I algebraic geometry

Research Programme in relation to the state-of-the-art:

In 1998, Connes and Moscovici discovered a new type of cyclic cohomology. Now our aim is to develop general cyclic theory with coefficients, to discover its geometrical meaning, and to apply it in the thus far unreachable examples (Hajac, Maszczyk, Zielinski).

The other main strand of our research is related to topological quantum groups. These have been initiated by the pioneering work of Woronowicz in late 1980s, reached maturity around 2000, and are now the subject of intensive investigation. In our group, the main focus is put on the study of actions of quantum groups on C*-algebras (Baum, Hajac, Rudnik, Soltan) and deep connections to classical and quantum probability (Skalski). In particular, Soltan and Skalski intend to exploit their complementary research experience and expertise to construct and analyse non-compact quantum symmetry groups. On the purely algebraic side, the categorical understanding of Hopf algebras seems to unify Hopf-Galois theory, theory of invariants, faithfully flat descend of Grothendieck and algebraic geometry. This path of investigations towards an appropriate framework for noncommutative geometry is pursued by Maszczyk.
Major research achievements:

1. 
   The invention of stable anti-Yetter-Drinfeld modules as coefficients of Hopf-cyclic cohomology by Hajac, Khalkhali, Rangipour, and Sommerhaeuser. The article "Hopf-cyclic homology and cohomology with coefficients" opened up a new area of research and introduced the term "Hopf-cyclic" that is now used as standard. Its main theorem links the anti-Yetter-Drinfeld property with the existence of the cyclic operator.
   
2. 
   The discovery of a pairing between super-Lie-Rinehart homology with coefficients in traces and periodic cyclic homology, extending a previously known special case defined by means of an invariant trace (Connes-Moscovici) generalized sufficiently to express celebrated index pairing as an above homological pairing (On a pairing between super-Lie-Rinehart and periodic cyclic homology, Maszczyk, Comm. Math. Phys., 2006).
   

Current research:

1. 
   The Peter-Weyl-Galois theory of classical and quantum compact principal bundles (Baum, Hajac, Szymanski)
   
2. 
   The invention of the Toeplitz quantum projective spaces (Hajac, Kaygun and Zielinski)
   
3. 
   Noncommutative join construction by Dabrowski, Hadfield, and Hajac
   
4. 
   Constructions of new cyclic modules by Boehm and Hajac
   
5. 
   The investigation of certain probabilistic aspects of (locally) compact quantum groups (Skalski)
   
6. 
   The study of connections between quantum group theories and noncommutative topological entropy (Skalski)
   
7. 
   The analysis of various phenomena related to topological quantum group actions on classical and quantum spaces (Soltan, Skalski)
   
8. 
   The construction of an adjunction between noncommutative algebras and quantum sets (Maszczyk)
   
9. 
   Monoidal categorical duality between Galois theory and spectral theory (Maszczyk)
   

Existing and previous research collaborations:

• 
  P.M. Hajac: 13 collaborators
  
• 
  P.F. Baum: 23 collaborators
  
• 
  T. Maszczyk: 3 collaborators
  
• 
  A. Skalski: 8 collaborators
  
• 
  P.M. Soltan: 3 collaborators
  
• 
  B. Zielinski: 6 collaborators
  

VIII. AFFINE ALGEBRAIC GEOMETRY AND SYMPLECTIC GEOMETRY. CRYPTOLOGY.
Team leader: prof. Zbigniew Jelonek
Members:

• 
  prof. Stanisław Janeczko
  
• 
  dr Robert Dryło
  
• 
  2 Ph.D. students
  

Outside members of the team:

• 
  prof. Krzysztof Kurdyka (Savoie University, France)
  

Research mission and objectives:

To attain/maintain a top world level in investigations of affine varieties, morphism and automorphisms of these varieties and spaces related to them, and applications to symplectic geometry. In particular:

1. 
   Bifurcation points of a complex polynomial
   
2. 
   K-uniruled varieties and the set of non-properness of morphisms of such varieties
   
3. 
   The automorphism group of affine variety
   
4. 
   The Abhyankar-Moh property and exotic embeddings of affine varieties
   
5. 
   Algebraic vector bundles and the Cancellation Problem
   
6. 
   Polynomial symplectic geometry
   
7. 
   Dealing with (and finding new roads) in some problems in cryptology
   

Relation to the state-of-the-art:

Affine algebraic geometry was initiated by Abhyankar and Nagata in late 1970 and 1980' and then developed by many researchers both in Europe and US. In the Objectives a), d) and e) the team obtained important results.

