Marián Fabian

Grants
- Non-linear analysis in Banach spaces (7AMB12FR003)
  from 01/01/2012 to 31/12/2013 investigator
- Programme type: MOBILITY - France Objectives: The goal of this project is to contribute to a better understanding of the following topics:
- Topological and geometrical properties of Banach spaces and operator algebras (P201/12/0290)
  from 01/01/2012 to 31/12/2016 main investigator
- We would like to study the structure of Banach spaces, spaces of continuous functions, C* algebras and their relationship. Main topics will include quantitative properties of Banach spaces, decompositions of Banach spaces to smaller subspaces, descriptive properties of the weak topology, James' boundaries of compact convex sets, Baire classes of strongly ane functions, noncommutative Choquet theory, weakly compact sets and spaces they generate, problems of Banach-Stone type for spaces of continuous functions, uniformly continuous functions and ane functions, various types of universal Banach spaces, existence of fixed points and approximate fixed points, noncommutative measure theory, structures on the set of all abelian subalgebras of a C* algebra and a Jordan algebra, representations of operator algebras using weights and completely positive maps, new types of orders on operators. We would like to pay special attention to mutual influence of particular structures and to subsequent connecting of dierent areas of functional analysis and clarifying their mutual relationships.
- Nonlinear functional analysis (P201/11/0345)
  from 01/01/2011 to 31/12/2015 investigator
- The subject of our proposal are abstract problems concerning nonlinear mappings between Banach spaces and their subsets. For easier orientation, it is convenient to divide the project into the following five interdependent areas. 1. General properties of uniform mappings, their reduction to Lipschitz mappings, and their metric properties. 2. Linearization properties of Lipschitz mappings, in particular the existence of derivatives. 3. Structural properties of participating spaces, linear theory. 4. Renormings of Banach spaces. 5. Applications to other areas of mathematics, such as fixed point theory and differential equations. Concrete examples of the proposed problems. Are the classical Banach function spaces linearly isomorphic to their uniformly homeomorphic images? Is the unit ball uniformly homeomorphic to the unit sphere? Are Lipschitz isomorphic separable Banach spaces linearly isomorphic? What are the complemented subspaces of the classical function Banach spaces? Do reflexive Banach spaces have a fixed point property for nonexpansive mappings?
- Variational inequalities in equilibrial optimization (MEB021024)
  from 01/01/2010 to 31/12/2011 main investigator
- Programme type: MOBILITY - France Objectives: Development of tools of variational analysis suitable for study of local properties of solutions of variational inequalities. Analysis of the stability and sensibility of multi-valued mappings which assign suitable sets of solutions to the data of problems or to perturbing parameters. Application of obtained results on selected equilibrial problems of continuum mechanics and of mathematical economy.
- Topological and geometrical structures in Banach spaces (IAA100190901)
  from 01/01/2009 to 31/12/2011 main investigator
- Stability of weak Asplund spaces and Gateaux differentiability spaces, Banach spaces with projectional skeleton, stability of Valdivia compacta, topological characterizations of some classes of Banach spaces, compact convex sets - duality of compact spaces and Banach spaces and topological classification, weak topology of weakly K-analytic Banach spaces, geometric properties of boundaries of compact convex sets, strongly affine functions and Baire classes of Banach spaces, isomorphic properties of L_1-preduals, descriptive properties of ranges of derivatives on Banach spaces, descriptive properties of norm and weak topologies on Banach spaces, binormality in Banach spaces, spaces of continuous functions, renormings and differentiability, weak-star uniformly Kadec-Klee norms, Lipschitz homeomorphisms, Markushevich bases, extensions of special Lipschitz mappings, retracts, absolutely minimal Lipschitz functions, Gurarij space, compacta with extra structure.
- Infinite dimensional analysis (201/07/0394)
  from 01/01/2007 to 31/12/2009 investigator
- We propose to work on problems in several areas of infinite dimensional analysis. Regarding the structure of Banach spaces, does every Banach space contain a monotone basic sequence? What is the optimal value of boundedness constant for an M-basis of a general Banach space. We conjecture the answer is 2. What are the structural consequences of a long Schauder basis. In connection with renorming theory, find an example of a LUR renormable space with M-basis, but no strong M-basis. Clarify the relationship between the separable complementation property and PRI. Clarify the relationship between Szlenk index and dentability index, for which there exists a nonconstructive proof of dependence. Try to generalize the Szlenk index technique in the context of topologies framented by a metric and beyond. In particular, as an application, does there exist a universal Corson compact space?
- Geometry of weakly Lindelöf determined spaces (IAA100190610)
  from 01/01/2006 to 31/12/2008 main investigator
- Renorming theorems in weakly Lindelöf determined Banach spaces for the so called uniformly Kadec-Klee smoothness and its variants. The use of Shlenk and dentability index for these renormings. Study of l_p generated Banach spaces for p>2. For a given Banach space with some geometrical property, finding a space with the same property and moreover having an unconditional basis, which is densely embedded in the original space. Investigation of the geometry of Banach spaces possessing an unconditional basis in the spirit of our recent results. Study of long Schauder bases.
- The structure of Banach spaces (IAA100190502)
  from 01/01/2005 to 31/12/2007 investigator
- The aim of the project is to contribute to the solution of some of the fundamental question regarding the structure of Banach spaces. In the contextof separable spaces it is especially the containment of copies of cO into quotients of polyhedral spaces. The question on the existence of smooth separating functions on asplund spaces. Problems of the density of smooth renormings and the existence of lipschitz retractions for certain C(K)spaces. The boundary problem, ie. generalization of Rainwaters theorem to the case of arbitrary boundary and dropping the sequentiality assumption. Structure of biorthogonal systems and Markushevich basas in nonseparable Banach spaces.