I algebraic geometry

Last update: 06.02.2012 godz. 12:10
I. ALGEBRAIC GEOMETRY
Team leader: Prof. dr hab. Piotr Pragacz.
Members:

• 
  prof. dr hab. Slawomir Cynk
  
• 
  dr hab. Slawomir Rams (Jagiellonian University)
  
• 
  dr hab. Halszka Tutaj-Gasinska (Jagiellonian University)
  
• 
  dr Jaroslaw Buczynski
  
• 
  dr Grzegorz Kapustka
  
• 
  dr Karol Palka
  

PIOTR PRAGACZ did his PhD Thesis in 1981 and Habilitation in 1990. He is the head of Department of Algebra and Algebraic Geometry at IMPAN since 2000. His current mathematical interests overlap: algebraic geometry (and especially intersection theory), algebraic combinatorics (and especially symmetric functions), and global singularity theory (and especially characteristic classes of singular varieties and Thom polynomials).

His Habilitation was devoted to Schubert varieties and degeneracy loci of morphisms of vector bundles, the main topic being P-ideals of degeneracy loci.

Major research achievements: P. Pragacz has published about 70 papers, some of them in top mathematical journals, e.g., Journal Amer. Math. Soc. (1), J. Differential Geom. (2), Advances in Math. (3). He has MathSciNet Math. Rev. 415 citations by 250 authors. In 1998, he published with

William Fulton the book "Schubert varieties and degeneracy loci", Springer LNM 1689.

He has pioneered in P-ideals of degeneracy loci, and co-pioniered in:

- application of Schur functions to algebra: In particular he proved with Sergeev a fundamental formula in the theory of supersymmetric functions,

- (with Lascoux) proving the Ribbon Formula,

- (with Srinivas) classyfying manifolds with the diagonal property,

- (with Lascoux) recently applying the Schur functions to Thom polynomials,

- (with Weber) recently proving results on positivity of Thom polynomials.

He was editor of 3 books: 1. "Topics in cohomological studies of algebraic varieties", Birkhauser, 2005, 2. "Algebraic cycles, sheaves, shtukas and moduli", Birkhauser, 2007, 3. "Hoene Wronski: zycie, matematyka i filozofia", IM PAN, 2008.
He is also the editor of the forthcoming volume "Contributions to Algebraic Geometry" - to appear in the EMS Publishing House, 2012. His most recent research is about Thom polynomials of Legendrian singularities.

The most recent interest of Pragacz is about characteristic classes of singular varieties, and on the Debarre conjecture about the characterization of Jacobians among principally polarized abelian varieties, via the diagonal property.

In 2000 he invented at IM PAN the seminar IMPANGA, which became an international forum of algebraic geometry.

Piotr Pragacz plans a 1 month visit at MPI in Bonn to work on characteristic classes and abelian varieties. He plans a 1 month visit in RIMS at the University of Kyoto to work on

mixed Hodge modules with Professor Morihiko Saito. He plans a 1 month visit at IMS Chennai to work with Professor D.S. Nagaraj on abelian varieties. This will be a continuation of an earlier contact. He also plans a 1 month visit at the Unversity of Washington in Seatle to work with Dr Dave Anderson on geometric and combinatorial aspects of equivariant cohomology.

A 1 year postdoctoral position will be opened for applicants to work on the

subject of characteristic classes.

SLAWOMIR CYNK is a middle stage experienced researcher (19 year after PhD).

He adviced PhD theses of G. Kapustka and M. Kapustka, both on the subject of Calabi-Yau manifolds. His research interests in last years concentrated on various aspects of Calabi-Yau manifolds including: certain geometric constructions of Calabi-Yau manifolds (especially double octics and generalized Kummer construction in higher dimensions), arithmetic properties of Calabi-Yau threefolds (modularity, Calabi-Yau threefolds in positive characteristic non-liftable to characteristic zero).

Another area of interest of S. Cynk is study of invariants (especially the Euler and Hodge numbers, dimension of the deformation space) of resolutions of singular hypersurfaces. He has recent joint papers with the following foreign mathematicians: E. Freitag, K. Hulek, R. Salvati Manni, M. Schuett, and D. van Straten. His current research (in collaboration with van Straten) concentrates on geometric construction of Calabi-Yau equations. Calabi-Yau equation is a special type of a Picard-Fuchs equation originally associated to a variation of Hodge structure on a Calabi-Yau threefold. This project aims on a construction of a large class of Calabi-Yau threefolds (preferably desingularized fiber products of elliptic surfaces) with Picard number one, or with a geometrically significant rank 4 submotive of the middle cohomology for which the Picard Fuchs equation can be computed.