In some aspects, e.g. for study of bifurcation points or Abhyankar-Moh Problem, results of Jelonek with co-authors [1, 3, 5, 7, 9] provide the best current results.

Objective f) has been little studied in the literature. The paper (24) of Jelonek and Janeczko seems to be the beginning of the story. Objective e) is based on an interesting observation of Jelonek [12] and is a good direction to solve Zariski Cancellation Problem in general.

Major research achievements:

Jelonek has published 50 papers which were cited 173 times by 94 authors. His major mathematical achievements are:

1. 
   Creation of a notation of the set of non-properness of a polynomial mapping [2, 4, 14]
   
2. 
   A breakthrough [1, 3] on algebraic analog of Whitney theorem on equivalent embeddings of manifolds into affine space
   
3. 
   The depth study, together with Krzysztof Kurdyka [5, 8], of the bifurcation points of polynomial mappings
   

Prizes:

• 
  2001- The award of the Polish Mathematical Society- The Sierpinski Prize
  
• 
  1991- The special award of the Polish Mathematical Society for young Polish mathematicians- The Kuratowski Prize.
  

Current research: Z. Jelonek launched studies in all the topics in the objectives section: a)-e).
Existing and previous research collaborations:

Prof. Jelonek collaborates with K. Kurdyka and Adam Parusinski (France), as well as J. Wlodarczyk from USA.

Future research plans are related to the main research objectives:

Continue existing projects and involving in new ones, as described in Section 2, in particular in cryptology:

1. 
   Arithmetic in finite fields and polynomial equations (normal basis, fast algorithms for arithmetical operations)
   
2. 
   Polynomial equations
   
3. 
   Elliptic and hyperelliptic curves
   
4. 
   Elements of number theory groups of ideals of orders
   

IX. DYNAMICAL SYSTEMS
Team leader: Prof. Feliks Przytycki
Members:

• 
  dr hab. Michal Rams
  
• 
  4 Ph.D students
  

On part time positions:

• 
  prof. Eugene Gutkin (Nicolaus Copernicus University, Torun)
  
• 
  prof. Maciej Wojtkowski (tentative from 2011, Olsztyn)
  
• 
  prof. Mariusz Lemanczyk (Nicolaus Copernicus University, Torun)
  

Outside members of the team:

• 
  dr Ludwik Jaksztas (Warsaw University of Technology)
  
• 
  prof. Janina Kotus (Warsaw University of Technology)
  
• 
  dr Boguslawa Karpinska (Warsaw University of Technology)
  
• 
  prof. Grzegorz Swiatek (Warsaw University of Technology)
  
• 
  prof. Anna Zdunik (University of Warsaw)
  
• 
  dr hab. Krzysztof Baranski (University of Warsaw)
  

Research objectives:

1. 
   To attain and maintain a top world level in investigations of non-uniformly hyperbolic (chaotic) dynamics in particular low-dimensional (real and complex) in application to celestial mechanics, differential geometry and population dynamics
   
2. 
   To develop the holomorphic iteration techniques in meromorphic functions and involve in conformal invariance of scale limits of physical processes (percolation, Ising model)
   
3. 
   Fractals vs. chaos. Multifractal spectra
   
4. 
   Tame (zero entropy) dynamics, billiards and ergodic theory of flow suspensions of interval exchange maps
   

Research programme in relation to the state-of-the-art:

Geometrical-analytic tools have been developed by leading scientists in low dimensional, interval and holomorphic dynamics and scaling limits: C. McMullen [McM], J.-Ch. Yoccoz, S. Smirnov (Fields medallists) [Smi], A. Avila [Avila].

