MATHEMATICAL MODELING THROUGH TOPOLOGICAL SURGERY AND APPLICATIONS : doctoral thesis accepted by the national technical university of athens

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. In this book the authors observe that it also occurs in natural phenomena of all scales: 1-dimensional surgery happens during DNA recombination and when cosmic magnetic lines reconnect; 2-dimensional surgery happens during tornado formation and cell mitosis; and they conjecture that 3-dimensional surgery happens during the formation of black holes from cosmic strings, offering an explanation for the existence of a black hole’s singularity. Inspired by such phenomena, the authors present a new topological model that extends the formal definition to a continuous process caused by local forces. Lastly, they describe an intrinsic connection between topological surgery and a chaotic dynamical system exhibiting a “hole drilling” behavior. The authors’ model indicates where to look for the forces causing surgery and what deformations should be observed in the local submanifolds involved. These predictions are significant for the study of phenomena exhibiting surgery and they also open new research directions. This novel study enables readers to gain a better understanding of the topology and dynamics of various natural phenomena, as well as topological surgery itself and serves as a basis for many more insightful observations and new physical implications. Front Matter ....Pages i-xvii Introduction (Stathis Antoniou)....Pages 1-3 Useful Mathematical Notions (Stathis Antoniou)....Pages 5-14 The Formal Definition of Surgery (Stathis Antoniou)....Pages 15-19 Continuity (Stathis Antoniou)....Pages 21-23 Dynamics (Stathis Antoniou)....Pages 25-32 Solid Surgery (Stathis Antoniou)....Pages 33-40 A Dynamical System Modeling Solid 2-Dimensional 0-Surgery (Stathis Antoniou)....Pages 41-48 The Ambient Space \(S^{3}\) (Stathis Antoniou)....Pages 49-55 Embedded Surgery (Stathis Antoniou)....Pages 57-65 3-Dimensional Surgery (Stathis Antoniou)....Pages 67-77 Conclusions (Stathis Antoniou)....Pages 79-80 Back Matter ....Pages 81-85