b-lab Seminar by Alessandro Tamai on April 19
Mar 23, 2021
The seminar has re-started. We will meet on google meet (you will get the google meet address by email in the weekly digest). If you are interested in receiving this information please write to pau.mir.garcia@upc.edu
On April 19 at 16h (google meet) AlessandroTamai (Trieste) will talk about
Title: Singular solutions spaces of rolling balls problem
Abstract: " The "Rolling Balls Model", the model describing a pair of spheres of different ray rolling one on another without slipping or twisting, is a classical example of sub-Riemannian problem. The symmetries of the distribution associated with the system depend on the ratio of the rays and radically change when the ratio equals 3. Indeed, for this value of the ratio (and only for this value) it extends to the exceptional simple Lie group G2 which acts, still for this value of the ratio, also on the singular solutions related to the problem.
In this talk we show how it is possible to describe the spaces of such singular solutions in a geometric way, as a family of 5-dimensional manifolds depending on the ratio. For rational values of the ratio such manifolds have a structure of SO(2)-principal bundles which are not topologically distinguished by their homology, homotopy and de Rham cohomology groups. In addition, we show that for integer values of the ratio the configuration manifold of the problem is a branched covering of each of such manifolds and how the covering maps associated allow to relate them with another known family of topological spaces, the lens spaces.This talk is based on the research works developed in my master thesis at University of Trieste, under the supervision of the professor Agrachev from SISSA (Trieste).
Title: Singular solutions spaces of rolling balls problem
Abstract: " The "Rolling Balls Model", the model describing a pair of spheres of different ray rolling one on another without slipping or twisting, is a classical example of sub-Riemannian problem. The symmetries of the distribution associated with the system depend on the ratio of the rays and radically change when the ratio equals 3. Indeed, for this value of the ratio (and only for this value) it extends to the exceptional simple Lie group G2 which acts, still for this value of the ratio, also on the singular solutions related to the problem.
In this talk we show how it is possible to describe the spaces of such singular solutions in a geometric way, as a family of 5-dimensional manifolds depending on the ratio. For rational values of the ratio such manifolds have a structure of SO(2)-principal bundles which are not topologically distinguished by their homology, homotopy and de Rham cohomology groups. In addition, we show that for integer values of the ratio the configuration manifold of the problem is a branched covering of each of such manifolds and how the covering maps associated allow to relate them with another known family of topological spaces, the lens spaces.This talk is based on the research works developed in my master thesis at University of Trieste, under the supervision of the professor Agrachev from SISSA (Trieste).
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