In the objectives a) and c) the team has already a substantial successes (see e.g. the monography [PU] by F. Przytycki and M. Urbanski) and is close to world leading edge. In b) (scaling limits) and d), it needs a substantial transfer of knowledge to catch up and unlock its technical potential and enthusiasm of its young researchers.
Major research achievements:

F. PRZYTYCKI has published 58 papers, some of them in top mathematical journals, e.g. Annals of Math. (1), Inventiones Math. (4). He has MathSciNet Math. Rev. 574 citations by 321 authors, one of highest in mathematics in Poland. His Hirsch index is 14.

He coordinated FP5 Centre of Excellence INCO `IMPAN-BC', FP5 MC-Training Site BANACH and FP6 ToK SPADE2. He has been a coordinator of the Warsaw node of FP6 RTN: CODY (Conformal Structures and Dynamics) 2007-2010, and the chair of the scientific committee of the network. He has (co)pioneered in:

1. 
   Estimates of entropy discoveries, with Misiurewicz and Marzantowicz, e.g. [MP]
   
2. 
   Ergodic properties of linked twists and related mappings, e.g.
   
3. 
   Hyperbolic endomorphisms, fundamentals of the theory, e.g.
   
4. 
   Boundary behaviour of univalent functions in presence of dynamics. Discovery of the very singular vs absolutely continuous dichotomy. E.g. Przytycki, Urbanski, Zdunik [PUZ]
   
5. 
   Proof of the conjectured topological invariance of the Collet-Eckmann condition. Fundamentals of the non-uniform hyperbolicity theory for maps of interval and holomorphic maps, thermodynamical formalism, see e.g. [PRS] by Przytycki, Rivera-Letelier and Smirnov.
   

F. Przytycki is an elected ordinary member of the Warsaw Scientific Society. He has been scientific director of IMPAS for 8 years. Since 2010 he is its director.

M. RAMS has about 15 papers published in renowned international journals; mostly cited (MathSciNet: 26) is [PRSS] by Peres, Rams, Simon and Solomyak.

Rams has invented new geometric methods to estimate fractal dimensions of limit fractals for non-conformal IFS' and their projections. Remarkable is his series of recent papers on fractal percolations. With E. Gutkin he has been working on the complexity of polygonal (polyhedral) billiards, see e.g. [GR]. In 2000 he has obtained Polish Prime Minister Award for the outstanding PhD thesis.

WOJTKOWSKI AND GUTKIN worked for many years in the USA, Tucson and UCLA respectively. Wojtkowski was the ICM Beijing 2002 invited speaker in the Mathematical Physics section.

SWIATEK was the ICM Berlin 1998 invited speaker.
Current research:

Multifractal studies of limit sets in interval holomorphic and low-dimensional non-holomorphic dynamics, with the use of methods from statistical mechanics.

Existing and previous research collaboration:

Within the EU programmes the team has cooperated with the University of Warwick, Manchester, Universite d'Orleans, PUC Chile, and individual mathematicians from many other places.

Gutkin and Wojtkowski settled in Poland after about 30 year stays in the USA (UCLA and Tucson Arizona).

Przytycki spent 1 year at IHES France, 1 year at IMPA Brasil and 1 year at SUNY at Stony Brook and Yale Universities and 1/2 year in Warwick and Hebrew University of Jerusalem, which has resulted in many publications.

M. Rams spent 1 year in France, INRIA Roquencourt, that resulted e.g. in [LR1], [LR2].

An example of a collaboration (Poland, Germany-Brasil, Chile) is the recent preprint by Gelfert, Przytycki, Rams, Rivera-Letelier [GPRR] or the remarkable [PRSS] (Poland, USA, Israel, Hungary)

The collaboration with other teams at IMPAS was successfully attempted within the FP6 ToK SPADE2 "Deterministic and Stochastic Dynamics, Fractals, Turbulence", see http://www.impan.pl/~spade/

Future research plans:

Together with further development of the application of thermodynamics ideas, mathematical analysis and stochastic processes, there is dynamics to understand the small scale structures. We plan to involve in leading edge developments (using in particular Stochastic Loewner Evolution), in scaling limits, growth processes.

