous fluids with free surface. Of special interest was the stability problem of equilibrium solutions for viscous, compressible fluids bounded by a free surface. Since the problem poses serious technical difficulties, I use function spaces with little regularity, mainly L2-spaces. I have proved existence of solutions near equilibria.

My present investigations concentrate on existence proofs in Lp-spaces and studying flows with large data (large initial velocity). I study incompressible fluids using new techniques.

#### Selected publications:

- On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993).
- (with E. Zadrzyńska), On a differential inequality for equations of a viscous compressible heat conducting fluid bounded by a free surface, Ann. Polon. Math. 61 (1995), 141-188.
- L∞-estimates for solutions of nonlinear parabolic systems, Banach Center Publ. 33 (1996), 465-490.
- (with E. Zadrzyńska), On global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting capillary fluid, J. Appl. Anal. 2 (1996), 125-169.
- (with G. Stroehmer), On the existence and properties of rotationally symmetric equilibrium states of compressible barotropic self-gravitating fluids, Indiana Univ. Math. J. 46 (1997), 1181-1220.
- (with G. Stroehmer), On dynamic stability of a certain equilibrium solution for compressible barotropic self-gravitating fluid motion bounded by a free surface, Nonlinear Anal., to appear.
- (with T. Kobayashi), On global motion of a compressible barotropic viscous fluid with boundary slip condition, Appl. An

### Staff:

**Tristan Bice**/ Assistant Professor / email

**Ryszard Frankiewicz**/ Professor / email

**Jakub Gismatullin**/ Assistant Professor / email

**Joanna Jaszuńska**/ Assistant Professor / email

**Piotr Koszmider**/ Professor / email

**Marcin Sabok**/ Assistant Professor / email

**Tin Lok Wong**/ Assistant Professor / email

### PHD Students:

**Michał Tomasz Godziszewski**

Research of the Section involves a rather wide spectrum of matters which are connected with the foundations of mathematics, such as set theory, model theory, foundations of arithmetic, cathegorical logic and their applications in analysis, topology, computability and algebra.

Zofia Adamowicz

She is working in foundations of arithmetic. Her main research is in "bounded arithmetic" together with its links to computational complexity.

#### Selected papers:

- An application of a reflection principle, with L. Kołodziejczyk and P. Zbierski, Fundamenta Mathematicae 180 (2003), 139-159
- Well-behaved principles alternative to bounded induction, with. L. Kołodziejczyk, Theoretical Computer Science 322 (2004), 5-16
- Partial Collapses of the $\Sigma_1$ Complexity Hierarchy In Models for fragments of Bounded Arithmetic, with L. Kołodziejczyk, Annals of Pure and Applied Logic 145 (2007) 91-95
- A note on the $\Sigma_1$ collection scheme and fragments of bounded arithmetic with L. Kołodziejczyk, Mathematical Logic Quarterly 56 (2010), 126-130
- Lower bounds for the provability of Herbrand consistency in weak arithmetics, with K. Zdanowski, Fund. Math. vol. 212 no. 3, 2011, s. 191-216
- Truth definitions without exponentiation and the $\Sigma_1$ collection scheme, with L.A.Kołodziejczyk, J.B.Paris, Journal of Symbolic Logic 77 (2):649-655
- Existentially closed models in the framework of Arithmetic, with A.Cordon-Franco, F.Lara-Martin,Journal of Symbolic Logic Volume 0, Number 0,xxxx 2015

### Piotr Koszmider

His research is focused on developing and applications of the methods of combinatorial set theory and logic such as forcing, stepping up, anti-Ramsey results, bookkeeping principles in analysis and topology in particular in Banach spaces, weak and weak* topology and in algebras of operators.

#### Selected papers:

- Piotr Koszmider, Cristobal Rodriguez-Porras, On automorphisms of the Banach space to appear in Fund. Math.
- Piotr Koszmider; On constructions with 2-cardinals; To appear in Arch. Math. Logic.
- T. Kania, P. Koszmider, N. J. Laustsen, A weak*-topological dichotomy with applications in operator theory; Trans. London Math. Soc. (2014) 1 (1): 1-28
- Antonio Aviles, Piotr Koszmider; A continuous image of a Radon-Nikodym compact space which is not Radon-Nikodym; Duke Math. J. 162, 12 (2013), 2285-2299.
- Piotr Koszmider, Saharon Shelah; Independent families in Boolean algebras with some separation properties; Algebra Universalis 69 (2013), no. 4, 305 - 312
- Piotr Koszmider; On large indecomposable Banach spaces; J. Funct. Anal. 264 (2013), no. 8, 1779–1805
- Christina Brech, Piotr Koszmider; On universal Banach spaces of density continuum, Israel J. Math. 190 (2012), 93–110.

### Marcin Sabok

His research is focused on descriptive set theory, Ramsey theory and forcing. Recently, he is also interested in applications and connections of logic to fields such as operator algebras and geometric group theory.

#### Selected papers:

- Sabok, Marcin, Completeness of the isomorphism problem for separable C*-algebras, Invent. math., to appear
- Sabok, Marcin, Automatic continuity for isometry groups, submitted
- Kanovei, Vladimir; Sabok, Marcin; Zapletal, Jindřich, Canonical Ramsey theory on Polish spaces, Cambridge Tracts in Mathematics, 202. Cambridge University Press, Cambridge, 2013.
- Sabok, Marcin, Extreme amenability of abelian L0 groups, J. Funct. Anal. 263 (2012), no. 10, 2978-2992
- Pawlikowski, Janusz, Sabok, Marcin, Decomposing Borel functions and structure at finite levels of the Baire hierarchy, Ann. Pure Appl. Logic 163 (2012), no. 12, 1748–1764.
- Sabok, Marcin, Complexity of Ramsey null sets, Adv. Math. 230 (2012), no. 3, 1184–1195
- Sabok, Marcin, Forcing, games and families of closed sets, Trans. Amer. Math. Soc. 364 (2012), no. 8,4011–4039.
- Sabok, Marcin, Zapletal, Jindřich, Forcing properties of ideals of closed sets, J. Symbolic Logic 76 (2011),no. 3, 1075–1095.

**Yuriy Tomilov**

pok. 419 / tel. 22 5228 238

### Staff:

**Rami Ayoush**/ Graduate Assistant / email

**Marek Bożejko**/ Professor / email

**Tadeusz Figiel**/ Professor / email

**Anna Kamont**/ Professor / email

**Tomasz Kochanek**/ Assistant Professor / email

**Adam Nowak**/ Associate Professor / email

**Jan Rozendaal**/ Assistant Professor / email

**Piotr Śniady**/ Professor / email

**Tomasz Zachary Szarek**/ Assistant Professor / email

**Bartosz Trojan**/ Assistant Professor / email

**Michał Wojciechowski**/ Associate Professor / email

**Przemysław Wojtaszczyk**/ Professor / email

**Wiesław Żelazko**/ Professor / email

**Jaroslav Zemánek**/ Professor / email

### PHD Students:

**Rami Ayoush**/ e-mail

**Bartłomiej Kielak**/ e-mail

**Maciej Rzeszut**/ e-mail

**Bartłomiej Zawalski**

#### About the Department

In the recent years the research activity of members of the Department has covered a wide range of topics concerning functional analysis and its relationships with other fields. Below we present the most important results obtained in recent years.

**Applications of interpolation theory to functional analysis**. A modern approach by interpolation of Banach spaces is presented in [DMM1] to prove the abstract type Littlewood inequalities for inclusion maps between Banach symmetric sequence spaces. This extends the famous analogues in L_{p}-spaces due to Littlewood, Orlicz, Bennett and Carl. These results have many different applications, e.g. to eigenvalue distributions of compact operators [DMM2], local theory of Banach spaces and theory of interpolation functors [M], s-numbers in finite-dimensional Schatten classes [DMM3].

**Approximation theory**. The study of generalized Franklin systems was undertaken and results were presented in the series of papers (see e.g.[GK1]). Non-linear m-term greedy approximation with respect to the Haar system and other wavelet systems was studied. In particular stability of greedy approximation in the space BV was obtained in [BDKPW]. Important results about quasi-greedy bases we obtained in [GK2],[W].

**Local theory of Banach spaces**. In [MT1] the geometry of random sections and projections of symmetric convex bodies was investigated. Relation between optimal radii of Euclidean balls inscribed in sections and superscribed on projections of symmetric convex bodies was given in [MT2]. A lower bound for Banach-Mazur distances between symmetric polytopes generated by subgaussian vectors was given in [LMOT].

**Methods of the theory of locally convex spaces and their applications to classical analysis**. Derived functors on locally convex spaces are applied to the problem of parameter dependence of solutions of linear partial differential operators [BD] and to the problem which composition operators on the space of real analytic functions have closed range [DL]. The structure of the corresponding spaces of functions or distributions is analyzed [DV].

**Sobolew spaces**. In [PW1] the unconditional structure of Sobolev spaces and spaces of functions of bounded variation are studied. Faliure of local unconditional structure of Soblev spaces in L^{1}-norms and spaces of functions with bounded variation are dscussed [PW2]. The bounded approximation property of the space of functons with bounded variation is established in [ACDP]. The relation between singularities of vector measure and constrains on directions of its Fourier transform is investigated in [RW].

**Topological algebras**. Several papers deal with ideals in F-algebras. In [Z1] it is shown that a unital F-algebra has all left (right) maximal ideals closed if and only if it is a Q-algebra, i.e. the group of its invertible elements is open. In [Z2] it is shown that a unital F-algebra has all one-sided ideals closed if and only if it is both left and right Noetherian. In [Z3] it is constructed an m-convex B_{0}-algebra in which all left but not all right ideals are closed. Other results concern topologically invertible elements and operators on locally convex spaces and their hyperinvariant subspaces.

**Handbook of the Geometry of Banach Spaces** Members of the Department contributed four survey articles to the Handbook of the Geometry of Banach Spaces describing "state of the art" in presented areas, [HB1], [HB2], [HB3], [HB4].

