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Formality Of The Little N Disks Operad
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Book Synopsis Formality of the Little $N$-disks Operad by : Pascal Lambrechts
Download or read book Formality of the Little $N$-disks Operad written by Pascal Lambrechts and published by American Mathematical Soc.. This book was released on 2014-06-05 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 .
Download or read book 2016 MATRIX Annals written by Jan de Gier and published by Springer. This book was released on 2018-04-10 with total page 667 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) by : Boyan Sirakov
Download or read book Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) written by Boyan Sirakov and published by World Scientific. This book was released on 2019-02-27 with total page 5393 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Homotopy Theory of Function Spaces and Related Topics by : Yves Félix
Download or read book Homotopy Theory of Function Spaces and Related Topics written by Yves Félix and published by American Mathematical Soc.. This book was released on 2010 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Handbook of Homotopy Theory by : Haynes Miller
Download or read book Handbook of Homotopy Theory written by Haynes Miller and published by CRC Press. This book was released on 2020-01-23 with total page 982 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Formality of the Little N-Disks Operad by : Pascal Lambrechts
Download or read book Formality of the Little N-Disks Operad written by Pascal Lambrechts and published by . This book was released on 2014-09-11 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Real Homotopy of Configuration Spaces by : Najib Idrissi
Download or read book Real Homotopy of Configuration Spaces written by Najib Idrissi and published by Springer Nature. This book was released on 2022-06-11 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Homotopy of Operads and Grothendieck-Teichmuller Groups by : Benoit Fresse
Download or read book Homotopy of Operads and Grothendieck-Teichmuller Groups written by Benoit Fresse and published by American Mathematical Soc.. This book was released on 2017-05-22 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Maurer–Cartan Methods in Deformation Theory by : Vladimir Dotsenko
Download or read book Maurer–Cartan Methods in Deformation Theory written by Vladimir Dotsenko and published by Cambridge University Press. This book was released on 2023-08-31 with total page 187 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Cubical Homotopy Theory by : Brian A. Munson
Download or read book Cubical Homotopy Theory written by Brian A. Munson and published by Cambridge University Press. This book was released on 2015-10-06 with total page 649 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Special Values of Automorphic Cohomology Classes by : Mark Green
Download or read book Special Values of Automorphic Cohomology Classes written by Mark Green and published by American Mathematical Soc.. This book was released on 2014-08-12 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study the complex geometry and coherent cohomology of nonclassical Mumford-Tate domains and their quotients by discrete groups. Their focus throughout is on the domains which occur as open -orbits in the flag varieties for and , regarded as classifying spaces for Hodge structures of weight three. In the context provided by these basic examples, the authors formulate and illustrate the general method by which correspondence spaces give rise to Penrose transforms between the cohomologies of distinct such orbits with coefficients in homogeneous line bundles.
Book Synopsis Transfer of Siegel Cusp Forms of Degree 2 by : Ameya Pitale
Download or read book Transfer of Siegel Cusp Forms of Degree 2 written by Ameya Pitale and published by American Mathematical Soc.. This book was released on 2014-09-29 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let be the automorphic representation of generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and be an arbitrary cuspidal, automorphic representation of . Using Furusawa's integral representation for combined with a pullback formula involving the unitary group , the authors prove that the -functions are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations have a functorial lifting to a cuspidal representation of . Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of to a cuspidal representation of . As an application, the authors obtain analytic properties of various -functions related to full level Siegel cusp forms. They also obtain special value results for and
Book Synopsis Critical Population and Error Threshold on the Sharp Peak Landscape for a Moran Model by : Raphaël Cerf
Download or read book Critical Population and Error Threshold on the Sharp Peak Landscape for a Moran Model written by Raphaël Cerf and published by American Mathematical Soc.. This book was released on 2014-12-20 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. The author considers a Moran model describing the evolution of a population of size of chromosomes of length over an alphabet of cardinality . The mutation probability per locus is . He deals only with the sharp peak landscape: the replication rate is for the master sequence and for the other sequences. He studies the equilibrium distribution of the process in the regime where
Book Synopsis Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture by : Joel Friedman
Download or read book Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture written by Joel Friedman and published by American Mathematical Soc.. This book was released on 2014-12-20 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the author establishes some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. He then uses these ideas to prove the Hanna Neumann Conjecture of the 1950s; in fact, he proves a strengthened form of the conjecture.
Book Synopsis Polynomial Approximation on Polytopes by : Vilmos Totik
Download or read book Polynomial Approximation on Polytopes written by Vilmos Totik and published by American Mathematical Soc.. This book was released on 2014-09-29 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial approximation on convex polytopes in is considered in uniform and -norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the -case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate -functional follows as a consequence. The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.
Book Synopsis A Geometric Theory for Hypergraph Matching by : Peter Keevash
Download or read book A Geometric Theory for Hypergraph Matching written by Peter Keevash and published by American Mathematical Soc.. This book was released on 2014-12-20 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, their main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, the authors apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in -graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi Theorem. Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.
Book Synopsis Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres by : J.-M. Delort
Download or read book Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres written by J.-M. Delort and published by American Mathematical Soc.. This book was released on 2015-02-06 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hamiltonian ∫X(∣∂tu∣2+∣∇u∣2+m2∣u∣2)dx, defined on functions on R×X, where X is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of u. The associated PDE is then a quasi-linear Klein-Gordon equation. The author shows that, when X is the sphere, and when the mass parameter m is outside an exceptional subset of zero measure, smooth Cauchy data of small size ϵ give rise to almost global solutions, i.e. solutions defined on a time interval of length cNϵ−N for any N. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on u) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.
