Author: Darby Alastair
Publisher: World Scientific
ISBN: 9813226587
Category : Mathematics
Languages : en
Pages : 448
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Book Description
This volume consists of introductory lectures on the topics in the new and rapidly developing area of toric homotopy theory, and its applications to the current research in configuration spaces and braids, as well as to more applicable mathematics such as fr-codes and robot motion planning. The book starts intertwining homotopy theoretical and combinatorial ideas within the remits of toric topology and illustrates an attempt to classify in a combinatorial way polytopes known as fullerenes, which are important objects in quantum physics, quantum chemistry and nanotechnology. Toric homotopy theory is then introduced as a further development of toric topology, which describes properties of Davis–Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. The book also displays the current research on configuration spaces, braids, the theory of limits over the category of presentations and the theory of fr-codes. As an application to robotics, the book surveys topological problems relevant to the motion planning problem of robotics and includes new results and constructions, which enrich the emerging area of topological robotics. The book is at research entry level addressing the core components in homotopy theory and their important applications in the sciences and thus suitable for advanced undergraduate and graduate students. Contents: Toric Homotopy Theory (Stephen Theriault)Fullerenes, Polytopes and Toric Topology (Victor M Buchstaber and Nikolay Yu Erokhovets)Around Braids (Vladimir Vershinin)Higher Limits, Homology Theories and fr-Codes (Sergei O Ivanov and Roman Mikhailov)Configuration Spaces and Robot Motion Planning Algorithms (Michael Farber)Cellular Stratified Spaces (Dai Tamaki) Readership: Advanced undergraduate and graduate students as well as researchers interested in homotopy theory and its applications in the sciences. Keywords: Toric Topology;Toric Homotopy;Configuration Space;Stratified Spaces;Braid Group;Fullerene;Polytope;Virtual Braid Group;Thompson Group;Robotics;Motion PlanningReview: Key Features: The first book in the area of toric homotopy theory consisting of introductory lectures on the topics and their applications to fr-codes and robot motion planning
Author: Alastair Darby
Publisher:
ISBN: 9789813226579
Category : Combinatorial topology
Languages : en
Pages : 435
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Book Description
"This volume consists of introductory lectures on the topics in the new and rapidly developing area of toric homotopy theory, and its applications to the current research in configuration spaces and braids, as well as to more applicable mathematics such as fr-codes and robot motion planning. The book starts intertwining homotopy theoretical and combinatorial ideas within the remits of toric topology and illustrates an attempt to classify in a combinatorial way polytopes known as fullerenes, which are important objects in quantum physics, quantum chemistry and nanotechnology. Toric homotopy theory is then introduced as a further development of toric topology, which describes properties of Davis-Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. The book also displays the current research on configuration spaces, braids, the theory of limits over the category of presentations and the theory of fr-codes. As an application to robotics, the book surveys topological problems relevant to the motion planning problem of robotics and includes new results and constructions, which enrich the emerging area of topological robotics. The book is at research entry level addressing the core components in homotopy theory and their important applications in the sciences and thus suitable for advanced undergraduate and graduate students."--Publisher's website.
Author: Victor M. Buchstaber
Publisher: American Mathematical Soc.
ISBN: 147042214X
Category : Mathematics
Languages : en
Pages : 534
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Book Description
This book is about toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a class of manifolds with torus actions defined in combinatorial terms. Construction of moment-angle manifolds relates to combinatorial geometry and algebraic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connections with classical and modern areas of symplectic, Lagrangian, and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and polyhedral products provides for a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate subject of homotopy theory. A new perspective on torus actions has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. This book includes many open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter this beautiful new area.
Author: Megumi Harada
Publisher: American Mathematical Soc.
ISBN: 0821844865
Category : Mathematics
Languages : en
Pages : 424
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Book Description
Toric topology is the study of algebraic, differential, symplectic-geometric, combinatorial, and homotopy-theoretic aspects of a particular class of torus actions whose quotients are highly structured. The combinatorial properties of this quotient and the equivariant topology of the original manifold interact in a rich variety of ways, thus illuminating subtle aspects of both the combinatorics and the equivariant topology. Many of the motivations and guiding principles of the fieldare provided by (though not limited to) the theory of toric varieties in algebraic geometry as well as that of symplectic toric manifolds in symplectic geometry.This volume is the proceedings of the International Conference on Toric Topology held in Osaka in May-June 2006. It contains about 25 research and survey articles written by conference speakers, covering many different aspects of, and approaches to, torus actions, such as those mentioned above. Some of the manuscripts are survey articles, intended to give a broad overview of an aspect of the subject; all manuscripts consciously aim to be accessible to a broad reading audience of students andresearchers interested in the interaction of the subjects involved. We hope that this volume serves as an enticing invitation to this emerging field.