Another project (with S. Rams) deals with nodal hypersurfaces and complete intersections in projective spaces. This project includes study of non-factorial complete intersections with small number of nodes.

Cynk plans a 6-12 month visit to the University of Mainz to collaborate with Duco van Straten on the above project. A 1 year postdoctoral position will be opened for applicants to work on the subject of Calabi-Yau manifolds, geometric properties and applications.

SLAWOMIR RAMS obtained his Ph.D. From the Jagiellonian University in 1999, and his habilitation at Universitaet Erlangen-Nuernberg in 2006. He is author (or co-author) of about 20 papers and preprints. His recent research interest overlaps the following two topics:

1. The geometry of Calabi-Yau varieties with special emphasis on methods of computing their Hodge numbers. Recent interesting results on this area (defect-type formulae for resolutions of hypersurfaces satisfying Bott-type conditions with A-D-Esingularities) were obtained in collaboration with S. Cynk. The important question how to generalize such results to the case of higher codimension or higher singularities remains open, but certain results for hypersurfaces in weighted projective spaces have been recently obtained by R.N. Kloosterman and K. Hulek.

2. The geometry of (resolutions of) surfaces with emphasis on properties of hypersurfaces that carry many singularities (respectively fixed-degree rational curves). After several papers on low-degree hypersurfaces in three-dimensional projective space, the most recent preprint by S. Rams on

this subject with M. Schuett from Hannover contains the proof of the fact that Barth quintic surface has Picard number 41. The latter result is obtained using positive-characteristic techniques and properties of K3 surfaces.

Rams cooperated with W.P. Barth (Erlangen), M. Schuett (Hannover) andA. Sarti (Poitiers).

Rams plans a 6-month visit to the University of Mainz to cooperate with Prof. D. van Straten on the topics 1. and 2. He plans a 1-year visit to the University of Hannover to cooperate with Prof. M. Schuett on the 2nd topic. He plans a 3-month visit to the University of Poitiers to cooperate with Prof. A. Sarti on the 2nd topic. He plans a 2-month visit to the Humboldt University in Berlin to cooperate with Prof. R.N. Kloosterman on the 1st topic.

A 1 year postdoctoral position will be opened for applicants to work on the subject of Calabi-Yau manifolds and the geometry of resolutions of surfaces.

HALSZKA TUTAJ-GASINSKA is a researcher working on topics concerning local and global positivity of line bundles on algebraic manifolds. In particular, she works on some connections between the positivity of line bundles on algebraic surfaces in general points and symplectic packing of the surfaces. She has visited mathematical institutions (e.g. in Essen, Barcelona, Ghent) and colaborated with mathematicians there.

Tutaj-Gasinska plans to visit the University of Barcelona for two-three months to work with Joaquim Roe on Seshadri constants and related topics.

A 1 year postdoctoral position will be opened for applicants to work on the subject of Seshadri constants.

JAROSLAW BUCZYNSKI is a young researcher 4 years after receiving his PhD, after a period of his postdoctoral mobility within the Marie Sklodowska Curie program International outgoing fellowship. His research so far concentrates on three subjects: contact manifolds, toric geometry, and secant varieties. Thanks to his publications and numerous travels and research visits to universities and research institutes in Europe, US and Australia, and South Korea, he has become a worldwide recognized expert in these subjects. His plans are to continue the research in these three areas, as well as to approach problems in the geometry of Calabi-Yau manifolds and hyperkaehler manifolds. He is certainly an attractive collaborator to numerous researchers and has some experience in mentoring mathematicians on earlier stages of their career. The subject of secant varieties is a classical subject of algebraic geometry, which has recently become important and popular due to its applications to signal processing and statistics. We plan to focus on an interaction between scientists and engineers, which is important as the two societies speak different language and for instance some well know problems for engineers have been solved by mathematicians a decade earlier. So definitely there is a need for communication and improving the dictionaries between the two areas.

Jaroslaw Buczynski plans a 3 month visit to University of Oslo to collaborate with Professor Kristian Ranestad on the following two topics: secant varieties and their generalizations, and Calabi-Yau manifolds. This would be a continuation of a collaboration that commenced in 2011. Another 3 month visit is planned to University of Zurich to collaborate with Dr Michal Kapustka on the subject of hyperkeahler manifolds and contact manifolds.

