Jiří Vanžura

Grants
- Algebraic methods in geometry and topology (201/08/0397)
  from 01/01/2008 to 31/12/2012 investigator
- The aim of the project is to bring together mathematicians working in diverse but closely related fields (algebra, topology, differential geometry), emphasizing the synthesis that takes place in contemporary mathematics. More concretely, we mean the following topics. (1) Applications of graph complexes to invariant differential operators, with particular attention paid to Riemann and symplectic geometry. (2) Investigation of Cartan connections and parabolic geometries. (3) Description of algebras of symmetries of differential operators and construction of operators of special types. (4) Study of questions related to classification of hypersurfaces in CR-geometry. (5) Construction of higher-dimensional analogs of the Dolbeaut complex as resolutions of the Dirac operator in several variables. (6) Appliacations of homotopy methods to formal solutions of differential relations. (7) Study of forms on low-dimensional manifolds and induced G-structures.
- Investigating manifolds with special structures from topolog (IAA100190701)
  from 01/01/2007 to 31/12/2010 investigator
- Manifolds with special structures are presently actively investigated by mathematicians and physicists. They are related to outstanding theoretical questions in geometry and appear as models in the string theory. Our aims are: 1. Finding necessary and sufficient conditions for the existence of a closed G2-structure (resp. a flat G2-structure among a given cohomology class) on a 7-manifold. 2. Finding global invariants of closed G2-structures. 3. Investigating the above problems for multi-symplectic forms in dimensions 6 and 8. 4. Finding necesary and sufficient conditions for the existence of symplectic or Kaehler submanifolds realizing a given homology class. 5. Develop techniques to deal with above problems in a general framework. Our main approach to this project has been guided by recent observations that it is not only possible to apply methods presently known. It must be here a bigger framework which unifies these problems and these methods.
- Algebraic methods in topology and geometry (201/05/2117)
  from 01/01/2005 to 31/12/2007 investigator
- The project aims at bringing together mathematicians working in diverse but closely related fields (algebra, topology, differential geometry), thus reflecting the synthesis which takes place in modern mathematics. More specifically, we propose: 1) Studytransfers of strongly homotopy Lie structures, with an attention paid to minimal models, properties of the moduli space of solutions of the Mauer-Cartan equation and deformation theory. 2) Investigate invariant differential operators for parabolic geometries, in particular in the case when fields correspond to representations with singular character. Apply the Lie theory to the geometry of manifolds with a given parabolic structure. Study local invariants of pseudo-convex CR manifolds. 3) Describegeometry and topology of orbits of 3-forms with respect to the action of the general linear group. Find necessary and sufficient conditions for the existence of 3-forms on low-dimensional manifolds.
- Eduard Čech Center for Algebra and Geometry (LC505)
  from 01/01/2005 to 31/12/2011 main investigator
- The Eduard Cech Center for Algebra and Geometry forms an institutional background for long term post-doc stays, stays of leading foreign experts, and direct collaboration of a large group of Czech and foreign institutes. The research will focus on interaction between algebraic and geometric approaches in Differential Geometry, Geometric Analysis, Category Theory, Algebraic Topology, Algebraic Number Theory, Coding Theory, and Proof Complexity. There will be international open calls for the post-doc positions and short time stays of small groups of leading experts will be organized. The activities of the center will be monitored and governed by the international Steering Board. Indicators The numbers of post-docs and their publications,The measurable outputs Public databases of publications,Critical conditions The interest in the opened positions.