Author: Hans J. Baues
Publisher: Walter de Gruyter
ISBN: 9783110124880
Category : Mathematics
Languages : en
Pages : 412
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Book Description
The bulk of the book is devoted to the algebraic theory of quadratic modules and their connections with 4-dimensional complexes, Pontrjagin squares, homotopy groups, the cohomology of categories, and algebraic K-theory. The first three chapters provide the background needed and may serve as an introduction to basic combinatorial homotopy theory. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Najib Idrissi
Publisher: Springer Nature
ISBN: 3031044282
Category : Mathematics
Languages : en
Pages : 201
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Book Description
This volume provides a unified and accessible account of recent developments regarding the real homotopy type of configuration spaces of manifolds. Configuration spaces consist of collections of pairwise distinct points in a given manifold, the study of which is a classical topic in algebraic topology. One of this theory’s most important questions concerns homotopy invariance: if a manifold can be continuously deformed into another one, then can the configuration spaces of the first manifold be continuously deformed into the configuration spaces of the second? This conjecture remains open for simply connected closed manifolds. Here, it is proved in characteristic zero (i.e. restricted to algebrotopological invariants with real coefficients), using ideas from the theory of operads. A generalization to manifolds with boundary is then considered. Based on the work of Campos, Ducoulombier, Lambrechts, Willwacher, and the author, the book covers a vast array of topics, including rational homotopy theory, compactifications, PA forms, propagators, Kontsevich integrals, and graph complexes, and will be of interest to a wide audience.
Author: Anthony Bahri
Publisher: Springer
ISBN: 9783031572036
Category : Mathematics
Languages : en
Pages : 0
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Book Description
This book explores toric topology, polyhedral products and related mathematics from a wide range of perspectives, collectively giving an overview of the potential of the areas while contributing original research to drive the subject forward in interesting new directions. Contributions to this volume were written in connection to the thematic program Toric Topology and Polyhedral Products held at the Fields Institute from January-June 2020. 16 original conributions were inspired or influenced by the program. Toric Topology arose as a subject in its own right about twenty-five years ago. It sits at the intersection of commutative algebra, topology, combinatorics, algebraic geometry, and symplectic and convex geometry. Polyhedral products are a functorial generalization of a construction that is at the centre of Toric Topology. They are of independent interest and unify several constructions that arise in a diverse range of areas, such as geometric group theory, homotopy theory, algebraic combinatorics and subspace arrangements.
Author: Anthony Bahri
Publisher: Springer Nature
ISBN: 3031572041
Category :
Languages : en
Pages : 325
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Book Description
Author: Anatoly M. Vershik
Publisher: American Mathematical Soc.
ISBN: 1470456648
Category : Education
Languages : en
Pages : 345
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Book Description
Vladimir Abramovich Rokhlin (8/23/1919–12/03/1984) was one of the leading Russian mathematicians of the second part of the twentieth century. His main achievements were in algebraic topology, real algebraic geometry, and ergodic theory. The volume contains the proceedings of the Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100, held from August 19–23, 2019, at The Euler International Mathematics Institute and the Steklov Institute of Mathematics, St. Petersburg, Russia. The articles deal with topology of manifolds, theory of cobordisms, knot theory, geometry of real algebraic manifolds and dynamical systems and related topics. The book also contains Rokhlin's biography supplemented with copies of actual very interesting documents.
Author: Weizhu Bao
Publisher: World Scientific
ISBN: 9811266069
Category : Mathematics
Languages : en
Pages : 361
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Book Description
The Institute for Mathematical Sciences at the National University of Singapore hosted a thematic program on Quantum and Kinetic Problems: Modeling, Analysis, Numerics and Applications from September 2019 to March 2020. As an important part of the program, tutorials and special lectures were given by leading experts in the fields for participating graduate students and junior researchers. This invaluable volume collects six expanded lecture notes with self-contained tutorials. The coverage includes mathematical models and numerical methods for multidimensional solitons in linear and nonlinear potentials; Bose-Einstein condensation (BEC) with dipole-dipole interaction, higher order interaction and spin-orbit coupling; classical and quantum turbulence; and molecular dynamics process based on the first-principle in quantum chemistry.This volume serves to inspire graduate students and researchers who will embark into original research work in these fields.