[ACDP] | G. Alberti, M. Csörneyi, A.Pełczyński, D. Preiss, BV has the Bounded Approximation Property, Journal of Geometric Analysis 15 (2005),1-7. |

[BDKPW] | P. Bechler, R. Devore, A. Kamont, G. Petrova, P. Wojtaszczyk, Greedy wavelet projections are bounded on BV. Trans. Amer. Math. Soc., 359 (2007), 619-635. |

[BD] | J. Bonet, P. Domański, The splitting of exact sequences of PLS-spaces and smooth depepndence of solutions of linear partial differential equations, Adv. Math., 217 (2008), 561-585. |

[DMM1] | A. Defant, M. Mastyło, C. Michels, Summing inclusion maps between symmetric sequence spaces, Trans. Amer. Math. Soc., 354 (2002), 4473-4492. |

[DMM2] | A. Defant, M. Mastyło, C. Michels, Eigenvalues estimates for operators on symmetric Banach sequence spaces, Proc. Amer. Math. Soc. 132 (2003), 513-521. |

[DMM3] | A. Defant, M. Mastyło, C. Michels, Summing norms of identities between unitary ideals, Math. Z. 252 (2006), 863-882. |

[DL] | P. Domański, M. Langenbruch, Coherent analytic sets and composition of real analytic functions, J. reine angew. Math., 582 (2005), 41-59. |

[DV] | P. Domański, D. Vogt, The space of real analytic functions has no basis, Studia Math., 142 (2000), 187-200. |

[GK1] | G. G. Gevorkyan, A. Kamont, General Franklin systems as bases in H[0,1]. Studia Math., 167 (2005), 259-292.^{1} |

[GK2] | G. G. Gevorkyan, A. Kamont, Two remarks on quasi-greedy bases in the space L. (Russian) Izv. Nats. Akad. Nauk Armenii Mat., 40 (2005), no. 1, 5-17._{1} |

[HB1] | T. Figiel, P. Wojtaszczyk, Special bases in function spaces, Handbook of the geometry of Banach spaces, vol I, North Holland, W.B.Johnson and J. Lindenstrauss editors, Amsterdam 2003, 561-590. |

[LMOT] | R. Latała, P. Mankiewicz, K. Oleszkiewicz, N. Tomczak-Jaegermann, Banach-Mazur distances and projections on random subgaussian polytopes, Discrete Comput. Geom., 38 (2007), 29-50. |

[HB2] | P. Mankiewicz, N. Tomczak-Jaegermann, Quotients of finite-dimensional Banach spaces; random phenomena, Handbook of the geometry of Banach spaces, vol II, North Holland, W.B.Johnson and J. Lindenstrauss editors, Amsterdam 2003, 1201-1246. |

[MT1] | P. Mankiewicz and N. Tomczak-Jaegermann, Geometry of Families of Random Projections of symmetric convex bodies, Geom. Funct. Anal., 11 (2001), 1282-1326. |

[MT2] | P. Mankiewicz, N. Tomczak-Jaegermann, Low Dimensional sections versus projections of convex bodies, Israel J. of Math., 153 (2006), 45-60. |

[M] | M. Mastyło, Interpolation methods of means and orbits, Studia Math. 17 (2005), 153-175. |

[HB3] | A. Pełczyński, M. Wojciechowski, Sobolev Spaces, ibidem, 1361-1425. |

[HB4] | P. Wojtaszczyk, Spaces of analytic functions with integral norm, ibidem, 1671-1702. |

[PW1] | A. Pełczyński, M. Wojciechowski, Spaces in several variables in L, Ark. Mat., 40 (2002) 363-382.^{1} norm are non isomorphic to Banach lattices |

[PW2] | A. Pełczyński, M. Wojciechowski, Spaces of functions with bounded variation and sobolev spaces without local unconditional structure, J. reine angew. Math., 558 (2003), 109-157. |

[RW] | M. Roginskaya, M. Wojciechowski, Singularity of vector valued measures in terms of Fourier transform, J. Fourier Analysis and Applications, 12, (2006), 213 - 223. |

[W] | P. Wojtaszczyk, Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107 (2000), 293-314. |

[Z1] | W. Żelazko, When a unital F-algebra has all left (right) ideals closed, Studia Math., 175 (2006), 279-284. |

[Z2] | W. Żelazko, A characterization of F-algebras with all one-sided ideals closed, Studia Math., 168 (2005), 135-145. |

[Z3] | W. Żelazko, An m-convex B, Coll. Math., 194 (2006), 317-324._{0}-algebra with all left but not all right ideals closed |

#### About the Department

The Katowice Branch of the Institute of Mathematics was founded in 1966. Jan Mikusiński was the head of the Branch until his retirement in 1984. From 1985 to 1994 the Branch was headed by Piotr Antosik, and then by Ryszard Rudnicki. The following mathematicians worked in the Branch: B. Aniszczyk, P. Antosik, J. Burzyk, T. Dłotko, C. Ferens, P. Hallala, A. Kamiński, W. Kierat, C. Kliś, S. Krasińska, M. Kuczma, A. Lasota, S. Lewandowska, Z. Lipecki, K. Łoskot, J. Mikusiński, P. Mikusiński, J. Mioduszewski, J. Pochciał, R. Rudnicki, Z. Sadlok, K. Skórnik, W. Smajdor, T. Szarek, Z. Tyc, J. Uryga and P. Uss.

The main line of research has been closely related to Prof. Mikusiński's interests. The dominating topics of investigations are sequential theory of distributions, Mikusiński operational calculus and convergence theory. The main results obtained in this area are: introduction of regular and irregular operations on distributions and local derivatives, functional description of the convergence in the field of Mikusiński operators, axiomatic theory of convergence, diagonal theorems and Paley-Wiener type theorems for regular operators. Moreover, several results concerning applications of operational calculus to differential equations, theory of controllability and special functions have been obtained.

Numerous results obtained by Mikusiński's team are presented in five books written by Mikusiński, Antosik, Sikorski and Boehme. Mikusiński's books have been translated into various languages, for example "Operational Calculus'' was published in Polish, English, Russian, German, Hungarian and Japanese.

In the early nineties a group of scientists connected with Prof. Andrzej Lasota began to work in the Branch. Their main research interests are in probability theory, partial differential equations and biomathematics. The main results obtained are: sufficient conditions for asymptotic stability of Markov operators and semigroups, asymptotic behaviour of solutions of generalized Fokker-Planck equations, constructions of semifractals and global properties of nonlinear models of population dynamics.

### Selected publications

#### Books:

- J. Mikusiński, Operational Calculus, Pergamon Press and PWN, 1967; 1983.
- J. Mikusiński and T.K. Boehme, Operational Calculus, Volume II, PWN and Pergamon Press, 1987.
- P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions, The Sequential Approach, Elsevier-PWN, 1973 (Russian edition 1976).
- J. Mikusiński, The Bochner Integral, Birkhäuser, 1987; Academic Press, 1978.
- P. Antosik and C. Swartz, Matrix Methods in Analysis, Springer, 1985.

#### Papers:

- P. Antosik, On the Mikusiński diagonal theorem, Bull. Acad. Polon. Sci. 20 (1972), 373-377.
- J. Burzyk, On convergence in the Mikusiński operational calculus, Studia Math. 75 (1983), 313-333.
- J. Burzyk, A Paley-Wiener type theorem for regular operators, Studia Math. 93 (1989), 187-200.
- H. Gacki, T. Szarek and S. Wędrychowicz, On existence and stability of solutions of stochastic integral equations with applications to control system, Indian J. Pure Appl. Math. 29 (1998), 175-189.
- A. Kamiński, On the Rényi theory of conditional probabilities, Studia Math. 79 (1984), 151-191.
- A. Kamiński, D. Kova?ević and S. Pilipović, The equivalence of various definitions of the convolution of ultradistributions, Trudy Mat. Inst. Steklov. 203 (1994) 307-322.
- C. Kliś, An example of a non-complete normed (K) space, Bull. Acad. Polon. Sci. 26 (1978), 415-420.
- A. Lasota and J. Myjak, Semifractals, Bull. Polish Acad. Sci. Math. 44 (1996), 5-21.
- A. Lasota and J. A. Yorke, When the long time behavior is independent of the initial density, SIAM J. Math. Anal. 27 (1996), 221-240.
- K. Łoskot and R. Rudnicki, Limit theorems for stochastically perturbed dynamical systems, J. Appl. Probab. 32 (1995), 459-469.
- J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 (1996), 1149-1164.
- M. C. Mackey and R. Rudnicki, Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548-566.
- M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89-109.
- J. Mikusiński, Sequential theory of the convolution of distributions, Studia Math. 29 (1968), 151-160.
- J. Mikusiński, A theorem on vector matrices and its applications in measure theory and functional analysis, Bull. Acad. Polon. Sci. 18 (1970), 151-155.
- J. Mikusiński, On full derivatives and on the integral substitution formula, Accad. Naz. Lincei Probl. Atti Sci. Cult. 217 (1975), 377-390.
- J. Mikusiński and P. Mikusiński, Quotients de suites et leurs applications dans l'analyse fonctionnelle, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 463-464.
- K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 (1997), 56-74.
- J. Pochciał, Sequential characterizations of metrizability, Czech. Math. J. 41 (1991), 203-215.
- R. Rudnicki, Asymptotical stability in L1 of parabolic equations, J. Differential Equations 102 (1993), 391-401.
- R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262.
- K. Skórnik, On fractional integrals and derivatives of a class of generalized functions, Soviet Math. Dokl. 22 (1980), 541-543.
- K. Skórnik and J. Wloka, m-reduction of ordinary differential equations, Colloq. Math. 78 (1998), 195-212.

### Staff:

**Michał Jóźwikowski**/ Assistant Professor / email

**Andrzej Królak**/ Professor / email

**Tatiana Shulman**/ Assistant Professor / email

#### Research activities:

- Principal structures of geometry and physics (Janusz Grabowski)
Canonical algebraic and geometric structures that play fundamental role in description of various systems in physics are studied in the language of differential geometry and super-geometry. Among them are Poisson and Jacobi structures (e.g. symplectic and contact), Lie and Courant algebroids, Dirac structures, generalized (e.g. complex) geometries, Nijenhuis tensors and the corresponding contractions, principles of variational calculus – all this with applications to Theoretical Mechanics, especially to frame-independent description of mechanical systems, foundations of Quantum Mechanics and the geometry of quantum states, quantum information and description of entanglement.

- J. Grabowski, K. Grabowska: Variational calculus with constraints on general algebroids, J. Phys A: Math. Theor. 41 (2008), to appear.
- J. Grabowski, G. Marmo and M. Kuś: On the relation between states and maps in infinite dimensions, Open Syst. Inf. Dyn. 14 (2007), 355-370

- Mathematical methods of physics (Witold Kondracki, Andrzej Królak)
Main activity are applications of methods of statistics and the theory of stochastic processes to the problem of detection of gravitational waves in the noise of the detector. In particuar in the paper [1] optimal statistic to search for modulated periodic gravitational waves from rotating neutron stars is derived. This optimal statistic is now commonly used in the gravitational wave data analysis in particular in the Einstein@Home project.

In paper [2] it is shown that the response of a detector to the superposition of tens of millions of gravitational wave signals from binary white dwarf systems is a cyclo-stationary random process. Using these methods real data from resonant bar detectors EXPLORER and NAUTILUS are analyzed. Also several problems of the Lorentzian geometry in the large, in particular properties of Cauchy horizons have been studied.

- P. Jaranowski, A. Królak, and B. F. Schutz: Data analysis of gravitational-wave signals from pulsars. I. The signal and its detection, Phys. Rev. D58 (1998) 063001.
- J. Edlund, M. Tinto, A. Królak, and G. Nelemans: The White Dwarf - White Dwarf galactic background in the LISA data, Phys. Rev. D71 (2005) 122003.

The main areas of research are geometrical structures of nonlinear field theories, in particular of the Einstein theory of gravitation and Yang-Mills gauge theories, and statistical methods of detection and estimation of signals of gravitational and electromagnetic origin.