A 1 year postdoctoral position will be opened for applicants to work on the subject of secant varieties and their interactions with applications.

GRZEGORZ KAPUSTKA is a young researcher 4 years after his PhD. His thesis concerned primitive contractions of Calabi-Yau manifolds where the main result is the counstruction of several examples of new Calabi-Yau threefolds with Picard group of rank one. His research is now concentrated on hypekahler manifolds. More precisely he works on the O'Grady program of classification of hyperkahler fourfolds, where he recently made a significant progress.

Kapustka after his numerous research visits in several Universities in Europe finds a collaboration with famous mathematicians in this area. He has also an experience in mentoring mathematicians on an earlier stage of their carrier (the advisor of two master students).

Kapustka plans a 2 month visit in the University of Roma to collaborate with Prof. Kieran O'Grady on his conjecture about hyperkeahler manifolds. He plans also a 2 month visit to the University Paris VI (or de Versailles Saint-Quentin-en-Yvelines) to collabotrate with Prof. Laurent

Gruson. This would be a continuation of a collaboration on a special case of the O'Grady conjecture.

A 1 year postdoctoral position will be opened for applicants to work on the subject of hyperkaehler manifolds.

KAROL PALKA is a young researcher, who obtained his Ph.D. from the University of Warsaw in 2009. In his thesis he studied the class of Q-homology planes, i.e. complex surfaces having rational homology of a plane. The main result was a major generalization of a theorem of Koras-Russell on contractible surfaces, which was a crucial step in the proof of the Linearization Conjecture for C* actions on C^3. Results from the thesis and his further research led to classification results for arbitrarily singular Q-homology planes. Palka collaborates with researchers in affine geometry in Japan, France and Canada. His research is now concentrated on the following three topics:

1. 
   Analysis of singularities and fundamental groups of Q-homology planes of general type.
   
2. 
   Study of homologically trivial complex threefolds, in particular the class of contractible threefolds, which is crucial for the Zariski Cancellation Conjecture.
   
3. 
   Classification of closed C* embeddings into the plane and extension of the methods to analyze cuspidal curves in P^2.
   

Palka plans a 6-month visit in RIMS, Kyoto, Japan to collaborate with Prof. Masayoshi Miyanishi and Prof. Takahashi Kishimoto (Saitama University) on contractible threefolds and methods of the log Minimal Model Program. He plans a 4-month visit in Dijon, France to collaborate with prof. Adrien Dubouloz on contractible threefolds. He also plans a 3-month visit in Montreal, Canada to continue his collaboration with prof. Peter Russell on plane C* embeddings and on the Coolidge-Nagata conjecture.

A 1 year postdoctoral position will be opened for applicants to work on the subject of complex affine geometry.

II. PARTIAL DIFFERENTIAL EQUATIONS
Team leader: prof. Wojciech Zajaczkowski
Members:

• 
  prof. Teresa Reginska
  
• 
  dr Joanna Renclawowicz
  
• 
  dr Tomasz Cieslak,
  
• 
  phd student Jan Burczak
  

Outside members:

From University of Warsaw:

• 
  dr hab. Piotr Gwiazda
  
• 
  prof. Piotr Mucha
  
• 
  dr Agnieszka Swierczewska – Gwiazda
  
• 
  prof. Grzegorz Lukaszewicz
  
• 
  prof. Piotr Rybka
  
• 
  prof. Dariusz Wrzosek
  
• 
  dr Bernard Nowakowski
  
• 
  
  

From Systems Research Institute:

• 
  prof. Irena Pawlow
  

From Warsaw University of Technology:

• 
  dr hab. Ewa Zadrzynska
  
• 
  dr Adam Kubica
  

Research objectives:

• 
  Analysis of weak and regular solutions to models arising from fluid mechanics, especially Navier-Stokes equations, magnetohydrodynamics, non-Newtonian fluids. Problems of existence of global solutions, regularity, optimal regularity, qualitative behavior, stability and asymptotics.
  
• 
  Analysis of models from the crystal growth theory and image processing. Looking for new approaches giving nice properties of solutions in order to have relations to computer simulations.
  
• 
  Free boundary problems, especially the dynamic contact angle problem. Existence, stability and asymptotic behavior.
  
• 
  Parabolic systems. Existence and regularity of solutions. Relation to non-Newtonian fluids.
  
• 
  Mathematical Biology. Studies of morphogen transport and chemotactic processes related to cancer growth simulations.
  
