The thesis is made up of three distinct parts. The first part deals with aspects related to relative wedge products and helicity, an integral invariant of ideal fluid flows. We provide a novel perspective of the helicity integrand in terms of the natural wedge product on a relative version of the de Rham complex, that can be associated to any smooth map. Next to the product structure on the relative de Rham complex, we also establish that smooth homotopies induce differential graded algebra isomorphisms between the relative de Rham complexes. The interpretation of helicity via relative de Rham theory not only allows to place more clearly helicity in the context of algebraic and differential topology, but we are also able to establish that the invariance of helicity under volume-preserving diffeomorphism can be understood in terms of the invariance under a special class of homotopies. Next to the considerations on helicity and exact vector fields, we also place phase functions and multi-valued potentials in the context of relative de Rham theory. The inspiration for this approach comes from Steenrod's interpretation of the Hopf invariant of a map in terms of cup products. We elaborate on the Hopf invariant via the relative de Rham complex and relate this to the work of Steenrod and Whitehead in an appendix. An extract of this will be prepared for submission soon. The second part of the thesis deals with aspects of geometric flocking dynamics. Flocking dynamics concerned with the emergent behaviour of a group of particles subject to dynamical models that produce particle velocity alignment, the most important such model being the Cucker-Smale model. Here we consider a generalization of this latter model from Euclidean case to complete Riemannian manifolds with non-vanishing radius of injectivity (Riemannian Cucker-Smale model). We also consider a further generalization of this later model which is defined on any Riemannian manifold (geometric Cucker-Smale model). These new models open up the possibility to study the interplay of flocking dynamics with geometry and topology. They further tackle what is called the flocking realization problem: given a manifold define a dynamical system whereby flocking behaviour can emerge. We can establish sufficient conditions for the emergence of flocking for both the geometric and Riemannian Cucker-Smale model. We derive the explicit form of the Riemannian Cucker-Smale model on the unit sphere in 3D and in other specific cases. For the hyperbolic plane, for example, we can establish the sufficient flocking conditions in some generality. For the unit circle case, we establish a relationship with the Kuramoto model and study the emergent dynamics. This part of the work has been done in collaboration with S.-Y. Ha and D. Kim in Seoul and an extract has now been submitted for publication. In the final part of this thesis, we construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincaré duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result. This part of the work originates from the author's Master thesis, after the submission of the thesis the work continued in collaboration with J.T. Essig, and it appeared in the Journal of Singularities (2019).
The thesis is made up of three distinct parts. The first part deals with aspects related to relative wedge products and helicity, an integral invariant of ideal fluid flows. We provide a novel perspective of the helicity integrand in terms of the natural wedge product on a relative version of the de Rham complex, that can be associated to any smooth map. Next to the product structure on the relative de Rham complex, we also establish that smooth homotopies induce differential graded algebra isomorphisms between the relative de Rham complexes. The interpretation of helicity via relative de Rham theory not only allows to place more clearly helicity in the context of algebraic and differential topology, but we are also able to establish that the invariance of helicity under volume-preserving diffeomorphism can be understood in terms of the invariance under a special class of homotopies. Next to the considerations on helicity and exact vector fields, we also place phase functions and multi-valued potentials in the context of relative de Rham theory. The inspiration for this approach comes from Steenrod's interpretation of the Hopf invariant of a map in terms of cup products. We elaborate on the Hopf invariant via the relative de Rham complex and relate this to the work of Steenrod and Whitehead in an appendix. An extract of this will be prepared for submission soon. The second part of the thesis deals with aspects of geometric flocking dynamics. Flocking dynamics concerned with the emergent behaviour of a group of particles subject to dynamical models that produce particle velocity alignment, the most important such model being the Cucker-Smale model. Here we consider a generalization of this latter model from Euclidean case to complete Riemannian manifolds with non-vanishing radius of injectivity (Riemannian Cucker-Smale model). We also consider a further generalization of this later model which is defined on any Riemannian manifold (geometric Cucker-Smale model). These new models open up the possibility to study the interplay of flocking dynamics with geometry and topology. They further tackle what is called the flocking realization problem: given a manifold define a dynamical system whereby flocking behaviour can emerge. We can establish sufficient conditions for the emergence of flocking for both the geometric and Riemannian Cucker-Smale model. We derive the explicit form of the Riemannian Cucker-Smale model on the unit sphere in 3D and in other specific cases. For the hyperbolic plane, for example, we can establish the sufficient flocking conditions in some generality. For the unit circle case, we establish a relationship with the Kuramoto model and study the emergent dynamics. This part of the work has been done in collaboration with S.-Y. Ha and D. Kim in Seoul and an extract has now been submitted for publication. In the final part of this thesis, we construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincaré duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result. This part of the work originates from the author's Master thesis, after the submission of the thesis the work continued in collaboration with J.T. Essig, and it appeared in the Journal of Singularities (2019).
Topics on Helicity, Geometric Flocking Dynamics and Intersection Space Cohomology
SCHLÖDER, FRANZ WILHELM
2020-06-29
Abstract
The thesis is made up of three distinct parts. The first part deals with aspects related to relative wedge products and helicity, an integral invariant of ideal fluid flows. We provide a novel perspective of the helicity integrand in terms of the natural wedge product on a relative version of the de Rham complex, that can be associated to any smooth map. Next to the product structure on the relative de Rham complex, we also establish that smooth homotopies induce differential graded algebra isomorphisms between the relative de Rham complexes. The interpretation of helicity via relative de Rham theory not only allows to place more clearly helicity in the context of algebraic and differential topology, but we are also able to establish that the invariance of helicity under volume-preserving diffeomorphism can be understood in terms of the invariance under a special class of homotopies. Next to the considerations on helicity and exact vector fields, we also place phase functions and multi-valued potentials in the context of relative de Rham theory. The inspiration for this approach comes from Steenrod's interpretation of the Hopf invariant of a map in terms of cup products. We elaborate on the Hopf invariant via the relative de Rham complex and relate this to the work of Steenrod and Whitehead in an appendix. An extract of this will be prepared for submission soon. The second part of the thesis deals with aspects of geometric flocking dynamics. Flocking dynamics concerned with the emergent behaviour of a group of particles subject to dynamical models that produce particle velocity alignment, the most important such model being the Cucker-Smale model. Here we consider a generalization of this latter model from Euclidean case to complete Riemannian manifolds with non-vanishing radius of injectivity (Riemannian Cucker-Smale model). We also consider a further generalization of this later model which is defined on any Riemannian manifold (geometric Cucker-Smale model). These new models open up the possibility to study the interplay of flocking dynamics with geometry and topology. They further tackle what is called the flocking realization problem: given a manifold define a dynamical system whereby flocking behaviour can emerge. We can establish sufficient conditions for the emergence of flocking for both the geometric and Riemannian Cucker-Smale model. We derive the explicit form of the Riemannian Cucker-Smale model on the unit sphere in 3D and in other specific cases. For the hyperbolic plane, for example, we can establish the sufficient flocking conditions in some generality. For the unit circle case, we establish a relationship with the Kuramoto model and study the emergent dynamics. This part of the work has been done in collaboration with S.-Y. Ha and D. Kim in Seoul and an extract has now been submitted for publication. In the final part of this thesis, we construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincaré duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result. This part of the work originates from the author's Master thesis, after the submission of the thesis the work continued in collaboration with J.T. Essig, and it appeared in the Journal of Singularities (2019).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