In particular we carried out research on the long-standing hypothesis of Roger Penrose - the cosmic censorship hypothesis, which asserts that the final state of gravitational collapse of a star of sufficiently large mass is always a black hole. To approach this problem we have used methods of geometry and differential topology. We work on theoretical and practical aspects of analysis of data from gravitational-wave detectors. Ground-based detectors of gravitational waves are currently working in Germany, France, Italy, Japan, and the USA. NASA and the European Space Agency are planning to put a gravitational wave detector (LISA project) in an orbit around the Sun. Detection of gravitational waves will be a final confirmation of Einstein’s theory of gravity and will open a new window on the Universe. Our main interest is detection of very weak, quasi-periodic signals in large parameter spaces. Using our theoretical methods and algorithms we currently analyse data from Italian NAUTILUS resonant bar detector. Analysis is performed on a large network of computers. We study theoretical methods and we develop data analysis tools to analyse the gravitation wave signal originating from superposition of many signals from binary systems in our Galaxy. This is the dominant gravitational wave signal that will be present in the data of a space-borne detector LISA.

We have taken part in organization of several conferences in Banach International Mathematical Center:

- Symplectic Singularities and Geometry of Gauge Fields
- Quantum Groups and Quantum Spaces
- Mathematics of Gravitation
- Mathematics of Gravitation II

The team has strong collabor

- W. Kondracki and J. Rogulski, Metody geometryczne w mechanice klasycznej i klasycznej teorii pola, Wydawnictwa Uniwersytetu Warszawskiego, 1978.
- C. J. S. Clarke and A. Królak, Conditions for the occurrence of strong curvature singularities, Journal of Geometry and Physics, Vol. 12 (1985), 127.
- W. Kondracki and J. Rogulski, On the stratification of the orbit space for the action of automorphisms on connections, Dissertationes Mathematicae, Vol. 250 (1986).
- A. Królak and B. F. Schutz, Coalescing binaries - probe to the Universe, General Relativity and Gravitation, Vol. 19 (1987), 1163. (Second Prize by Gravity Research Foundation, Ma, USA).
- A. Królak, Towards the proof of the cosmic censorship hypothesis , Classical and Quantum Gravity Vol.3 (1986), 267.
- S. Klimek and W. Kondracki, A construction of two-dimensional quantum chromodynamics, Commun. Math. Phys. Vol. 113 (1987), 389.
- P. Jaranowski and A. Królak, Optimal solution of the inverse problem for the gravitational wave signal of a coalescing compact binary, Phys. Rev. D49 (1994), 1723.
- P. Jaranowski, A. Królak, and B. F. Schutz, Data analysis of gravitational-wave signals from pulsars. I. The signal and its detection, Phys. Rev. D58 (1998), 063001.
- R. Budzyński, W. Kondracki, and A. Królak, New properties of Cauchy and event horizons, Nonlinear Analysis Vol. 47 (2001), 2983.
- A. Królak, Cosmic Censorship Hypothesis, Contemporary Mathematics, Vol.359 (2004), 51.
- P. Jaranowski and A. Królak, Gravitational Wave Data Analysis: Formalism and Applications, Living Reviews, lrr-2005-3, 2005

- Algebraic Analysis, algebras with logarithms (Danuta Przeworska-Rolewicz)
New approach to logarithmic mappings as invertible selectors of multifunctions is proposed together with some applications of logarithmic and antilogarithmic mappings to linear and nonlinear equations in algebras of matrices. Also problems of algebraic analysis of algebraic structures with right-invertible operators.

- D. Przeworska-Rolewicz: Logarithms and Antilogarithms. An Algebraic Analysis Approach, Kluwer Academic Publishers, Dordrecht, 1998.
- [2] D. Przeworska-Rolewicz: Algebraic Analysis in structures with Kaplansky-Jacobson property, Studia Math. 168(2005), 165-186.

- Applications of Functional Analysis and Theory of Metric Spaces to Optimization. Paraconvex analysis. (Stefan Rolewicz)
Problems concerning strongly paraconvex functions on an open convex subset in a Banach space X are studied, e.g. their differentiability [1], as well as an extension of the notion ofstrongly paraconvex (uniformly approximate convex) functions on differentiable manifolds and Φ-convexity in metric spaces [2].

- S. Rolewicz: On differentiability of strongly α(.)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167, (2005), 235 – 244.
- S. Rolewicz: Φ-convex functions de¯ned on metric spaces, Journal of Mathematical Sciences (ed. Plenum) 115 (2003), 2631 - 2652.

- Wavelets and Data Analysis (Piotr Wojdyłło)
Wilson systems play an essential role in the construction of orthonormal bases from the Gabor tight frames with certain redundancy. The research shows that there is a strong connection between the Wilson system construction and the automorphisms of the underlying system of operators, especially as they are unitarily implemented.

The wavelet analysis of time series is investigated, showing the so called Long-Range Dependence. This behaviour is observed, e.g. in packet load in web traffic or in economical processes. To this group of stochastic processes belong, in particular, 1/f noises.

- P.Wojdyłło:Characterization of Wilson Systems for General Lattices, International Journal of Wavelets, Multiresolution and Information Processing, 6/2(2008), 305-314.
- P.Wojdyłło: Modified Wilson Orthonormal Bases, Sampling Theory in Signal and Image Processing, 6 (2007), 223-235.

- Symplectic geometry (Aleksy Tralle, Bogusław Hajduk)
Explicit constructions of symplectic manifolds with prescribed properties are performed. The most important problem in this area is the existence problem for symplectic structures on closed manifolds. It is generally believed that closed symplectic manifolds do not possess any special topological properties, however, the main problem in this area (Thurston's conjecture) still is not solved. Several questions of the same type were aked about some other homotopic properties. For example, it was asked if there are relations between hard Lefschetz property, formality and Betti numbers of closed symplectic manifolds.

- B. Hajduk, A. Tralle, Diffeomorphisms and almost complex structures on tori, Annals Global Anal. Geom.28 (2005), 337-349.
- B. Hajduk, A. Tralle, Exotic smooth structures and symplectic forms on closed manifolds, Geom. Dedicata 132 (2008), 31-42.

- Bihamiltonian Systems and Integrability (Andriy Panasyuk)
Integrability, and Poisson pensils are studied in the context of bihamiltonian systems.

- A. Panasyuk:Reduction by stages and the Raïs-type formula for the index of a Lie algebra with an ideal, Ann. Global Anal. Geom. 33 (2008), 1-10.
- A. Panasyuk:Algebraic Nijenhuis operators and Kronecker Poisson pencils, Differential Geom. Appl. 24 (2006), 482-491.

#### Annual Scientific Report

- Scientific Report 2007 rok (PDF)
- Scientific Report 2006 (PDF)
- Scientific Report 2005 (PDF)

### Staff:

**Szymon Majewski**/ Graduate Assistant / email

**Błażej Miasojedow**/ Assistant Professor / email

**Tomasz Rychlik**/ Professor / email

#### About the Department

The Section of Mathematical Statistics was established on April 1, 2004, as a unification of Section of Mathematical Statistics and Its Applications and Section of Applied Probability. For brief history of the two last mentioned Sections see below.

The following researchers have been members of the Section of Mathematical Statistics: Tadeusz Bednarski, Przemysław Biecek, Katarzyna Danielak, Tadeusz Inglot, Teresa Ledwina, Ryszarda Rempała, Tomasz Rychlik, Grzegorz Wyłupek and Ryszard Zieliński. The main scientific interests have been focused on the following topics:

- asymptotic optimality and efficiency of tests,
- characterizations by means of ordered statistical data,
- data driven tests,
- general methodology for robust estimation and testing,
- methods of model selection,
- multiple testing procedures,
- optimal bounds on statistical functionals,
- optimal nonparametric quantile estimation,
- sequential decision problems for controlling production, distribution and inventory processes.

#### Selected publications:

- T. Bednarski (2004).
*Robust estimation in the generalized Poisson model*, Statistics 38, 149-159. - M. Bogdan, F. Frommlet, P. Biecek, R. Cheng, J.K. Ghosh, R.W. Doerge (2008).
*Extending the modified Bayesian Information Criterion (mBIC) to dense markers and multiple interval mapping*, Biometrics 64, 1162-1169. - K. Danielak (2005).
*Distribution-free bounds for expectations of increments of records*, J. Statist. Plann. Inf. 133, 239-247. - T. Inglot, T. Ledwina (2004).
*On consistent minimax distinguishability and intermediate efficiency of Cramér-von Mises test*, J. Statist. Plann. Inf. 124, 453-474. - T. Inglot, T. Ledwina (2006).
*Data driven score tests for a homoscedastic linear regression model: asymptotic results*, Probab. Math. Statist. 26, 41-61. - J. Navarro, T. Rychlik (2007).
*Reliability and expectation bounds for coherent systems with exchangeable components*, J. Multivar. Anal. 98, 102-113. - W. Niemiro, R. Zieliński (2007).
*Uniform asymptotic normality for the Bernoulli scheme*, Appl. Math. 34, 215-221. - P. Jaworski, T. Rychlik (2008).
*On distributions of order statistics for absolutely continuous copulas with applications to reliability problems*, Kybernetika 44, 757-776. - G. Wyłupek (2010).
*Data driven k-sample tests*, Technometrics 52, 107-123. - T. Ledwina, J. Mielniczuk (2010).
*Variance function estimation via model selection*, Appl. Math. 37, 387-411. - J. Navarro, T. Rychlik (2010).
*Comparisons and bounds for expected lifetimes of reliability systems*, Europ. J. Operational Res. 207, 309-317. - K. Jasiński, T. Rychlik (2012).
*Bounds on dispersion of order ststistics based on dependent symmetrically distributed random variables*, J. Statist. Plann. Inf. 142, 2421-2429. - T. Ledwina, G. Wyłupek (2012).
*Two-sample test against one-sided alternatives*, Scand. J. Statist. 39, 358-381. - P. Miziuła, T. Rychlik (2014).
*Sharp bounds for lifetime variances of reliability systems with exchangeable components*, IEEE Trans. Reliab. 63, 850-857. - T. Ledwina, G. Wyłupek (2014).
*Validation of positive quadrant dependence*, Insurance Math. Econom. 56, 38-47. - P. Miziuła, T. Rychlik (2015).
*Extreme dispersions of semicoherent and mixed system lifetimes*, J. Appl. Probab. 52, 117-128.

#### Seminars:

- Asymptotic Statistics (T. Inglot, T. Ledwina; Wrocław, since 2004),
- Mathematical Statistics and other Probabilistic Applications (M. Męczarski, J. Mielniczuk - until June 2010, P. Jaworski - since October 2010, T. Rychlik; Warszawa),
- Mathematical Statistics (W. Niemiro, T. Rychlik; Toruń, 2008-2014),
- Statistics in Medicine (P. Biecek; Wrocław, 2007-2008).

### The Section of Mathematical Statistics and Its Applications

The Section of Mathematical Statistics and Its Applications stemmed from the Department of Applied Mathematics of the State Institute of Mathematics, which was established in November 20, 1948. In 1949 the Department consisted of six sections. Two of them: the General Section of Applications, headed by Hugo Steinhaus, and the Section of Technical Applications, directed by Jan Mikusiński, were located in Wrocław. These two sections formed later on the Section of Applications to Biology, Economics and Technology, directed by Hugo Steinhaus until his retirement in 1960. Julian Perkal (1960-1965) and Stefan Zubrzycki (1965-1968) were later directors. The Section has played a very important role in the Wrocław scientific community, in particular through many collaborations with various professionals and educational work in applied probability and statistics. The journal Zastosowania Matematyki (the subtitle Applicationes Mathematicae added in 1965), founded by Hugo Steinhaus in 1953, with help of Jan Oderfeld, was a significant platform for the activity of the group. This activity was further supported by Julian Perkal, who founded the Listy Biometryczne journal in 1964. The journal is currently published as Biometrical Letters.