• 
  To develop numerical analysis for solving unstable differential problems which occur in many science and engineering applications and which can be classified as the inverse - or ill-posed ones. They are among the most complicated ones. Thermoviscoelasticity and Cahn-Hilliard equations. Qualitative theory and asymptotic analysis of solutions to equations of fluid mechanics, attractors.
  

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

• 
  Fluid mechanics delivers still plenty of challenges for mathematicians. The knowledge of stability of solutions is very poor. The regularity problem of weak solutions for the Navier-Stokes equations in the three-dimensional case is an open problem, one of the most important issues in mathematics ('Millenium problem')
  
• 
  The regularity theory for general parabolic systems, including non-diagonal ones, and non-Newtonian Navier-Stokes equations are still open problems.
  
• 
  The rudiments of theory for the free boundary problem are well investigated but there are still many open questions related to physically reasonable systems which cannot be solved using standard techniques.
  
• 
  The field of inverse and ill-posed problems has certainly been one of the fastest growing areas in applied mathematics.
  
• 
  Many important new results on the regularity of global solutions to thermoviscoelasticity and Cahn-Hilliard equations have been proved recently.
  
• 
  The studies on qualitative properties of solutions of hydrodynamics are core interest for mathematicians and there is substantial progress in this field.
  

Major research achievements:

W. M. ZAJACZKOWSKI has 136 papers in leading journals like: Comm. Math. Physics, SIAM Jornal of Math. Analysis, Comm. Pure and Applied Analysis, Math. Methods in Applied Sciences, MathSciNet citation: 247 by 108 authors. His main research directions are:

• 
  Parabolic and elliptic equations in the Sobolev weighted spaces
  
• 
  Global regularity of solutions to the Navier-Stokes equations in domains with slip boundary conditions
  
• 
  Free boundary problems for compressible NSE .
  

P. B. MUCHA has published 54 papers in top journals like: Comm.Math.Phys., J.Differ.Eqs., IMRN. MathSciNet Index citation: 119 by 57 authors. He is a head of the National PhD Programme in Mathematical Sciences 2009-2015. Born in 1973, he has become one of the youngest full professors in Poland. He pioneered in:

• 
  qualitative behavior of systems with slip boundary conditions
  
• 
  monodimensional results for the models from the crystal growth theory
  
• 
  weak solutions to compressible steady flows. Many rewards, fellowships and grants as Humboldt fellowship, the prize of the Minister for Science for the best young scientists.
  

T. REGINSKA is a head of a small laboratory of numerical analysis at IMPAN. She published 30 papers; mostly cited is joint with L.Elden and F. Berntsson Wavelet and Fourier methods for solving the sideway heat equation, SIAM J.Sci.Comput (2000)(/ISI Web of Sciences: 62 citations).

T. PIASECKI has 3 publications in journals like J.Diff.Equations, JMAA. J.Renclawowicz has published 18 papers. MathSciNet Index citation: 18 by 39 authors.

P. GWIAZDA has published 26 papers. MathSciNet Index citation: 55 by 29 authors He is the head of the International PhD Programme Mathematical Methods in Natural Sciences.

A. SWIERCZEWSKA-GWIAZDA has 12 publications. MathSciNet Index citation: 15 by 10 authors.

T. CIESLAK has 9 publications, MathSciNet Index: 36 by 24 authors. He specializes in mathematical theory of chemotaxis and asymptotic analysis of parabolic systems.

P. RYBKA has 39 publications, MathSciNet Index: 203 by 120. His main field of interest is viscoelasticity and crystal growth models.

G. LUKASZEWICZ has 52 publications, MathSciNet Index: 197 by 119 authors. He is renowned specialist in theory of asymptotic behavior of solutions of partial differential equations.

D. WRZOSEK has 32 publications, MathSciNet Index: 209 by 114 authors. He is working on mathematical biology, chemotaxis.

I. PAWLOW has 67 publications, MathSciNet Index: 216 by 107 authors. She is working in thermoviscoelasticity, Kelvin-Voigt model and Cahn-Hilliard equations.

E. ZADRZYNSKA has 26 publications, MathSciNet Index: 41 by 11 authors. Her research concerns free boundary problems for equations of fluid mechanics.

Current Research:

W. M. Zajaczkowski is working on Ladyzhenskaya-Prodi-Serrin-type conditons for axially symmetric solutions to NSE. He is also investigating the stability of solutions to compressible NSE.

P. B. Mucha is ivestigating the regularity of weak solutions to compressible NSE. He is also working on the problems of crystal growth.

