Algebraic Modeling of Topological and Computational Structures and Applications

This interdisciplinary book covers a wide range of subjects, from pure mathematics (knots, braids, homotopy theory, number theory) to more applied mathematics (cryptography, algebraic specification of algorithms, dynamical systems) and concrete applications (modeling of polymers and ionic liquids, video, music and medical imaging). The main mathematical focus throughout the book is on algebraic modeling with particular emphasis on braid groups.

The research methods include algebraic modeling using topological structures, such as knots, 3-manifolds, classical homotopy groups, and braid groups. The applications address the simulation of polymer chains and ionic liquids, as well as the modeling of natural phenomena via topological surgery. The treatment of computational structures, including finite fields and cryptography, focuses on the development of novel techniques. These techniques can be applied to the design of algebraic specifications for systems modeling and verification.

This book is the outcome of a workshop in connection with the research project Thales on Algebraic Modeling of Topological and Computational Structures and Applications, held at the National Technical University of Athens, Greece in July 2015. The reader will benefit from the innovative approaches to tackling difficult questions in topology, applications and interrelated research areas, which largely employ algebraic tools.

Contenu

PART ONE: Konstantinos Karvounis and Sofia Lambropoulou, Link invariants from the Yokonuma Hecke algebras.- Dimos Goundaroulis, A survey on Temperley-Lieb-type quotients from the Yokonuma-Hecke algebras.- Maria Chlouveraki, Representation theory of framisations of knot algebras.- Loic Poulain d'Andecy, Invariants for links from classical and affine Yokonuma- Hecke algebras.- Marcelo Flores, On the framization of the Hecke algebra of type B.- Bostjan Gabrovsek and Maciej Mroczkowski, Link diagrams in Seifert manifolds and applications to skein modules.- Ioannis Diamantis and Sofia Lambropoulou, "On the HOMFLYPT skein module of the lens spaces $L(p,1)$ via braids.- Dimitrios Kodokostas and Sofia Lambropoulou, Some Hecke-type algebras derived from the braid group with two fixed strands.- Valerij G. Bardakov, Braid groups in handlebodies and corresponding Hecke algebras.- Nondas E. Kechagias, Infinite loop spaces, Dyer-Lashof algebra, cohomology of the infinite symmetric group and modular invariants.- PART TWO: Kenneth Millett and Eleni Panagiotou, Linking in systems with one-dimensional periodic boundaries.- Neslihan Gügümcü and Louis H. Kauffman, On the height of knotoids.- Stephan Klaus, Fourier braids.- Niki Vergadou, Molecular simulation of ionic liquids: complex dynamics and structure.- Stathis Antoniou and Sofia Lambropoulou, Topological surgery in nature.- PART THREE: Jiannis A. Antoniadis and Aristides Kontogeorgis, Automorphisms of curves.- Nicola Angius, Maria Dimarogkona and Petros Stefaneas, Building and integrating semantic theories over Institutions.- Sergey V. Sudoplatof, Yiannis Kiouvrekis and Petros Stefaneas Generic constructions and generic limits.- Katerina Ksystra, Nikos Triantafyllou and Petros Stefaneas, On combining algebraic specifications with first-order logic via Athena.- Theodoros Mitsikas, Petros Stefaneas and Iakovos Ouranos, A rule-based approach for air traffic control in the vicinity of the airport.- arianthiBozapalidou Sequential Machines and Affine Musical Contours.- Antonios Kalampakas, Nikolaos Triantafyllou, Katerina Ksystra and Petros Stefaneas, A formal representation of video content with the picture hyperoperation,- Evi Karali, Novel approaches to medical information processing and analysis.