From 1968 to 1991 the Section was headed by Witold Klonecki and was renamed to the Section of Mathematical Statistics and Its Applications in 1973. At that time, the main activity was focused on PhD studies in modern statistical methodology, initially with the kind aid of colleagues from Wrocław University. Members of the group participated very actively in the organization of a series of conferences, beginning with one in Wisła in 1973. Many of these conferences were international. Symposium to honour Jerzy Neyman (1974), the European Meeting of Statisticians (1981) and Banach Center Semester on Nonparametric and Robust Methods (1984) are prominent examples. Witold Klonecki was also one of the founders of the Probability and Mathematical Statistics journal (1980).

Since 1991 until 2004 the Section was headed by Tadeusz Bednarski. Apart from research in mathematical statistics, much attention was focused on direct applied work.

#### Selected publications:

- K. Florek, J. Łukaszewicz, J. Perkal, H. Steinhaus, S. Zubrzycki (1951). Sur la liaison et la division des points d'un ensemble fini, Colloq. Math. 2, 282-285.
- H. Steinhaus (1953/1954). On establishing paternity [in Polish], Zastosow. Mat. I, 67-82.
- A. Zięba (1953/1954). Elementary theory of pursuit [in Polish], Zastosow. Mat. I, 273-298.
- S. Drobot, M. Warmus (1954). Dimensional analysis in sampling inspection of merchandise, Dissertationes Math. [Rozprawy Mat.] V.
- S. Gładysz, A. Rybarski (1954/1956). On modelling three-dimensional fields by a plane field of current [in Polish], Zastosow. Mat. II, 150-160.
- S. Zubrzycki (1954/1956). On the optimal method of water meter acceptance [in Polish], Zastosow. Mat. II, 199-209.
- A. Huskowska (1954/1956). On the accuracy of some natural science measurements, [in Polish], Zastosow. Mat. II, 426-430.
- J. Battek, J. Perkal (1956/1958). Quality and shape of forest stands [in Polish], Zastosow. Mat. III, 285-306.
- J. Łukaszewicz, M. Warmus (1956). Numerical and Graphical Methods [in Polish], PWN, Warszawa.
- S. Paszkowski i M. Warmus (1956). On some mathematical method in anthropology [in Polish], Przegląd Antropologiczny XXII, 627-650.
- H. Steinhaus (1957). The problem of estimation, Ann. Math. Statist. 28, 633-648.
- S. Zubrzycki (1957). On estimating gangue parameters [in Polish], Zastosow. Mat. III, 105-153.
- E. Marczewski, H. Steinhaus (1958/1959). On the systematic distance of biotopes, [in Polish], Zastosow. Mat. IV, 195-203.
- H. Steinhaus, S. Trybuła (1958/1959). Measurement by successive comparison [in Polish], Zastosow. Mat. IV, 204-212.
- H. Steinhaus, K. Urbanik (1959). Poissonsche Folgen (Leon Lichtenstein zum Gedáchtnis), Math. Zeitsch. 72, 127-145.
- T. Dalenius, J. Hájek, S. Zubrzycki (1961). On plane sampling and related geometrical problems, in : Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1, 125-130.
- S. Trybuła (1961). On the paradox of three random variables, Zastosow. Mat. V, 321-332.
- A. Bartkowiakowa, B. Gleichgewicht (1962). On the syllable characteristic of the Polish language [in Polish], Zastosow. Mat. VI, 309-319.
- A. Krzywicki, A. Rybarski (1962). On a linearization of an equation of an elastic rod, Zastosow. Mat. VI, 321-332.
- J. Perkal (1962). A design for a genetic test [in Polish], Zastosow. Mat. VI, 257-285.
- H. Steinhaus (1963). Probability, credibility, possibility, Zastosow. Mat. VI, 341-361.
- L. Zubrzycka (1963). On the adaptation of the typewriter keyboard to the structure of the language [in Polish], Zastosow. Mat. VI, 419-439.
- K. Florek (1964). On a certain method of graphical integration and graphical harmonic analysis, Zastosow. Mat. VII, 353-370.
- B. Kopociński, L. Zubrzycka (1964). Remark on division of systems of features into harmonized subsystems [in Polish], Zastosow. Mat. VII, 317-321.
- S. Zubrzycki (1966). Explicit formulas for minimax admissible estimators in some cases of restrictions imposed on the parameter, Zastosow. Mat. IX, 31-52.
- B. Bednarek-Kozek (1973). On estimation in the multidimensional Gaussian model, Zastosow. Mat. XIII, 511-520.
- H. Drygas, J. Srzednicka (1976). A new result on Hsu's model on regression analysis, Bull. Acad. Polon. Sci., Sér. Sci. Math. Phys. Astronom. 24, 1133-1136.
- S. Gnot (1976). The mean efficiency of block designs, Math. Operationsforsch. Statist. 7, 75-84.
- W. Klonecki (1977). Optimal C($\alpha$) tests for homogeneity, in : Proceedings of the Symposium to honour Jerzy Neyman (Warsaw 1974), PWN, 161-175.
- A.S. Kozek (1977). Efficiency and Cramér-Rao type inequalities for convex loss functions, J. Multivariate Anal. 7, 89-106.
- M. Musiela (1977). Sequential estimation of parameters of a stochastic differential equation, Math. Operationsforsch. Statist. Ser. Statist. 8, 483-498.
- T. Ledwina (1978). Admissible tests for exponential families with finite support, Math. Operationsforsch. Statist. Ser. Statist. 9, 105-118.
- R. Zmyślony (1980). A characterization of best unbiased estimators in the general linear model, in : Mathematical Statistics and Probability Theory (Proc. Sixth International Conf. Wisła 1978), Lecture Notes in Statist. 2, Springer-Verlag, 365-378.
- W. Wawrzyniak (1981). A characterization of minimum variance unbiased estimators in the general linear model with restrictions on parameter space. Math. Operationsforsch. Statist. Ser. Statist. 12, 456-477.
- T. Bednarski (1982). Binary experiments, minimax tests and 2-alternating capacities, Ann. Statist. 10, 226-232.
- A.S. Kozek (1982). Towards a calculus for admisibility, Ann. Statist. 10, 825-837.
- S. Gnot, J. Kleffe, R. Zmyślony (1985). Nonnegativity of admissible invariant quadratic estimates in mixed linear models with two variance components, J. Statist. Plann. Inference 12, 249-258.
- J. Zabrzeski, R. Zmyślony (1987). Experimental description of the rate of platinum and rodium losses in the process of ammonia oxidation, Appl. Catalisis 35, 13-22.
- R.H. Farrel, W. Klonecki, S. Zontek (1989). All admissible linear estimators of the vector of gamma scale parameters with application to random effects models, Ann. Statist. 17, 268-281.
- W. Klonecki, S. Zontek (1992). Admissible estimators of variance components obtained via submodels, Ann. Statist. 20, 1454-1467.
- T. Bednarski (1993). Robust estimation in the Cox regression model, Scand. J. Statist. 20, 213-225.
- T. Bednarski, S. Zontek (1996). Robust estimation of parameters in mixed unbalanced models, Ann. Statist. 24, 1493-1510.
- K. Drosik, A. Michalski, S. Sadowski, A. Tukiendorf, R. Zmyślony (1998). Neoplasmatic Disease Incidence in Opole Province, Wydawnictwo Uniw. Opolskiego.
- T. Inglot, W.C.M. Kallenberg, T. Ledwina (1998). Vanishing shortcoming of data driven Neyman's tests, in : Asymptotic Methods in Probability and Statistics, A Volume in Honour of Miklós Csörgő, North-Holland, 811-829.
- T. Bednarski, W. Florczak (1999). On a local uniform bootstrap validity, Statist. Neerl. 53, 111-121.
- W.C.M. Kallenberg, T. Ledwina (1999). Data driven rank tests for independence, J. Amer. Statist. Assoc. 94, 285-301.
- A. Janic-Wróblewska, T. Ledwina (2000). Data driven rank test for two-sample problem. Scand. J. Statist. 27, 281-297.
- T. Inglot, T. Ledwina (2001). Intermediate approach to comparison of some goodness-of-fit tests, Ann. Inst. Statist. Math. 53, 810-834.
- G.R. Ducharme, T. Ledwina (2003). Efficient and adaptive nonparametric test for the two-sample problem, Ann. Statist. 31, 2036-2058.
- T. Inglot, W.C.M. Kallenberg (2003). Moderate deviations of minimum contrast estimators under contamination, Ann. Statist. 31, 852-879.

### Section of Applied Probability

The Section of Applied Probability originated in a long process rooted in the Department of Applied Mathematics of the State Mathematical Institute, founded in 1948. Two Warsaw's sections of this department were forerunners of it. Namely, the Section of Statistical Quality Control and the Actuarial Section. In accordance with the evolving interests of the members, the Section of Statistical Quality Control was renamed to the Industrial Applications Section in 1962. In the period from 1952 to 1970 these sections were directed by Jan Oderfeld, with short break in 1959 when Czesław Rajski was in charge of it. At that time, the research was mainly focused on statistical quality control, operations research and reliability theory. The Actuarial Section evolved into the Section of Mathematical Statistics. In 1951, Oskar Lange was put in charge of the section and he did it until 1958. In 1958-1960 Marek Fisz was the leader of it while later on the section was headed by Wiesław Sadowski. Interests of the group were focused on statistics, decision making and econometry.

These two evolving groups were the predecessors the Applied Probability Section, formally founded in 1972.

The section was headed by Robert Bartoszyński, Ryszard Zieliński and Tomasz Rychlik in the periods 1972-1985, 1985-2002 and 2002-2004, respectively. In the early seventies, the scientific interests of the group switched to probabilistic modelling (mostly, but not only, of biological phenomena), controlled Markov chains, stochastic approximation and random search. In recent years, various issues of mathematical statistics, including robust statistics, fixed precision estimation and moment bounds have become major focuses of the section's research.

The section was active in the organization of courses in applied mathematics, enhancing education in a wide spectrum of applied probability. Also, starting from 1971, Eugeniusz Fidelis annually organized the very popular National Conferences on Applications of Mathematics. Members of the group organized many international conferences in statistics, including Banach Center Semesters on Mathematical Statistics (1976) and Sequential Methods in Statistics (1981), among others.