T. Piasecki is investigating the existence of regular solutions to stationary compressible NSE.

J. Renclawowicz is investigating the existence of solutions for inflow-outflow problems for NSE.

P. Gwiazda and A. Swierczewska- Gwiazda are working on non-Newtonian fluids with

nonstandard rheology.

T. Reginska has studied an extension of the inverse problem theory to the case of unbounded operators on the basis of the model Cauchy problem for the Helmholtzequation.

B. Nowakowski is working on micropolar and magnetohydrodynamics equations.

J. Burczak is doing research on regularity of solutions to non-newtonian Stokes systems. E. Zadrzynska is studying the stability of special solutions to compressible Navier-Stokes equations.

I. Pawlow is investigating sixth-orded Cahn-Hilliard equations.
Existing collaboration:

Praha, Paris 12, Hannover, Kiel, Ulm, Essen, Freiburg, Chemnitz, RICAM Institute (Austria), Darmstadt, Sankt-Petersburg, Oxford, Suwon (Korea), Iowa City, Tokyo, Ferrara, Naples.

Future Research Plans:

• 
  Regularity of weak solutions to Navier-Stokes Equations (Ladyzhenskaya-Prodi-Serrin conditions). Existence of global regular solutions under some geometrical and analytical restrictions.
  
• 
  Stability of special solutions to incompressible and compressible NSE. Existence of stationary solutions. Problems of inflow-outflow.
  
• 
  Shape optimization for incompressible and compressible flows.
  
• 
  The free boundary problems for magnetohydrodynamics, which can be treated as a starting point for the pinch and the fusion problems.
  
• 
  Measure-valued, weak and regular solutions for non-newtonian fluids and general parabolic problems. Problems of the flow of non-Newtonian fluids with grom conditions given by an N-function (solutions in Orlicz spaces).
  
• 
  Analysis of the flow of polymers. Combining the equations for macroscopic quantities with microscopic structure of the fluid. Existence of solutions and singular limits to the system with implicit rheology.
  
• 
  Theoretical foundation and numerical methods of inverse problems for differential equations on bounded domains related to problems in current interdisciplinary science.
  

III. GEOMETRY AND TOPOLOGY WITH EMPHASIS ON GEOMETRIC GROUP THEORY
Team leader: prof. Tadeusz Januszkiewicz
Members:

• 
  dr Piotr Przytycki
  
• 
  1 Phd student
  

Outside members:

• 
  prof. Jan Dymara (Wroclaw University)
  
• 
  prof. Jacek Swiatkowski (Wroclaw University)
  

Research mission and objectives:

To attain and maintain a top world level in investigations of infinite groups by geometric methods, and spaces related to them.

• 
  Study of various notions of nonpositive curvature: CAT(0) spaces, Simplicial Non Positive Curvature and other forms of Combinatorial Nonpositive curvature.
  
• 
  Applications of nonpositive curvature tools to the study of special classes of groups: Coxeter groups, Artin groups, groups acting on buildings.
  
• 
  Topological study of various boundaries of groups and spaces: Gromov boundaries of hyperbolic groups, ending laminations spaces.
  
• 
  Introducing nonpositive curvature tools into the study of projective varieties: asphericity and hyperbolicity of ramified coverings.
  
• 
  Applications: metric graph theory; statistical mechanics and billiards, configuration spaces and topological robotics.
  

Relation to the state-of-the-art:

Ideas from negative and nonpositive curvature were introduced into group theory first by Margulis and Gromov in late 1970 and 1980' and then developed by many researchers both in Europe and US. The developments covered very diverse topics, but were packaged as "Geometric group theory".

In the Objectives 1 and 3 the team is on the leading edge of research worldwide. Januszkiewicz was invited speaker in Section Geometry at ICM 2010. Old paper of Januszkiewicz with Davis [HoP] and recent papers of Swiatkowski provide the richest source of examples. Objective 2 is a vast and diffuse field, and the "edge" difficult to define. In some aspects, for example L2-cohomology computations for special groups Dymara and Januszkiewicz with co-authors [DDJO] provide best current results. Objective 4 has been little studied in the literature. Perhaps the best results are due to Dimitri Panov, Kings' College London (former student of M. Kontsevich, IHES), with whom the team members are in close contact. Objective 5 would require significant transfer of knowledge to catch up. The team is in contact with V. Chepoi (Marseille, a leading expert on metric graph theory) and with M. Farber (Durham, a leading expert in topological robotics). E. Gutkin and M. Wojtkowski, attached to the dynamical systems team of this project, DS, are both experts on billiards, and we hope to be able to use their expertise and their contacts with other experts.
Major research achievements:

