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.