References

• 
  [Avila] A. Avila, Dynamics of renormalization operators, ICM 2010 Hyderabad, Plenary Lecture.
  
• 
  [PRS] 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.
  
• 
  [Smi] S. Smirnov, Discrete Complex Analysis and Probability, ICM 2010, Hyderabad, Plenary Lecture by Fields Medallist.
  

Tematyka zespołu Układów Dynamicznych, w której chcemy gości w celu ich wytrenowania:

1. 
   Multifractal spectra, fractal percolation. Scientist-in-charge: Michał Rams
   
2. 
   Holomorphic iteration techniques in entire and meromorphic functions. Scientist-in-charge: Feliks Przytycki, in cooperation with Janina Kotus (outside member of the team)
   
3. 
   Thermodynamical formalism in 1D-dynamics, real and complex. Scientist-in-charge: Feliks Przytycki
   
4. 
   Tame (zero entropy) dynamics, billiards and its complexity. Scientists-in-charge: Michał Rams and Eugene Gutkin.
   

Experience:

FELIKS PRZYTYCKI (full Professor) coordinated FP5 Centre of Excellence INCO `IMPAN-BC', FP5 MC-Training Site BANACH, FP6 ToK SPADE2, and the Warsaw node of FP6 RTN: CODY (Conformal Structures and Dynamics) 2007-2010 (he chaired the scientific committee of the network).

MICHAŁ RAMS (Ph.D 1999, habilitation 2008) is one of world leaders in multifractal analysis, deterministic and random.
The group hosted several Ph.D students and postdocs for 1/2-1 year terms, working in either of the above topics. The group runs Polish Ministry grant: Chaos, fractals and conformal dynamics.
Chętnie przyjmiemy stażystów w każdym z 4 ww. tematów, po 1 roku.
We cooperate mainly with:

• 
  University of Warsaw: Krzysztof Barański, Anna Zdunik,
  
• 
  Warsaw University of Technology: Janina Kotus, Grzegorz Świątek
  
• 
  University of Warwick, that coordinated the network CODY.
  

X. GEOMETRIC METHODS IN ODE'S AND CONTROL THEORY
Team Leader: prof. Bronislaw Jakubczyk
Members:

• 
  prof. Stanislaw Janeczko (also in team IX)
  
• 
  Pawel Nurowski (visiting associate professor)
  
• 
  dr hab. Piotr Mormul
  
• 
  prof. Witold Respondek
  
• 
  dr Wojciech Krynski
  
• 
  dr Gabriel Pietrzkowski
  
• 
  dr Marek Grochowski
  

Outside member of the team:

• 
  prof. Witold Respondek (INSA-Rouen)
  

Research mission and objectives:

Ordinary Differential Equations (ODEs) and Control Theory (CT) provide adequate language and tools for stating and solving many problems in Mechanics, Mathematcal Physics, Engineering and other applied fields. When nonlinear problems are considered, the use of differential-geometric methods is natural. Such methods play an increasing role in understanding and solving some of the basic problems in these fields.

We intend to combine and extend the existing results in geometric theories of both fields (ODEs and CT) and in Symplectic Geometry in order to create more effective tools for solving some of the problems appearing in applications. In particular, we want to extend the notion of curvature so that it can be used for the analysis of control systems and optimality of extremals.

Invariants non-holonomic distributions and of mechanical control systems are to be analysed and exploited. Intrinsic geometries hidden in the geometric structures defined by ODEs are hoped to be revealed.

Research Programme and future research plans in relation to the state-of-the-art:

Geometric Control Theory started in late 70s’ with the use of Lie bracket (R. Hermann, P. Brunovsky, H. Hermes, A. Krener, C. Lobry, H. Sussmann). Transitivity and accessibility criteria were found, the local conrollability problem was understood, existence and uniqueness of realizations criteria were given, functional expansions were introduced and used.