JANUSZKIEWICZ has published 36 papers which were cited 310 times by 243 authors. His Hirsh index is 11. He was the only Polish invited speaker at the recent International Congress of Mathematicians in Hyderabad in 2010. In 2009 he received the main prize of the Polish Mathematical Society – Banach Prize. His major mathematical research achievements are:

1. 
   Creation of a small but active field of "toric topology" with the paper [CO]
   
2. 
   Elaborating in [HoP] on hyperbolization procedure of Gromov, still the richest source of examples of CAT0 spaces.
   
3. 
   A breakthrough paper with Jan Dymara [CB] on cohomology computations of buildings. The sequel to this was the series of 5 papers starting with[DDJO] dealing with other cohomology computations of buildings and related spaces.
   
4. 
   Invention and in depth study, together with Jacek Swiatkowski, of Simplicial Nonpositive curvature [SNPC 1,2,3]. There are about 20 follow up papers listed on http://www.math.uni.wroc.pl/~elsner/snpc.html
   

Januszkiewicz collaborated for more than 20 years with Michael W. Davis from Ohio State University. Items 1, 2, 3 above are joint with Davis (and others).

P. PRZYTYCKI (PhD in 2008) has together 11 papers published or accepted for publication.

His achievements are:

1. 
   Study of Simplicial Nonpositive curvature (hyperbolicity, fixed-point theorem, boundaries-with Daman Osajda)
   
2. 
   Study of boundaries of Gromov-hyperbolic spaces: Coxeter groups with Jacek Swiatkowski, random groups with Dahmani and Guirardel, the curve complex with Sebastian Hensel
   
3. 
   Partial solution of the isomorphism problem for Coxeter groups with Pierre-Emmanuel Caprace
   

In 2009 he received Polish Mathematical Society award for young mathematicians and Prime Minister award for his PhD thesis.
Current research:

T. Januszkiewicz has following active projects in advanced stages:

1. 
   collaboration with Davis, Dymara, and Okun on weighted cohomology of Coxeter groups, trying to go beyond results of [DDJO].
   
2. 
   2. collaboration with Ian Leary (Ohio State U) aiming at constructions of new high dimensional Kazhdan groups. Techniques used are related to papers [Fix]
   
3. 
   collaboration with Swiatkowski aimed at the study of "mock buildings": a class of cubical CAT0 spaces generalizing buildings. It rests on the work of Davis Januszkiewicz and Scott and the work of Rick Scott on "mock reflection groups" He also has initial stages of projects related to items 1,2,4 of the Research Objectives above.
   

P. Przytycki continues his project with P.-E. Caprace to completely solve the isomorphism problem for Coxeter groups. He also collaborates with S.Hensel (Bonn), A. Nagorko (Warsaw), F. Gueritaud (Lille) in a project to understand the topology of ending lamination space.
Existing and previous research collaborations:

Januszkiewicz was not involved recently in collaborations within EU, outside Poland. He spent years 2002-2010 in USA, where he has past and current collaboration projects with Mike Davis and Ian Leary. He held 3 NSF grants there, jointly with Davis. Prior to 2003 Januszkiewicz was working at Wroclaw University, and has past and current close collaborations with Dymara and Swiatkowski. He was an investigator in several Polish grants in Wroclaw continuously since 1994.

He co-authored papers with among others G. Arzhantseva (Vienna), M. Bridson (Oxford), M. Davis (Ohio State), A. Dranishnikov (Florida State), I. Leary (Ohio State), W. Neumann (Columbia), R. Spatzier (Ann Arbor), S. Weinberger (U of Chicago).

Piotr Przytycki carried out in 2008-2010 few month post-doc stays in Bonn, Toulouse, Paris and Vienna. He co-authored among others with P.-E. Caprace (Louvain-la-Neuve), F. Dahmani (Toulouse), V. Guirardel (Toulouse), S. Hensel (Bonn).

Future research plans are related to the main research objectives:

1. 
   Continue existing projects, as described in Section 5.
   
2. 
   Develop further various forms of Combinatorial Nonpositive Curvature. Explore an strenghten links with metric graph theory.
   
3. 
   Apply CAT0 methods to study projective and quasiprojective varieties. Big challenge here is to prove the Asphericity Cojecture for Artin groups using ramified covers and CAT0 methods.
   

