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Mathematical modeling through topological surgery and applications

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Mathematical modeling through topological surgery and applications (EN)
Antoniou, Stathis (EN)
ntua (EL)
Kauffman, Louis H. (EL)
Gordon, Cameron (EL)
Kodokostas, Dimitrios (EL)
Lambropoulou, Sofia (EN)
Apostolatos, Theocharis (EN)
Charalambopoulos, Antonios (EN)
Adams, Colin (EN)
doctoralThesis
2018-01-23T11:02:03Z
2017-12-18
2018-01-23
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)
Τοπολογική χειρουργική, μαθηματική μοντελοποίηση, τοπολογία, δυναμικά συστήματα, μελανές οπές (EL)
Topological surgery, mathematical modeling, topology, dynamical systems, black holes (EN)
English
Mathematics (EL)
Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών (EL)
Αναφορά Δημιουργού 3.0 Ελλάδα
http://creativecommons.org/licenses/by/3.0/gr/
National Technical University of Athens
Digital Library of National Technical University of Athens | Dspace@NTUA
Κεντρική Βιβλιοθήκη Ε.Μ.Π.
Ιδρυματικό Αποθετήριο
Διδακτορικές Διατριβές



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