Objective
The aim of the project is to carry out researches in the domains of the Algebraic Topology, Low Dimensional Topology and Cohomology of Groups with emphasis on the connections between these areas. Each of these domains of Mathematics has its own problems and methods of solving them. Connections between these fields arise naturally. One of them is based on homology of braid groups and configuration spaces. 9So the work is supposed to be done in the following interconnected directions:
Homotopy Theory, homological invariants of topological and geometrical
objects:
-1.1. To study cobordism theories, connected with generalized braid
-groups.
-1.2. To develop connection homology and homology between sets and to
-give a generalized form of Thom isomorphism.
-1.3. To consider the structure of Lp-de Rham complex for Riemannian
-manifolds.
-1.4. To investigate Lp-cohomology of warped products and warped flat
-bundles, normal solvability of the operator of exterior differentiation
-in the complex of Lp-differential forms on Riemannian manifolds.
-1.5. To study the topology of manifolds admitting Riemannian metrics of
-positive sectional curvature.
-1.6. To develop the application of equivariant Euler classes to Borsuk -
-Ulam type theorems for fibre-wise preserving mappings of bundles with an
-action of compact group Lie.
Low Dimensional Topology, Theory of Configuration Spaces:
-2.1. To give a new simple proof of the immersion theorem of Hirsch which
-involves configuration spaces and leads to new obstructions to immersion
-and embedding.
-2.2. To investigate analogues of the Vassiliev algebra for braids in
-handlebodies. To study the properties of Vassiliev invariants for knots
-in handlebodies.
-2.3. To prove Hurwitz type theorems for curves in Euclidean space and
-its discrete variants.
-2.4. To study the cohomology groups of various configuration spaces
-(determined by an action of finite group) and to obtain its applications
-to Bourgin--Yang type theorems.
Cohomology of Groups, Geometrical Group Theory:
-3.1. To study the generalizations of braids from homological point of
-view.
-3.2. To investigate the problem of pairings for generalized braid
-groups.
-3.3. To generalize results of Bestvina and Brady to arbitrary Artin
-groups.
-3.4. To investigate the invariants of group action on the space of ends.
-To characterize the actions of groups of cohomological dimension one on
-the space of ends. To describe the structure of the set of homomorphisms
-between two free groups. To examine the question whether a hyperbolic
-group can act by isometries on a simply connected CAT(O) complex.
-3.5. To study actions of compact groups on topological and Menger
-manifolds.
-3.6. To get a qualitative description of the deformation space of some
-Kleinian groups.
-3.7. To investigate the properties of finitely generated Kleinian groups
-acting in the real hyperbolic space of constant negative curvature. All results will be published as papers in mathematical journals and as preprints.
Topic(s)
Data not availableCall for proposal
Data not availableFunding Scheme
Data not availableCoordinator
08193 Bellaterra
Spain