#### Selected publications:

- J. Oderfeld (1951). On the dual aspects of sampling plans. Colloq. Math. 2, 89-97.
- M. Fisz (1954).
*Probability Theory and Mathematical Statistics*(first edition; in Polish), PWN. - T. Czechowski, W. Sadowski, W. Zasępa (1954/1956).
*On determining the safety factor*[in Polish], Zastosow. Mat. 2, 190-198. - W. Sadowski (1954/1956).
*On a non-parametric test of comparing dispersions*[in Polish], Zastosow. Mat., 2, 161-171. - C. Rajski (1954/1956).
*On the verification of hypotheses concerning two populations consisting of items marked by attributes*[in Polish], Zastosow. Mat., 2, 179-189. - M. Fisz, K. Urbanik (1956).
*Analytical characterization of the composed non-homogeneous Poisson process*, Stud. Math. 15, 328-336. - M. Fisz (1958).
*A limit theorem for empirical distribution functions*, Stud. Math. 17, 71-77. - O. Lange (1959).
*Introduction to Econometrics*(first edition; in Polish), PWN. - J. Łukaszewicz, Sadowski, W. (1960/1961).
*On comparing several populations with a control population*. Zastosow. Mat. 5, 309-320. - R. Bartoszyński (1961). A characterization of the weak convergence of measures. Ann. Math. Statist. 32, 561-576.
- J. Oderfeld, E. Pleszczyńska (1961). A linear estimate of the mean deviation in normal population [in Polish], Zastosow. Mat. VI, 111-117.
- E. Fidelis, J. Oderfeld (1962). Two-step control taking into account the measurement errors [in Polish], Zastosow. Mat. VI, 249-256.
- W. Rudzki (1962). Estimation of mean properties of shapeless products [in Polish], Zastosow. Mat. VI, 235-248.
- R. Bartoszyński (1967). Branching processes and the theory of epidemics, Proc. Fifth Berkeley Symp. on Math. Statist. Probab. IV, 615-618.
- E. Pleszczyńska (1973). Trend estimation problems in time-series analysis. Dissertationes Math. (Rozprawy Mat.) 104.
- R. Zieliński (1973). A new class of estimators with an application to statistical quality control. Zastosow. Mat. XIII, 279-300.
- R. Bartoszyński (1974). On a metric structure derived from subjective judgements: scaling under perfect and imperfect discrimination, Ecomometrica 42, 55-71.
- R. Bartoszyński (1975). On risk of rabies, Math. Biosci. 24, 357-377.
- R. Zieliński (1977). Global stochastic approximation. Dissertationes Math. (Rozprawy Mat.) 142.
- R. Bartoszyński, W.J. Bühler (1978). On survival in hostile environment, Math. Biosci. 38, 293-301.
- R. Bartoszyński, B.W. Brown, C.M. McBridge, J.R. Thompson (1981). Some nonparametric techniques for estimating the intensity function of a cancer related to nonstationary Poisson process. Ann. Statist. 9, 150-160.
- R. Zieliński (1983). Robust statistical procedures: a general approach, in: ``Stability Problems for Stochastic Models'' (V.V. Kalashnikov and V.M. Zolotarev, eds.), Lecture Notes in Mathematics 982, Springer-Verlag, Berlin, 283-295.
- B. Gołdys, M. Męczarski, R. Zieliński (1986). An asymptotic fixed-precision confidence interval for the minimum of a quadratic regression function. Probab. Math. Statist. 7, 7-11.
- J. Koronacki, W. Wertz (1988). A global stopping rule for recursive density estimators. J. Statist. Plann. Inference 20, 23-39.
- J. Koronacki (1989). Stochastic Approximation. Optimization Methods under Random Conditions [in Polish]. WNT, Warsaw.
- B. Gołdys (1990). Regularity properties of solutions to stochastic evolution equations, Colloq. Math. 58, 327-338.
- R. Rempała (1991). Forecast horizon in a dynamic family of one-dimensional control problems, Dissertationes Math. (Rozprawy Mat.) 315.
- R. Zieliński (1991). Fixed precision estimation in the Blum-Rosenblatt time series. Amer. J. Management Sci. 11, 233-239.
- T. Rychlik (1993). Bounds for expectation of L-estimates for dependent samples, Statistics 24, 1-7.
- L. Gajek, D. Zagrodny (1995). Geometric mean value theorems for the Dini derivative. J. Math. Anal. Appl. 191, 56-76.
- W. Niemiro (1995). Estimation of nuisance parameters for inference based on least absolute deviations. Zastosow. Mat. 22, 515-529.
- W. Niemiro, P. Pokarowski (1995). Tail events of some nonhomogeneous Markov chains, Ann. Appl. Probab. 5, 261-293.
- L. Gajek, T. Rychlik (1996). Projection method for moment bounds on order statistics from restricted families. I. Dependent case. J. Multivar. Anal. 57, 156-174.
- L. Gajek, T. Rychlik (1998). Projection method for moment bounds on order statistics from restricted families. II. Independent case. J. Multivar. Anal. 64, 156-182.
- R. Zieliński (1998). Uniform strong consistency of sample quantiles. Statist. Probab. Lett. 37, 115-119.
- T. Rychlik (2001). Projecting Statistical Functionals. Lecture Notes in Statistics 160, Springer-Verlag, New York.
- S. Bylka, R. Rempała (2003). Selected Problems in the Mathematical Inventory Theory [in Polish], Akademicka Oficyna Wydawnicza Exit, Warsaw.

#### About the Department

Research during the last few years concerned different branches of number theory as well as fields and polynomials, which will be reviewed in order adopted by Mathematical Reviewers.

In elementary number theory a lower estimate has been proved under certain conditions in [9] for the number of solutions of a linear homogeneous congruence in a multidimensional box. The same, best possible, estimate under different conditions is in the course of publications.

A problem on diophantine equations over rational integers has been solved in [1], binary forms over an arbitrary field have been considered in [7] and forms in many variables in [14]. An almost explicit construction of a point on an elliptic curve over a finite field has been given in [17].

A problem on the length (a kind of a height) of polynomials with real coefficients, related to diophantine approximation has been studied in [8] and [12], a similar study of polynomials with complex coefficients is in course of publication. Paper [6] studies a problem in geometry of numbers.

Elementary analytic number theory is represented by [16] and a more advanced on by [5]. [15] concerns elementary algebraic number theory [2] K-theory and [3] finite fields.

In field theory and polynomials papers [3] and [11] are concerned with localization of zeros of polynomials in one variable, papers [10] and [11] with reducibility of symmetric polynomials.

The following topics have been studied in the period 1999--2008:

- Representation of integer vectors as a linear combination of shorter integer vectors (I. Aliev, A. Schinzel)
- The Milnor group K
_{2F}(J. Browkin) - Distribution of primitive roots (A. Paszkiewicz, A. Schinzel)
- Pseudoprimes and their generalizations (A. Rotkiewicz, A. Schinzel)
- Reducibility of polynomials (A. Schinzel)
- The number of non-zero coefficients of the greatest common divisor of two polynomials with given numbers of non-zero coefficients (A. Schinzel)
- The Mahler measure and other measures of polynomials (A. Schinzel)
- Weak automorphs of binary forms (A. Schinzel)
- Number of solutions of a linear homogeneous congruence in a box (A. Schinzel)
- Representation of a multivariate polynomial as a sum of univariate polynomials in linear forms (A. Schinzel)
- Polynomial and exponential congruences to a prime modulus (M. Skałba, A. Schinzel)
- Congruences for L-functions (J. Urbanowicz)
- Divisibility of a generalized Vandermonde determinant by powers of two (J. Urbanowicz)

Several people not employed by the Number Theory Section have collaborated in the study of the above topics, namely W. Schmidt in 1), M. Zakarczemny in 9), A. Białynicki in 10), K. Williams in 12) and S. Spież in 13).

#### Research papers published in 2005--2008 (March)

- J. Browkin (with J. Brzeziński),
*On sequences of squares with constant second differences*, Canadian Math. Bulletin 48 (2006), 481--491. - J. Browkin,
*Elements of small order in K*, Chin. Ann. Math. Ser. B 28 (2007), 507--520._{2}F, II - A. Schinzel,
*Self-inversive polynomials with all zeros on the unit circle*, Ramanujan Journal 9 (2005), 19--23. - A. Schinzel (with T. Bolis),
*Identities which imply that a ring is Boolean*, Bull. Greek Math. Soc. 48 (2003), 1--5 (antedated). - A. Schinzel (with S. Kanemitsu and Y. Tanigawa),
*Sums involving the Hurwitz zeta-function values*, Zeta Function, Topology and Quanture Physics, 81-90, Springer 2005. - A. Schinzel (with I. Aliev and W. M. Schmidt),
*On vectors whose span contains a given linear subspace*, Monatsh. Math. 144 (2005), 177-191. - A. Schinzel,
*On weak automorphs of binary forms over an arbitrary field*, Dissert. Math. 434 (2005), 48 pp. - A. Schinzel,
*On the reduced length of a polynomial*, Functiones et Approximatio 35 (2006), 271--306. - A. Schinzel (with M. Zakarczemny),
*On a linear homogeneous congruence*, Colloq. Math. 106 (2006), 283--292. - A. Schinzel,
*Reducibility of symmetric polynomials*, Bull. Polish Acad. Sci. Mathematics 53 (2005), 251--258 (antedated). - A. Schinzel (with L. Losonczi),
*Self-inverse polynomials of odd degree*, Ramanujan Journal 14 (2007), 305--320. - A. Schinzel,
*On the reduced length of a polynomial with real coefficients, II*, Functiones et Approximatio 37 (2007), 445--459. - A. Schinzel,
*Reducibility of a special symmetric form*, Acta Math. Universitatis Ostraviensis 14 (2006), 71--74 (antedated). - A. Schinzel (with A. Białynicki-Birula),
*Representations of multivariate polynomials by sums of univariate polynomials in linear forms*, Colloq. Math. 112 (2008), 201--233. - M. Skałba,
*On sets which contain a q-th power residue for almost all prime modules*, Colloq. Math. 102 (2005), 67--71. - M. Skałba,
*Primes dividing both 2*, Arch. Math. 84 (2005), 485--495.^{n}and 3^{n}-2 are rare - M. Skałba,
*Points on elliptic curves over finite fields*, Acta Arith. 117 (2005), 293--301.

Besides the following book has been published

**A. Schinzel**, *Selecta (2 vols)*, ed. H. Iwaniec, W. Narkiewicz, J. Urbanowicz, Zürich 2007.

### Staff:

**Radosław Adamczak**/ Associate Professor / email

**Tomasz Byczkowski**/ Professor / email

**Adam Gregosiewicz**/ Assistant Professor / email

**Tomasz Komorowski**/ Professor / email

**Stanisław Kwapień**/ Professor / email

**Szymon Peszat**/ Professor / email

**Jerzy Zabczyk**/ Professor / email

#### About the Department

Main research results of the members of the Department are described below and a list of papers, selected by the authors, is presented. Complete lists of publications can be found through links.

The Department seminar

### Z. Ciesielski

In probability: Simple construction of Brownian motion in terms of Schauder bases. Determining the exact Hausdorff measure of the Brownian trajectories (with S. J. Taylor). Discovering the principle of not feeling the boundary in heat conduction. The Wiener measure is concentrated on a suitable Hölder-Orlicz class with exponent ½. Application of Schauder spline bases to calculating the fractal dimension of realizations of random fields.