IV. RISK –MODELING AND CONTROL OF STOCHASTIC PHENOMENA - COMPREHENSIVE ANALYSIS OF RISK IN FINANCIAL MARKETS

Team leader: prof. Łukasz Stettner
Members:

• 
  prof. Jerzy Zabczyk
  
• 
  prof. Szymon Peszat
  
• 
  dr Mateusz Kwaśnicki
  
• 
  4 PhD students
  
  
  

Outside members of the team:
University of Warsaw:

• 
  prof. Andrzej Palczewski
  
• 
  prof. Jacek Jakubowski
  
• 
  dr Jan Palczewski
  

Warsaw University of Technology:

• 
  dr Mariusz Niewęgłowski
  

Warsaw School of Economics:

• 
  dr Łukasz Delong
  

Warsaw University of Life Sciences:

• 
  dr Marek Kociński
  

Research mission and objectives:

The objective of the research is to study various properties of stochastic processes, and the use of them in modeling of risk. The study on risk, its modeling and even control under risk is one of the most important challenges of quantitative methods in economy. The following aspects of quantitative methods in economy. The topic of the research is closely related with real applications. Industry, in particular financial and insurance industry is interested in special departments which are monitoring risk.

Risk is an inevitable part of all human activity. It appears almost in everywhere: in construction works, production, investments, finance and insurance and in our everyday life. The notion of “risk” carries connotations of chaos, the unexpected and undesired behaviour of an observed phenomenon. It is difficult to anticipate how a chaotic model will behave, since- as the time itself indicates- such a model does not have predictable dynamics. But while it is hard to say how a chaotic process will behave at any specific moment, things are quite different if we take a longer-term perspective, chiefly looking at the mean value of certain functions that hinge on the process. For example, the investors of a bank or insurance company are much more interested in the overall market situation than in individual transactions they might stand to lose on. The point is to be sure they “come out on top” in the appropriate long-term perspective, regardless of short-term fluctuations. Calculating risk requires a certain language (or mathematical model) to describe the observed phenomena. In this project we shall concentrate ourselves on the mathematical aspects of risk with potential applications to finance and insurance. We are also interested in proper pricing of financial derivatives which should take into account various aspects of risk.

To study risk it is important to investigate various properties of stochastic processes. The following specific aspects of stochastic processes in particular will be studied: solutions to stochastic differential and partial differential equations, possibly with Levy noise with applications to mathematics of finance and insurance. We will be interested in regularity (strong Feller property, existence of densities and existence (uniqueness) of invariant measures) of solutions to such equations.

Another objective is to study ergodic properties of various Markov processes: solutions to stochastic differential and partial differential equations, filtering processes and also to use these properties to control models with ergodic type criteria.

Further topic is a comprehensive analysis of  risk measures and financial markets under various kinds of risk. In particular an illiquid markets and markets with frictions.

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

It is generally known that modeling of asset prices and their derivatives requires to consider more complicated noise consisting of both continuous part (Brownian motion) and discontinuous (jump part). Consequently, models with Levy has been considered as basic in mathematics of finance independently of the fact that such models include main insurance risk models. Such extension requires a number of technical problems to overcome. In particular, such necessity arises in term structure modeling and credit risk modeling (which are recently of great interest (see papers published in recent issues of Financial Mathematics and Finance and Stochastics).

Stability and closely related uniqueness of invariant measure are fundamental aspects considered in the study of long run asymptotics of stochastic models. In particular, asymptotics of filtering processes is the one of fundamental problems of recent systems theory (see papers by Van Handel 2008 and 2010).

There has been a lot of activities in mathematics of finance devoted to modeling and pricing of derivatives on idealistic liquid market without friction (see Delbaen Schachermayer "Mathematics of Arbitrage", Springer 2006). To make our modeling feasible, we have to consider more complicated models which involve transaction costs and an influence on the market of large transactions. Risk modeling, construction of proper measures of risk one of the most important problems of modern mathematics of finance.