The most important for applications class of linear systems was geometrically characterised (Jakubczyk and Respondek 1980) (feedback linearizable systems). A challenging open problem is to characterize flat systems. In a different setting the problem was already stated by E. Cartan and D. Hilbert. New developments in understanding the geometry of non-integrable (non-holonomic) distributions, especially identification of the role of abnormal curves, give hopes that such criteria will be found. Collaboration of mathematicians from Poland and France would be highly desirable in this context.
Major research achievements:

1. 
   Characterization of feedback linearizable systems for continuous time (Jakubczyk and Respondek 1980), and discrete time systems (Jakubczyk 1986)
   
2. 
   Introducing a geometric language for analysis of nonlinear discrite-time systems (Jakubczyk and Sontag, SIAM J. Control and Optimiz. 1990). This language was later used in similar contexts by other authors
   
3. 
   Finding criteria for existence of nonlinear realisations (Jakubczyk, SIAM J. Control and Optimiz. 1980 and 1986). The approach was later used by J.P. Gauthier et al. for systems on Lie groups. Another approach using noncommutative formal power series was proposed by M. Fliess (Invent. Math. 1983) and was later completed by Jakubczyk (Ann. Polon. Math. 2000)
   
4. 
   Presenting effective criteria for quasi-homogeneity of singular varieties, concluding from them a version of relative Poincare lemma needed for solving a Moser homotopy equation for closed differential 2-forms (W. Domitrz, S. Janeczko and M. Zhitomirskii 2004)
   
5. 
   Finding basic symplectic invariants and symplectic classification of singular curves (Domitrz, Janeczko and Zhitomirskii, J. Reine Angew. Math. 2008). These results considerably extend earlier ideas and results of V.I. Arnold
   
6. 
   Discovering natural conformal geometries hidden in ODEs considered modulo contact or point transformations (Nurowski, J. Geometry and Physics 2005). Identifying a nonstandard irreducible SO (3) geometry in dimension five (Bobienski and Nurowski, J. Reine Angew. Math. 2007)
   
7. 
   A general solution to an optimal motion planning problem for 1-step generating distributions (Gauthier, Jakubczyk and Zakalyukin, SIAM J. Control and Optimiz. 2010). Identification of canonical, fast oscillating controls in the case of free nilpotent approximation
   
8. 
   Identifying invariants of distributions of corank 2 (Jakubczyk, Krynski and Pelletier, Ann. Inst. H. Poincare 2009, Krynski and Zelenko 2010).
   

Some current research:

1. 
   Identification of basic invariants of dynamic pairs, including dynamic pairs
   
2. 
   Understanding complete invariants of the motion planning problem for 1-step (and k-step) generating distributions (Krynski, Jakubczyk, Pietrzkowski)
   
3. 
   Finding fast oscillating, asymptotically optimal controls for the motion planning problem with distributions of non-free nilpotent approximation (Pietrzkowski, Jakubczyk).
   

Existing and previous research collaborations include:

B. Jakubczyk: W. Respondek (INSA-Rouen, France), E. Sontag (Rutgers University, USA), J.P. Gauthier (France), V.Zakalyukin (Steklov Institute, Russia), M. Zhitomirskii (Technion, Israel).

S. Janeczko: C. Ishikawa (Japan), M. Roberts (United Kingdom), V. Zakalyukin (Russia), M. Zhitomirskii (Technion, Israel).

P. Nurowski: A. Rod Gover (New Zeeland), G.A.J. Sparling (USA), J.F. Plebanski (Mexico), S.Frittelli and E.T. Newman (USA), C. Kozameh (Argentina).

P. Mormul: M. Zhitomirskii (Technion, Israel), M. Cheaito, W. Pasillas and W. Respondek (France).

W. Krynski: I. Zelenko (Texas A&M University, USA), A. Cap (Vienna, Austria).

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