In approximation theory: Characterization of Hölder classes by the coefficients of Schauder and Franklin expansions. Description of the basic properties of the Franklin orthogonal system; exponential estimates. Positive solution to the Banach problem on existence of a basis in the space of continuously differentiable functions on the square. Building up (jointly with J. Domsta and T. Figiel) a theory of spline bases on compact sets whose idea preceded the wavelet theory.

### S. Peszat

In infinite dimensional stochastic analysis: Establishing the Freidlin-Ventsel large deviation estimates for a general class of diffusions in Hilbert spaces. Proving (jointly with J. Zabczyk) that the transition semigroup corresponding to a stochastic evolution equation is strong Feller and irreducible, provided that the nonlinearities are Lipschitz continuous. Consequently, the invariant measure for infinite dimensional diffusion is unique. Establishing sufficient conditions for existence and properties of solutions of infinite dimensional stochastic equations with Lipschitz nonlinearities (jointly with Zabczyk) and polynomial nonlinearities (jointly with Z. Brzeźniak). Formulating (with J. Zabczyk) necessary and sufficient conditions for the existence of function-valued solutions to multidimensional stochastic wave and heat equations.

Establishing (jointly with M. Capiński and Z. Brzeźniak) the existence and uniqueness of solutions to stochastic Navier-Stokes and Euler equations. Proving (with T. Komorowski) the uniqueness in law of the solution to the passive tracer problem in an irregular velocity field. Establishing (jointly with J. Zabczyk) basis for SPDEs with Lévy noise.

### Ł. Stettner

- In stochastic control:
- existence of solutions to the Bellman equations corresponding the problems: partially observed control problem with average cost per unit time criterion ([3]), risk sensitive control problem with infinite time ergodic cost criterion with complete and partial observations ([5], [6], [9]),
- construction of nearly optimal strategies with applications to adaptive control ([2], [4]);
- in filtering theory:
- conditions for ergodicity of filtering processes ([1], [8], [10]);
- in mathematics of finance:
- existence of optimal strategies for general utility maximization in discrete time ([7]),
- existence of optimal strategies for growth optimal and risk sensitive growth optimal portfolios with transaction costs ([11]),

**Related papers:**

[1] Ł. Stettner, *On Invariant Measures of Filtering Processes*, Proc. 4th Bad Honnef Conf. on Stochastic Differential Systems, Ed. N. Christopeit, K. Helmes, M. Kohlmann, Lect. Notes in Control Inf. Sci. 126, Springer 1989, 279 - 292.

[2] W. J. Runggaldier and Ł. Stettner, *Nearly Optimal Controls for Stochastic Ergodic Problems with Partial Observation*, SIAM J. Control Optimiz. 31 (1993), 180 - 218.

[3] Ł. Stettner, *Ergodic Control of Partially Observed Markov Processes with Equivalent Transition Probabilities*, Applicationes Mathematicae 22.1 (1993), 25 - 38.

[4] T. Duncan, B. Pasik-Duncan, Ł. Stettner, *Discretized Maximum Likelihood and Almost Optimal Adaptive Control of Ergodic Markov Models*, SIAM J. Control Optimiz. 36 (1998), 422 - 446.

[5] G. B. Di Masi, Ł. Stettner, *Risk sensitive control of discrete time Markov processes with infinite horizon*, SIAM J. Control Optimiz. 38 (2000), 61 - 78.

[6] G. B. Di Masi, Ł. Stettner, *Risk sensitive control of discrete time partially observed Markov processes with infinite horizon*, Stochastics and Stochastics Rep. 67 (1999), 309 - 322.

[7] M. Rasonyi, Ł. Stettner, *On utility maximization in discrete - time market models*, Annals of Applied Prob. 15 (2005), 1367 - 1395.

[8] G. Di Masi, Ł. Stettner, *Ergodicity of Hidden Markov Models*, Math. Control Signals Systems 17 (2005), 269 - 296.

[9] G. B. Di Masi, Ł. Stettner, *Infinite horizon risk sensitive control of discrete time Markov processes under minorization property*, SIAM J. Control Optimiz. 46 (2007), 231 - 252.

[10] G. Di Masi, Ł. Stettner, *Ergodicity of filtering process by vanishing discount approach*, Systems and Control Letters 57 (2008), 150 - 157.

[11] Ł. Stettner, *Discrete Time Infinite Horizon Risk Sensitive Portfolio Selection with Proportional Transaction Costs*, Banach Center Publications, to appear.

### J. Zabczyk

In stochastic processes: Polar sets do not coincide with null sets for Lévy processes. Strong Feller property is equivalent to null controllability for linear stochastic evolution equations (with G. Da Prato). Stochastic factorization and continuity of stochastic convolution (with G. Da Prato and S. Kwapień). Smoothing properties of transition semigroups in Hilbert spaces (with G. Da Prato). Characterization of linear stochastic systems with function valued solutions (with A. Karczewska). Existence of solutions to non-linear heat and wave equations with spatially homogeneous noise (with S. Peszat). Extending Liouville theorem for non-local operators (with E. Priola).

In deterministic control: Location of spectrum does not determine the growth of a linear system. Detectability implies uniqueness of algebraic Riccati equation in infinite dimensions. Analysis of spectral properties of null-controllable systems with vanishing energy (with E. Priola).

In stochastic control: Analysis of algebraic Riccati equation of discrete time stochastic infinite-dimensional systems. Continuous time version of the best choice (with R. Cowan).

### Selected publications

#### Z. Ciesielski:

*On the isomorphisms of the spaces H*, Bull. Acad. Polon. Sci. 8 (1960), 217 - 222._{α}and m- (with S. J. Taylor),
*First passage times and sojourn times for the Brownian motion in the space and the exact Hausdorff measure of the sample path*, Trans. Amer. Math. Soc. 103 (1962), 434 - 450. *Properties of the orthonormal Franklin system. I, II*, Studia Math. 23 (1963), 141 - 157; 27 (1966), 87 - 121.*Brownian motion, capacitory potentials and semi-classical sets. I-III*, Bull. Acad. Polon. Sci. 12 (1964), 265 - 270; 13 (1965), 147 - 150, 215 - 219.*A construction of basis in C*, Studia Math. 33 (1969), 243 - 247.^{1}(I^{2})*Constructive function theory and spline systems*, Studia Math. 53 (1975), 277 - 302.- (with T. Figiel),
*Spline bases in classical function spaces on compact C*, Studia Math. 76 (1983), 1 - 58, 95 - 136.^{∞}manifolds. I, II *Asymptotic nonparametric spline density estimation*, Probab. Math. Statist. 12 (1991), 1 - 24.*Orlicz spaces, spline systems and Brownian motion*, Constr. Approx. 9 (1993), 191 - 208.*Fractal functions and Schauder bases*, Comput. Math. Appl. 30 (1995), 283 - 291.

#### S. Peszat:

*Large deviation principle for stochastic evolution equations*, Probab. Theory Related Fields 98 (1994), 113 - 136.- (with J. Zabczyk),
*Strong Feller property and irreducibility for diffusions on Hilbert spaces*, Ann. Probab. 23 (1995), 157 - 172. *Existence and uniqueness of the solution for stochastic equations on Banach spaces*, Stochastics Stochastics Rep. 55 (1995), 167 - 193.- (with J. Zabczyk),
*Stochastic evolution equations with a spatially homogeneous Wiener process*, Stochastic Process. Appl. 72 (1997), 187 - 204. - (with Z. Brzeźniak),
*Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process*, Studia Math. 137 (1999), 261 - 299. - (with J. Zabczyk),
*Nonlinear stochastic wave and heat equations*, Probab. Theory Related Fields 116 (2000), 421 - 443. - (with M. Capiński),
*On the existence of a solution to stochastic Navier-Stokes equations*, Nonlinear Anal. 44 (2001), 141 - 177. *The Cauchy problem for a nonlinear stochastic wave equation in any dimension*, J. Evol. Equ. 2 (2002), 383 - 394.- (with T. Komorowski),
*Transport of a passive tracer by an irregular velocity field*, J. Statist. Phys. 115 (2004), 1383 - 1410. - (with F. Russo),
*Large noise asymptotics for one-dimensional diffusions*, Bernoulli 11 (2005), 247 - 262. - (with J. Zabczyk),
*Stochastic Partial Differential Equations with Lévy Noise*, Cambridge Univ. Press, 2007.

#### Ł. Stettner:

*Ergodic control of partially observed Markov processes with equivalent transition probabilities*, Appl. Math. 22 (1993), 25 - 38.- (with G. B. Di Masi),
*Risk sensitive control of discrete time partially observed Markov processes with infinite horizon*, in: Proc. 37th IEEE CDC, Tampa, 1998, 3467 - 3472. - (with W. Runggaldier),
*Approximations of Discrete Time Partially Observed Control Problems*, Appl. Math. Monographs CNR, Giardini Ed., Pisa, 1994. - (with T. Duncan and B. Pasik-Duncan),
*Discretized maximum likelihood and almost optimal adaptive control of ergodic Markov models*, SIAM J. Control Optim. 36 (1998), 422 - 446. - (with G. B. Di Masi),
*Bayesian ergodic adaptive control of discrete time Markov processes*, Stochastics Stochastics Rep. 54 (1995), 301 - 316. - (with G. B. Di Masi),
*Bayesian adaptive control of discrete-time Markov processes with long run average cost*, Systems Control Lett. 34 (1998), 55 - 62. - (with D. Gątarek),
*On the compactness method in general ergodic impulsive control of Markov processes*, Stochastics Stochastics Rep. 31 (1990), 15 - 26. - (with G. B. Di Masi),
*Risk sensitive control of discrete time Markov processes with infinite horizon*, SIAM J. Control Optim. 38 (1999), 61 - 78.

#### J. Zabczyk:

*On optimal stochastic control of discrete-time parameter systems in Hilbert spaces*, SIAM J. Control Optim. 13 (1975), 1217 - 1234.*A note on C*, Bull. Acad. Polon. Sci. 23 (1975), 895 - 898._{0}-semigroups*Remarks on algebraic Riccati equation in Hilbert spaces*, J. Appl. Math. Optim. 2 (1976), 251 - 258.- (with R. Cowan),
*An optimal selection problem associated with the Poisson problem*, Theory Probab. Appl. 23 (1978), 606 - 614. - (with G. Da Prato and S. Kwapień),
*Regularity of solutions of linear stochastic equations in Hilbert spaces*, Stochastics Stochastics Rep. 23 (1987), 1 - 23. - (with G. Da Prato),
*Smoothing properties of the Kolmogoroff semigroups in Hilbert spaces*, Stochastics Stochastics Reports 35 (1991), 63 - 77. *Mathematical Control Theory. An Introduction*, Birkhäuser, 1992.*Chance and Decision. Stochastic Control in Discrete Time*, Quaderni Scuola Norm. Sup. Pisa, 1992.- (with G. Da Prato),
*Stochastic Equations in Infinite Dimensions*, Cambridge Univ. Press, 1992. - (with G. Da Prato),
*Regular densities of invariant measures in Hilbert spaces*, J. Funct. Anal. 130 (1995), 427 - 449. - (with S. Peszat),
*Strong Feller property and irreducibility for diffusions on Hilbert spaces*, Ann. Probab. 23 (1995), 157 - 172. - (with G. Da Prato),
*Ergodicity for Infinite Dimensional Systems*, Cambridge Univ. Press, 1996. - (with S. Peszat),
*Stochastic evolution equations with a spatially homogeneous Wiener process*, Stochastic Process. Appl. 72 (1997), 187 - 204. - (with G. Da Prato, B. Gołdys),
*Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces*, C. R. Acad. Sci. Paris Série I 325 (1997), 433 - 438. - (with A. Karczewska),
*Stochastic PDEs with function-valued solutions*, Proceedings of the Colloquium ”Infinite-Dimensional Stochastic Analysis” of the Royal Netherlands Academy of Arts and Sciences, Amsterdam 1999, Eds. Ph. Clement, F. den Hollander, J. van Neerven and B. de Pagter, North Holland, 197 - 216. - (with S. Peszat),
*Nonlinear stochastic wave and heat equations*, PTRF 116 (2000), 421 - 443. - (with G. Da Prato),
*Second Order Partial Differential Equations in Hilbert Spaces*, Cambridge Univ. Press, 2002. - (with E. Priola),
*Null controllability with vanishing energy*, SIAM Journal on Control and Optimization 42 (2003), 1013 - 1032. - (with E. Priola),
*Liouville theorems for non-local operators*, Journal on Functional Analysis 216 (2004), 455 - 490. - (with S. Peszat),
*Stochastic Partial Differential Equations with Lévy Noise*, Cambridge Univ. Press, 2007.