Major research achievements:

ŁUKASZ STETTNER wrote 89 research papers and two monographs in stochastic control (ergodic control problems, control of manufacturing systems, risk sensitive control, filtering and control with partial observation) stochastic processes (large deviations, ergodicity of Markov processes) and mathematics of finance (arbitrage theory, portfolio analysis, pricing of financial derivatives, risk theory). He supervised 11 PhD theses (2 defended in economics). Prof. Stettner is an editor of Applications Mathematicae and is on editorial board of 3 other international journals. He organized over 38 of international and national conferences. Prof. Stettner has 193 MathSciNet Math. Rev. citations. His main results concern:

• 
  uniqueness of invariant measures for filtering processes, (Stettner 1989, Di Masi - Stettner 2005, 2008)
  
• 
  solutions to partially Bellman equations corresponding to  partially observed control problems with average cost per unit time criterions (Stettner 1993)
  
• 
  risk sensitive control with ergodic criterion (Di Masi - Stettner 2000, 2007)
  
• 
  regularity of stopping problems (Stettner - Zabczyk 1981, Stettner 2010, 2011, J. Palczewski - Stettner 2010)
  
• 
  asymptotics of portfolio optimization (Stettner 2010, 2011)
  

JERZY ZABCZYK wrote 87 papers and nine monographs in probability theory (stochastic PDEs, probabilistic potential theory), control theory (nonlinear systems, infinite dimensional systems, stochastic control) and applications of mathematics (mathematical finance, engineering). He wrote two fundamental monographs: S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise, Cambridge University Press, 2007, XII + 419 pp. and G. Da Prato, J. Zabczyk Stochastic Equations In Infinite Dimensions, Cambridge University Press, 1992, XVIII + 454 pp.). He supervised 8 PhD theses. Prof. Zabczyk is on editiorial board of 6 international journals. He organized 8 international conferences. Prof. Zabczyk has 2021 MathSciNet Math. Rev. citations (including 1450 citations to prof. Zabczyk's monographs).
Prof. Zabczyk is a corresponding member of the Polish Academy of Sciences since 2002. His main results concern:

• 
  regularity of stochastic linear equations in Hilbert spaces (Da Prato - Kwapień - Zabczyk 1987)
  
• 
  nonlinear stochastic wave and heat equations (Peszat - Zabczyk 1997, 2000)
  
• 
  strong Feller property and regularity of invariant measures corresponding to solutions to stochastic evolution equations (Da Prato - Zabczyk 1995, Peszat - Zabczyk 1995, Da Prato - Elworthy - Zabczyk 1997)
  
• 
  filtering equations in infinite dimensions (Ahmed - Fuhrman - Zabczyk 1997)
  
• 
  HJM models with jumps (J. Jakubowski - Zabczyk 2007)
  

SZYMON PESZAT wrote 34 papers and two monographs (jointly with prof. J. Zabczyk) on stochastic evolution equations and in infinite dimensional stochastic analysis. Prof. Peszat has 335 MathSciNet Math. Rev. citations. His main results concern:

• 
  large deviation principle for stochastic evolution equations (Peszat 1993)
  
• 
  stochastic Navier Stokes equations (Brzezniak - Peszat 2000, Capiński - Peszat 1997 and 2000)
  
• 
  erodicity of Markov processes in Polish spaces (Komorowski - Peszat - Szarek 2010)
  
  
  

Current research:

Prof. Stettner is currently interested in properties of the penalty method for discontinuous stopping problems, existence of unique invariant measures for hidden Markov models, aspects of arbitrage for simple strategies on markets with friction, financial markets with delayed information.

Prof. Zabczyk current research focusses on strong Feller property and large deviations of stochastic evolutional equations with Levy noise and on the study on the Musiela model of term structure with Levy noise.

Prof. Peszat is currently interested in regularity and analytical properties of the solutions to stochastic evolution equations with Levy noise and stochastic partial differential equations with noise in boundary conditions.

Existing and previous research collaboration:

Prof. Stettner has current collaborations with Humboltd University in Berlin, Munich University of Technology, Kaiserslautern, University and University of Padova.

Prof. Zabczyk has current collaborations with Torino University, University of Leipzig, University of York and Humboltd University in Berlin.

Prof. Peszat has current collaborations with University of Nancy, University of York and Torino University.

Future research plans:

Future research plans are closely related to the study of the research objectives formulated in the point 2. In particular the team wants to study risk measures in financial various  markets; to find sufficient conditions for strong Feller property and existence of a density of the solution to stochastic evolution equation with Levy noise, characterize invariant measures for hidden Markov non controlled and controlled models, characterize illiquid markets and markets with friction for simple (feasible) investment strategies.

V. FUNCTIONAL ANALYSIS, PROBABILITY AND THEIR APPLICATIONS
Team leader: prof. P. Mankiewicz
Members:

• 
  T. Figiel
  
• 
  A. Kamont
  
• 
  P. Śniady
  
• 
  M. Wojciechowski
  
• 
  J. Zemanek
  
• 
  Yu Tomilov