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| Author | : Pascal Lambrechts |
| Publisher | : |
| Total Pages | : 130 |
| Release | : 2014-09-11 |
| Genre | : MATHEMATICS |
| ISBN | : 9781470416690 |
Download Formality of the Little N-Disks Operad Book in PDF, ePub and Kindle
The little $N$-disks operad $\mathcal B$ along with its variants is an important tool in homotopy theory. It is defined in terms of configurations of disjoint $N$-dimensional disks inside the standard unit disk in $\mathbb{R} DEGREESN$ and it was initially conceived for detecting and understanding $N$-fold loop spaces. Its many uses now stretch across a variety of disciplines including topology algebra and mathematical physics. In this paper the authors develop the details of Kontsevich's proof of the formality of little $N$-disks operad over the field of real numbers. More precisely one can consider the singular chains $\operatorname{C}_*(\mathcal B; \mathbb{R})$ on $\mathcal B$ as well as the singular homology $\operatorname{H}_*(\mathcal B; \mathbb{R})$ of $\mathcal B$. These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras.The authors additionally prove a relative version of the formality for the inclusion of the little $m$-disks operad in the little $N$-disks operad when $N\ge
| Author | : Pascal Lambrechts |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 130 |
| Release | : 2014-06-05 |
| Genre | : Mathematics |
| ISBN | : 0821892126 |
Download Formality of the Little $N$-disks Operad Book in PDF, ePub and Kindle
The little -disks operad, , along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint -dimensional disks inside the standard unit disk in and it was initially conceived for detecting and understanding -fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics. In this paper, the authors develop the details of Kontsevich's proof of the formality of little -disks operad over the field of real numbers. More precisely, one can consider the singular chains on as well as the singular homology of . These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. The authors additionally prove a relative version of the formality for the inclusion of the little -disks operad in the little -disks operad when .
| Author | : Jan de Gier |
| Publisher | : Springer |
| Total Pages | : 667 |
| Release | : 2018-04-10 |
| Genre | : Mathematics |
| ISBN | : 3319722999 |
Download 2016 MATRIX Annals Book in PDF, ePub and Kindle
MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: - Higher Structures in Geometry and Physics - Winter of Disconnectedness - Approximation and Optimisation - Refining C*-Algebraic Invariants for Dynamics using KK-theory - Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.
| Author | : Sirakov Boyan |
| Publisher | : World Scientific |
| Total Pages | : 5396 |
| Release | : 2019-02-27 |
| Genre | : Mathematics |
| ISBN | : 9813272899 |
Download Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) Book in PDF, ePub and Kindle
The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.
| Author | : Yves Félix |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 246 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821849298 |
Download Homotopy Theory of Function Spaces and Related Topics Book in PDF, ePub and Kindle
This volume contains the proceedings of the Workshop on Homotopy Theory of Function Spaces and Related Topics, which was held at the Mathematisches Forschungsinstitut Oberwolfach, in Germany, from April 5-11, 2009. This volume contains fourteen original research articles covering a broad range of topics that include: localization and rational homotopy theory, evaluation subgroups, free loop spaces, Whitehead products, spaces of algebraic maps, gauge groups, loop groups, operads, and string topology. In addition to reporting on various topics in the area, this volume is supposed to facilitate the exchange of ideas within Homotopy Theory of Function Spaces, and promote cross-fertilization between Homotopy Theory of Function Spaces and other areas. With these latter aims in mind, this volume includes a survey article which, with its extensive bibliography, should help bring researchers and graduate students up to speed on activity in this field as well as a problems list, which is an expanded and edited version of problems discussed in sessions held at the conference. The problems list is intended to suggest directions for future work.
| Author | : Haynes Miller |
| Publisher | : CRC Press |
| Total Pages | : 1043 |
| Release | : 2020-01-23 |
| Genre | : Mathematics |
| ISBN | : 1351251600 |
Download Handbook of Homotopy Theory Book in PDF, ePub and Kindle
The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.
| Author | : Najib Idrissi |
| Publisher | : Springer Nature |
| Total Pages | : 201 |
| Release | : 2022-06-11 |
| Genre | : Mathematics |
| ISBN | : 3031044282 |
Download Real Homotopy of Configuration Spaces Book in PDF, ePub and Kindle
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 | : Benoit Fresse |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 743 |
| Release | : 2017-05-22 |
| Genre | : Mathematics |
| ISBN | : 1470434822 |
Download Homotopy of Operads and Grothendieck-Teichmuller Groups Book in PDF, ePub and Kindle
The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
| Author | : Vladimir Dotsenko |
| Publisher | : Cambridge University Press |
| Total Pages | : 187 |
| Release | : 2023-08-31 |
| Genre | : Mathematics |
| ISBN | : 1108965644 |
Download Maurer–Cartan Methods in Deformation Theory Book in PDF, ePub and Kindle
Covering an exceptional range of topics, this text provides a unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.
| Author | : Brian A. Munson |
| Publisher | : Cambridge University Press |
| Total Pages | : 649 |
| Release | : 2015-10-06 |
| Genre | : Mathematics |
| ISBN | : 1316351939 |
Download Cubical Homotopy Theory Book in PDF, ePub and Kindle
Graduate students and researchers alike will benefit from this treatment of classical and modern topics in homotopy theory of topological spaces with an emphasis on cubical diagrams. The book contains 300 examples and provides detailed explanations of many fundamental results. Part I focuses on foundational material on homotopy theory, viewed through the lens of cubical diagrams: fibrations and cofibrations, homotopy pullbacks and pushouts, and the Blakers–Massey Theorem. Part II includes a brief example-driven introduction to categories, limits and colimits, an accessible account of homotopy limits and colimits of diagrams of spaces, and a treatment of cosimplicial spaces. The book finishes with applications to some exciting new topics that use cubical diagrams: an overview of two versions of calculus of functors and an account of recent developments in the study of the topology of spaces of knots.