**Henryk Toruńczyk**

pok. 104 / tel. 22 5228 104

### Staff:

**Yeong Chyuan Chung**/ / email

**Si Tiep Dinh**/ Assistant Professor / email

**Lech Tadeusz Januszkiewicz**/ Professor / email

**Zbigniew Jelonek**/ Professor / email

**Michał Lasoń**/ Assistant Professor / email

**Piotr Nowak**/ Associate Professor / email

**Tomasz Odrzygóźdź**/ Graduate Assistant / email

**Damian Orlef**/ Graduate Assistant / email

**Damian Osajda**/ Assistant Professor / email

**Eric Reckwerdt**/ Assistant Professor / email

**Damian Sawicki**/ Graduate Assistant / email

**Marithania Silvero**/ Assistant Professor / email

**Stanisław Spież**/ Professor / email

**Krzysztof Ziemiański**/ Assistant Professor / email

### PHD Students:

**Paweł Bilski**/ e-mail

**Tomasz Odrzygóźdź**/ e-mail

**Damian Orlef**/ e-mail

**Damian Sawicki**/ e-mail

#### About the Department

Below are the descriptions of the main research results or research areas of the full time employees.

### Sylwia Antoniuk

My main areas of research are Combinatorics and Geometric Group Theory with special interest in random structures such as random groups, random simplicial complexes, random graphs and hypergraphs. My main results concern the study of the evolution of the random triangular group.

Selected papers:

Sylwia Antoniuk, Ehud Friedgut, and Tomasz Łuczak, A sharp threshold for collapse of the random triangular group, arXiv:1403.3516

Sylwia Antoniuk, Tomasz Łuczak, and Jacek Świątkowski, Random triangular groups at density 1/3, Compositio Mathematica, vol. 151, issue 01, p. 167-178. (arXiv:1308.5867)

Sylwia Antoniuk, Tomasz Łuczak, and Jacek Świątkowski, Collapse of random triangular groups: a closer look, Bull. Lond. Math. Soc., vol. 46, issue 4 (2014), p. 761-764 (arXiv:1304.3583)

### Łukasz Garncarek

My research interests revolve around geometric group theory and representation theory of groups. Roughly speaking, I investigate unitary representations arising from geometric group theory. For instance, in my PhD thesis I constructed a large family of irreducible unitary representations of an arbitrary Gromov hyperbolic group, using its action on the visual boundary. Among other things, I am interested in extending this construction into analogues of the principal series and complementary series, appearing in the representation theory of Lie groups.

### Tadeusz Januszkiewicz

I am interested in topology aspherical spaces, especially manifolds. Proving asphericity is often done by exhibiting a metric, or a similar combinatorial structure, with properties resembling nonpositive curvature. There are several abstract schemes and constructions in this direction, but there are also natural candidates for such approach arising in nature, such as iterated blowups or ramified covers.

### Piotr Nowak

I am interested mainly in geometric and analytic properties of discrete groups and their applications to other areas, such as higher index theory. The main topics include rigidity properties of groups, such as various strengthening of property (T), as well as the opposite properties viewed as versions of amenability. These notions often can be expressed in terms of (co)homological properties of groups or spaces and have a significant intersection with large scale geometry and topology of non-compact manifolds.

### Damian Osajda

I am working mostly in Geometric Group Theory. My main area of interest is studying complexes with some nonpositive curvature features, and groups acting on them. This includes objects that are: Gromov hyperbolic, small cancellation, CAT(0) cubical, (weakly) systolic, bucolic, weakly modular etc.

### Stanisław Spież

My major areas of research are shape theory (and indirectly homotopy theory), dimension theory, theory of embeddings and game theory.

Several of my papers in shape theory are devoted to studying movable spaces (their role is similar to that of CW-complexes in homotopy theory) and deformation dimension (which corresponds to the homotopical dimension). Some of my results in that area are related to the classical Whitehead and Hurewicz theorems in homotopy theory. Also I investigated the possibility of representing the strong shape category in the homotopy category.

Another area of my research is related to the following questions: "When can a pair of mappings of compact metric spaces X and Y into Rn be approximated by mappings with disjoint images, and also when can a map X→Rn be approximated by embeddings?'' Since the 1930's the standard answer to the latter has been "It suffices that 2 dim X < n" turns out that it is sufficient that dim(X×X) < n. Some other papers of mine concern the first question (which is more general).

Also I was interested in the questions of embedding polyhedra into Euclidean spaces, which were related to the van Kampen and Haefliger-Weber theorems. Recently I am also involved in research in game theory. Some results on the existence of equilibria in a class of games can be proved by using topological tools, such as coincidence theorems of Borsuk-Ulam type.

Several of the above results were obtained in collaboration with the following mathematicians: B. Günther, S. Nowak, J. Segal, R. Simon, A. Skopenkov and H. Toruńczyk.

### Henryk Toruńczyk

Major part of my research concerned topological properties of infinite-dimensional spaces, such as the Hilbert cube or Banach spaces. I consider the following my main results:

- developing a method of constructing smooth partitions of unity on Banach spaces in the absence of separability [20];
- proving that a product of an absolute retract with an appropriate normed linear space becomes homeomorphic to that space [21];
- proving, simultaneously with S. Ferry, that the homeomorphism group of a Hilbert cube manifold is a manifold [22];
- characterizing infinite-dimensional manifolds topologically (see R. D. Edwards' article in SLN 770, 278-302). As a consequence it turned out that infinite-dimensional Banach spaces of the same weight are homeomorphic;
- examining, jointly with S. Spież, when mappings X, Y→Rk can be ε-approximated by mappings with disjoint images [18];
- establishing, jointly with R. Simon and S. Spież, the existence of equilibria in a class of infinitely repeated games. (The proof in [19] depended on developing an appropriate topological aparatus.)

### Staff:

**Artem Dudko**/ prof. wizyt. / email

**Vasiliki Evdoridou**/ Assistant Professor / email

**Jonatan Gutman**/ Assistant Professor / email

**Lein Jin**/ / email

**Olena Karpel**/ Assistant Professor / email

**Michał Rams**/ Associate Professor / email

#### About the Department

The Laboratory exists since 2006. Before it was a part of the Laboratory of Functional Analysis. Its research staff includes: Feliks Przytycki (the head, permanent position), Michal Rams (long term), Pawel Walczak (part time), Krzysztof Frączek (part time), Carlos Cabrera (2006/2007), Neil Dobbs (2007/2008). The main fields of research run in the Laboratory are holomorphic dynamics in dimension 1, dynamical fractals and geometric dimensions, foliations and other topics in dynamical systems. It runs dynamical systems seminar and cooperates with Dynamical Systems Laboratory at Warsaw University. It is involved in the European FP6 Marie Curie programs ToK*Deterministic and Stochastic Dynamics, Fractals, Turbulence* (SPADE2) and RTN *Conformal Structures and Dynamics* (CODY).

#### 1. Iteration of holomorphic maps

**F. Przytycki**,*Conical limit sets and Poincaré exponent for iterations of rational functions*, Transactions of the AMS, 351.5 (1999), 2081-2099.**F. Przytycki, J. Rivera-Letelier, S. Smirnov**,*Equivalence and topological invariance of conditions for non-uniform hyperbolicity in iteration of rational maps*, Inventiones Mathematicae 151 (2003), 29-63.**F. Przytycki, J. Rivera-Letelier**,*Statistical properties of Topological Collet-Eckmann maps*. Annales Scientifiques de l'École Normale Superieure. 4^{e}série, t.40, (2007), 135-178.**N. Dobbs, B. Skorulski**,*Non-existence of absolutely continuous invariant probabilities for exponential maps*, Fund. Math. 198 (2008), 283-287

#### 2. Dynamics, complexity, ergodic theory

**K. Frączek, L. Polterovich**,*Growth and mixing*, to appear in Journal of Modern Dynamics.**E. Gutkin, M. Rams**,*Growth rates for geometric complexities and counting functions in polygonal billiards*, to appear.**F. Przytycki**, W. Marzantowicz,*Entropy Conjecture for continuous maps of nilmanifolds*, to appear in Israel Journal of Mathematics.

#### 3. Fractals, iterated function systems, thermodynamical formalism and geometric measure theory point of view

**M. Rams, Y.Peres, K.Simon, B.Solomyak**,*Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets*, Proc. of Amer. Math. Soc., 129 (2001), 2689-2699.**M. Rams**,*Packing dimension estimation for exceptional parameters*Israel J. of Math., 130 (2002), 125-144.**M. Rams**,*Measures of maximal dimension for linear horseshoes*, Real Anal. Exchange, 31 (2005/06), 55-62.

#### 4. Structures on manifolds, group actions, foliations

**B. Hajduk, R. Walczak**,*Symplectic forms invariant under free circle actions on 4-manifolds*, Trans. AMS 358 (2006), 1953-1970**P. Walczak**,*Dynamics of Foliations, Groups and Pseudogroups*, Monografie Matematyczne Vol.64 New Series, Birkhäuser, 2004.

#### About the Section

The main topics of investigations in recent 15 years were centered around theory of unbounded hyponormal operators, unbouded Toeplitz operators and (mostly) unbounded Jacobi matrices.

The main results are the following:

- Description of basic properties of unbounded hyponormal operators. In particular a characterization of hyponormal generators of $ C-0$ semigroups and generators of cosine functions.(Janas)
- Detailed analysis of various classes of unbounded Teoplitz operators in the Segal-Baragmann space. Various decompositions and spectral properties of these operators were found (Janas-Stochel).
- Precise analysis of asymptotic behaviour of some bases of solutions of the second oredr linear diffrence equations were proved (Janas-Naboko and Janas-Moszynski). Among the most interesting is formulation of the asymptotic formulas for the above bases in the case of double root situation. this was previously known practically only for rational coefficients(Janas).All these asymptotic formulkas found interesting applications in delicate spectral analysis of unbounded (or not) Jacobi operators.

