Mathematical modeling through topological surgery and applications

Antoniou, Stathis

dc.contributor.author	Antoniou, Stathis	en
dc.date.accessioned	2018-01-23T11:02:03Z
dc.date.available	2018-01-23T11:02:03Z
dc.date.issued	2018-01-23
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/46263
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.2819
dc.rights	Αναφορά Δημιουργού 3.0 Ελλάδα	*
dc.rights.uri	http://creativecommons.org/licenses/by/3.0/gr/	*
dc.subject	Topological surgery, mathematical modeling, topology, dynamical systems, black holes	en
dc.subject	Τοπολογική χειρουργική, μαθηματική μοντελοποίηση, τοπολογία, δυναμικά συστήματα, μελανές οπές	el
dc.title	Mathematical modeling through topological surgery and applications	en
dc.contributor.department	Mathematics	el
heal.type	doctoralThesis
heal.secondaryTitle	Μαθηματική Μοντελοποίηση μέσω Τοπολογικής Χειρουργικής και Εφαρμογές	el
heal.generalDescription	Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we enhance topological surgery with the observed forces and dynamics. We then generalize these low-dimensional cases to a model which extends the formal definition to a continuous process caused by local forces for an arbitrary dimension m. Next, for modeling phenomena which do not happen on arcs, respectively surfaces, but are 2-dimensional, respectively 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further present a dynamical system as a model for both natural phenomena exhibiting a `hole drilling' behavior and our enhanced notion of solid 2-dimensional 0-surgery. Moreover, we analyze the ambient space (which we consider as being the 3-sphere) in order to introduce the notion of embedded topological surgery in the 3-sphere. This notion is then used for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effects of the process lie beyond the initial manifold, such as the formation of tornadoes. Moreover, we present a visualization of the 4-dimensional process of 3-dimensional surgery by using the new notion of decompactified 2-dimensional surgery and rotations. Finally, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.	en
heal.classification	Mathematics	el
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2017-12-18
heal.abstract	Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we enhance topological surgery with the observed forces and dynamics. We then generalize these low-dimensional cases to a model which extends the formal definition to a continuous process caused by local forces for an arbitrary dimension m. Next, for modeling phenomena which do not happen on arcs, respectively surfaces, but are 2-dimensional, respectively 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further present a dynamical system as a model for both natural phenomena exhibiting a `hole drilling' behavior and our enhanced notion of solid 2-dimensional 0-surgery. Moreover, we analyze the ambient space (which we consider as being the 3-sphere) in order to introduce the notion of embedded topological surgery in the 3-sphere. This notion is then used for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effects of the process lie beyond the initial manifold, such as the formation of tornadoes. Moreover, we present a visualization of the 4-dimensional process of 3-dimensional surgery by using the new notion of decompactified 2-dimensional surgery and rotations. Finally, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.	en
heal.sponsor	I am grateful to my advisor Sofia Lambropoulou who inspired me to change my career to science. She has supported me throughout the whole duration of my thesis in many ways. Her guidance and enthusiasm pushed me towards higher goals, taught me how to properly write in a scientific way and opened up new research directions. I wish to express my gratitude to Louis H.Kauffman for his commitment to our meetings (real and virtual) which taught me how to tackle difficult problems and gave me the opportunity to expand not only my mathematical knowledge but also the way I am thinking. I would also like to thank Colin Adams, Cameron Gordon and Antonios Charalambopoulos for many insightful discussions and their support during my thesis. I am grateful to my parents Ioannis and Aggeliki and my aunt Lena who supported me in deciding to change my career to science; this decision which was undoubtedly the right one. Further, I would like to thank Danai, Orfeas, Rea and all my family and friends. All of them supported me, directly or indirectly. Finally, I am grateful for having the privilege of doing a funded Ph.D. during these unstable economic times for Greece. More precisely, I would like to thank the European Union (European Social Fund - ESF) and the Greek national funds for their funding through the Operational Program "Education and Lifelong Learning" and my teacher Sofia Lambropoulou who coordinated the Research Funding Program Thales. I would also like to thank the Papakyriakopoulos foundation and the Department of Mathematics for the Papakyriakopoulos scholarship.	en
heal.advisorName	Lambropoulou, Sofia	en
heal.committeeMemberName	Lambropoulou, Sofia	en
heal.committeeMemberName	Kauffman, Louis H.	el
heal.committeeMemberName	Charalambopoulos, Antonios	en
heal.committeeMemberName	Adams, Colin	en
heal.committeeMemberName	Apostolatos, Theocharis	en
heal.committeeMemberName	Gordon, Cameron	el
heal.committeeMemberName	Kodokostas, Dimitrios	el
heal.academicPublisher	Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών	el
heal.academicPublisherID	ntua
heal.numberOfPages	103
heal.fullTextAvailability	true