The Section was founded in the 70-ties with Professor Włodzimierz Mlak as head. The main results obtained in the 70-ties and 80-ties are: a general result on partition of spectral sets (now known as the Lautzenheiser-Mlak theorem) (W. Mlak), an operator version of the von Neumann inequality (called also the Arveson-Mlak-Parrot inequality) (W. Mlak), a complete description of circular operators (W. Mlak), canonical decompositions of general operator valued functions (W. Szymański), a characterization of C^{*}-algebras generated by Toeplitz operators (J. Janas), models for subnormal operators with infinitely connected complement of the spectrum (K. Rudol).

In the 90-ties interesting results were found in the theory of unbounded Toeplitz operators in the Segal-Bargmann space (J. Janas) and in applications of bundle shift models for subnormal operators to boundary values of holomorphic functions (K. Rudol).

In recent years spectral analysis of Jacobi matrices is the main field of investigation. J. Janas (in collaboration with S. Naboko) found some new and interesting theorems in this area.

### Selected publications in the 90-ties

- J. Janas,
*An extension of Berezin's approximation method*, J. Operator Theory 29 (1993), 43-56. - J. Janas and J. Stochel,
*Unbounded Toeplitz operators in the Segal-Bargmann space II*, J. Funct. Anal. 126 (1994), 418-446. - J. Janas,
*An operator version of the Newman-Shapiro isometry theorem*, Integral Equations Operator Theory 26 (1996), 188-201. - K. Rudol,
*Spectra of subnormal Hardy type operators*, Ann. Polon. Math. 45 (1997), 213-222.

#### Research activities:

- Numerical analysis of spectral problems (Pokrzywa, Regińska)
- Numerical analysis of methods for solving partial differential equations (Wakulicz, Deriaz)
- Applications of wavelets to problems of numerical analysis (Deriaz, Pokrzywa, Regińska, Wakulicz)
- Discrete ill-posed problems (Regińska)
- Inverse problems for partial differential equation and regularization methods (Regińska, Wakulicz, Deriaz)

A seminar organized by the Laboratory covers a wide scope of numerics and attracts attention of mathematicians from other institutes.

The laboratory also organized at the Banach Center mini-schools, directed especially towards young researchers: Regularization methods for ill-posed problems of analysis and statistics – lectures of S. Pereverzyev, (15-25 May 2007); Course on inverse and ill-posed problems – lectures of Andreas Neubauer (26-29 March 2008).

#### Selected publications:

- S. Piszczatowski, K. Skalski, G. Sugocki and A. Wakulicz,
*Finite element method formulation for the interactions between various elastic-viscoelastic structures in biomechanical model*, in: Computer Methods in Biomechanics & Biomedical Engineering-2, J. Middleton, M. L. Jones and G. N. Pande (eds.), Gordon and Breach, (1998), 313-320. - L. Eldén F Berntsson, T. Regińska,
*Wavelet and Fourier methods for solving the sideways heat equation*, SIAM J. Sci. Comput. Vol 21, No.6, pp. 2187-2205, (2000) - T.Regińska, L.E ldén,
*Stability and convergence of wavelet-Galerkin method for sideways heat equation*, J. Inverse and Ill-Posed Problems, vol. 8, no.1, pp. 31-49 (2000) - T. Regińska,
*Application of wavelet shrinkage to solving sideways heat equation*, BIT vol.41, no 5, pp. 1101-1110 (2001) - T.Regińska,
*Regularization parameters choosing for discrete ill-posed problems*, in "Inverse Problems in Engineering Mechanics IV" (Proceedings of the International Symposium on ISIP2003) M.Tanaka (ed.), Elsevier 2003, pp. 457-464. - T. Regińska,
*Regularization of discrete ill-posed problems*, BIT Numerical Mathematics vol.44, pp. 119-133 (2004) - W. Grzesikiewicz, A. Wakulicz,
*Axiomatic formulation of thermodynamics ideal gas laws*, KONES Journal of Powertrian and Transport vol .13. no. 103-110 (2006) - T. Regińska, K. Regiński,
*Approximate solution of a Cauchy problem for the Helmholtz equation*, Inverse Problems 22, pp. 975-989 (2006) - Erwin Deriaz, Valérie Perrier
*"Direct Numerical Simulation of turbulence using divergence-free wavelets"*,Preprint IMPAN nr 684, June 2007, SIAM Multiscale Modeling & Simulation, in print - T. Regińska, A. Wakulicz,
*Wavelet moment method for Cauchy problem for the Helmholtz equation*, Journal of Comp. and Appl. Math, (2008), doi:10.1016/j.cam.2008.01.005. - W. Arendt, T. Regińska,
*An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domain*, Ulmer Seminare 2007, Journal of Inverse and Ill-Posed Problems, in print

### Staff:

**Samuel Evington**/ Graduate Assistant / email

**Andrzej Sitarz**/ Associate Professor / email

**Adam Skalski**/ Associate Professor / email

**Mateusz Wasilewski**/ Graduate Assistant / email

**Stanisław Lech Woronowicz**/ Professor / email

**Safoura Zadeh**/ Assistant Professor / email

#### About the Section

Noncommutative geometry entered the research programme of IMPAN in 1999. Five years later, with the help of the Warsaw University transfer-of-knowledge grant *Quantum Geometry*, this branch of IMPAN's mathematics gained an international dimension. Since 2004, there are about 10-20 visitors per year who contribute their research experience and give talks at the weekly *Noncommutative Geometry Seminar* held in the Institute. Among our invitees were Alain Connes and Maxim Kontsevitch, and the seminar talks are announced to about 200 mathematicians worldwide.

The aforementioned scientific activity helped to cristalize a local research team consisting of Piotr M. Hajac, Ulrich Krähmer, Tomasz Maszczyk and Bartosz Zieliński. Ulrich Krähmer was a Marie Curie fellow in the years 2005-2007. In January 2008, the Noncommutative Geometry Research Unit was formally created by the Institute Scientific Council. In October 2008, the team was enlarged by Emily Burgunder, who chose IMPAN for her European Postdoctoral Institute fellowship.

Another chapter of Noncommutative Geometry at IMPAN opened in 2009 with the EU- project *Geometry and Symmetry of Quantum Spaces*. Co-sponsored by the Polish Government, this 4-year international research staff exchange programme established a transcontinental network of 12 nodes with IMPAN as the co-ordinating node. In particular, we welcomed in our group Paul F. Baum who joined us as a Visiting Professor working at IMPAN a month each year. Our mathematical environment was further enriched by Adam Skalski who came as another Marie Curie Postdoctoral Fellow for the years 2010-2012. Together with Paweł Kasprzak, Andrzej Sitarz, and Piotr M. Sołtan employed on short-term position, our Research Unit got top expertise in topological quantum groups and spectral geometry. On the other hand, a Ph.D.-student Jan Rudnik started his collaboration with Baum and Hajac on computing the K-theory of triple-pullback C*-algebras.

The key words characterizing IMPAN's research in noncommutative geometry are: K-theory of operator algebras and free actions of compact quantum groups on unital C*-algebras, multi-pullback C*-algebras and free distributive lattices of ideals, index theory of Fredholm modules and spectral geometry of Dirac operators, locally compact quantum groups and universal (free) quantum groups, Hopf-cyclic homology with coefficients and Chern-Galois character, corings and monoidal categories. The assumed research strategy is to explore the feedback between solving concrete difficult problems and developing new mathematical structures. The proposed approach is to unite rather than separate different fields of mathematics by taking advantage of complemetary tools that they offer. To this end, a large scale and intensive international collaboration is currently sustained and planned for the future.

A more systematic and detailed description of the aforementioned research profile is as follows:

**1. K-theory of operator algebras.** Computing K-theory of C*-algebras of quantum projective spaces of Toeplitz-type. Noncommutative version of the Borsuk-Ulam for a family quantum spheres. Non-existence of Z/2-equivariant homomorphisms from a unital C*-algebra A into its unital suspension SA. Computing the K-theory of noncommutative Bieberbach manifolds. An example of a published paper in this area of research is: P. F. Baum, P. M. Hajac, R. Matthes, W. Szymański, *The K-theory of Heegaard-type quantum 3-spheres,* K-Theory (2005) 35:159-186.

**2. Hopf-cyclic homology and Chern-Galois character.** Applying new coefficients of Hopf-cyclic homology and cohomology. Studying relationships between De Rham cohomology with coefficients in flat vector bundles and cyclic homology with coefficients. Version of the local index formula of Connes-Moscovici for twisted cyclic cohomology. Extending cyclic (co)homology with coefficients from algebras to monoidal functors. Among pivotal papers for this area of research are: P. M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhäuser, *Hopf-cyclic homology and cohomology with coefficients,* C. R. Acad. Sci. Paris, Ser. I 338 (2004) 667-672; T. Brzeziński, P. M. Hajac, *The Chern-Galois character,* C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113-116.

**3. Spectral geometry and index theorem.** Proving an analogue of the 2-dimensional Gauss-Bonnet for spectral triples. Stability of spectral triples and regular spectral geometries. Constructing spectral triples on cross-products of C*-algebras and studying their topological properties. Studying contact structures by means of noncommutative geometry. An example of a published paper related to this area of research is: P. M. Hajac, R. Matthes, W. Szymański, *Noncommutative index theory of mirror quantum spheres,* C. R. Acad. Sci. Paris, Ser. I 343 (2006) 731-736.

**4. Quantum group actions.** Equivalence of principality of actions by compact quantum groups on unital C*-algebras and the Hopf-Galois condition for induced coactions of corresponding polynomial Hopf algebras. Applying non-contractibility of compact quantum groups to prove non-triviality of noncommutative principal bundles obtained by means of the join construction of compact quantum groups. Proving principality of piecewise principal actions. Constructing examples of quantum spaces without group structure and actions of non-compact locally compact quantum groups. Quotienting of locally compact quantum groups by their closed subgroups. Studying quantum symmetry groups of group C*-algebras, especially focused on their representation theory. In particular, classification of compact group actions on the C*-algebra of 2 by 2 matrices. Finding non-classical quantum permutations on two elements. Extending Hopf-Galois theory to monoidal categories. An example of a published paper in this area of research is: P. M. Sołtan: *Examples of non-compact quantum group actions,* J. Math. Anal. Appl. 372 (2010), 224-236.

**5. Quantum spaces, sets, and cohomology.** Noncommutative deformations of complex projective spaces glued from Toeplitz cubes and other multi-pullback constructions of algebras. Constructing of a category of quantum sets (or quantum algebraic sets) and extenting of the contravariant adjunction between the category of sets and the category of commutative algebras to noncommutative sets and associative algebras. Generalizing the notion of a discrete (or algebraic) group to the corresponding notion making sense for quantum sets (or quantum algebraic sets). Constructing a recursive algorithm for computing genus zero Gromov-Witten invariants of some Fano varieties, generalizing the formula of Kontsevich. An example of a published paper in this area is: P. M. Hajac, A. Kaygun, Bartosz Zieliński, *Quantum complex projective spaces from Toeplitz cubes,* J. Noncommutative Geometry, 6: 3 (2012) 603-621.