A Foundation For Props Algebras And Modules

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A Foundation for PROPs, Algebras, and Modules

Donald Yau, Mark W. Johnson
A Foundation for PROPs Algebras and Modules Book PDF

Author : Donald Yau, Mark W. Johnson
Publisher : American Mathematical Soc.
Page : 311 pages
File Size : 42,9 Mb
Release : 2015-05-28
Category : Algebra, Homological
ISBN : 9781470421977

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A Foundation for PROPs, Algebras, and Modules by Donald Yau, Mark W. Johnson Pdf

PROPs and their variants are extremely general and powerful machines that encode operations with multiple inputs and multiple outputs. In this respect PROPs can be viewed as generalizations of operads that would allow only a single output. Variants of PROPs are important in several mathematical fields, including string topology, topological conformal field theory, homotopical algebra, deformation theory, Poisson geometry, and graph cohomology. The purpose of this monograph is to develop, in full technical detail, a unifying object called a generalized PROP. Then with an appropriate choice of pasting scheme, one recovers (colored versions of) dioperads, half-PROPs, (wheeled) operads, (wheeled) properads, and (wheeled) PROPs. Here the fundamental operation of graph substitution is studied in complete detail for the first time, including all exceptional edges and loops as examples of a new definition of wheeled graphs. A notion of generators and relations is proposed which allows one to build all of the graphs in a given pasting scheme from a small set of basic graphs using graph substitution. This provides information at the level of generalized PROPs, but also at the levels of algebras and of modules over them. Working in the general context of a symmetric monoidal category, the theory applies for both topological spaces and chain complexes in characteristic zero. This book is useful for all mathematicians and mathematical physicists who want to learn this new powerful technique.

Sugawara Operators for Classical Lie Algebras

Alexander Molev:
Sugawara Operators for Classical Lie Algebras Book PDF

Author : Alexander Molev:
Publisher : American Mathematical Soc.
Page : 304 pages
File Size : 42,9 Mb
Release : 2018-02-28
Category : Affine algebraic groups
ISBN : 9781470436599

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Sugawara Operators for Classical Lie Algebras by Alexander Molev: Pdf

The celebrated Schur-Weyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody algebras. The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical -algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the Harish-Chandra isomorphism connects the Sugawara operators with the classical -algebras, which play the role of the Weyl group invariants in the finite-dimensional theory.

Operads of Wiring Diagrams

Donald Yau
Operads of Wiring Diagrams Book PDF

Author : Donald Yau
Publisher : Springer
Page : 308 pages
File Size : 42,6 Mb
Release : 2018-09-19
Category : Mathematics
ISBN : 9783319950013

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Operads of Wiring Diagrams by Donald Yau Pdf

Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads. The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of the relational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images. Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.

Foundations of Arithmetic Differential Geometry

Alexandru Buium
Foundations of Arithmetic Differential Geometry Book PDF

Author : Alexandru Buium
Publisher : American Mathematical Society
Page : 357 pages
File Size : 49,7 Mb
Release : 2023-11-20
Category : Mathematics
ISBN : 9781470475772

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Foundations of Arithmetic Differential Geometry by Alexandru Buium Pdf

The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is “intrinsically curved”; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.

Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

Zhenbo Qin
Hilbert Schemes of Points and Infinite Dimensional Lie Algebras Book PDF

Author : Zhenbo Qin
Publisher : American Mathematical Soc.
Page : 336 pages
File Size : 44,6 Mb
Release : 2018-02-26
Category : Hilbert schemes
ISBN : 9781470441883

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Hilbert Schemes of Points and Infinite Dimensional Lie Algebras by Zhenbo Qin Pdf

Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes of collections of points (zero-dimensional subschemes) in a smooth algebraic surface . Schemes turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of , including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of and the Gromov–Witten correspondence. The last part of the book presents results about quantum cohomology of and related questions. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.

Infinity Operads And Monoidal Categories With Group Equivariance

Donald Yau
Infinity Operads And Monoidal Categories With Group Equivariance Book PDF

Author : Donald Yau
Publisher : World Scientific
Page : 486 pages
File Size : 44,5 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9789811250941

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Infinity Operads And Monoidal Categories With Group Equivariance by Donald Yau Pdf

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.

Persistence Theory: From Quiver Representations to Data Analysis

Steve Y. Oudot
Persistence Theory From Quiver Representations to Data Analysis Book PDF

Author : Steve Y. Oudot
Publisher : American Mathematical Soc.
Page : 218 pages
File Size : 50,8 Mb
Release : 2017-05-17
Category : Electronic
ISBN : 9781470434434

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Persistence Theory: From Quiver Representations to Data Analysis by Steve Y. Oudot Pdf

Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work. The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.

Grid Homology for Knots and Links

Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó
Grid Homology for Knots and Links Book PDF

Author : Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 47,6 Mb
Release : 2015-12-04
Category : Homology theory
ISBN : 9781470417376

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Grid Homology for Knots and Links by Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó Pdf

Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

Fokker–Planck–Kolmogorov Equations

Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov
Fokker Planck Kolmogorov Equations Book PDF

Author : Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov
Publisher : American Mathematical Society
Page : 495 pages
File Size : 47,7 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781470470098

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Fokker–Planck–Kolmogorov Equations by Vladimir I. Bogachev,Nicolai V. Krylov,Michael Röckner,Stanislav V. Shaposhnikov Pdf

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker–Planck–Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

The Ricci Flow: Techniques and Applications

Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther, James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni
The Ricci Flow Techniques and Applications Book PDF

Author : Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther, James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni
Publisher : American Mathematical Soc.
Page : 374 pages
File Size : 44,7 Mb
Release : 2015-10-19
Category : Electronic
ISBN : 9780821849910

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The Ricci Flow: Techniques and Applications by Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther, James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni Pdf

Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton's theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives. This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow.

The Economic Philosophy of the Internet of Things

James Juniper
The Economic Philosophy of the Internet of Things Book PDF

Author : James Juniper
Publisher : Routledge
Page : 246 pages
File Size : 49,8 Mb
Release : 2018-06-27
Category : Business & Economics
ISBN : 9781351069236

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The Economic Philosophy of the Internet of Things by James Juniper Pdf

To properly understand the nature of the digital economy we need to investigate the phenomenon of a "ubiquitous computing system" (UCS). As defined by Robin Milner, this notion implies the following characteristics: (i) it will continually make decisions hitherto made by us; (ii) it will be vast, maybe 100 times today’s systems; (iii) it must continually adapt, on-line, to new requirements; and, (iv) individual UCSs will interact with one another. This book argues that neoclassical approaches to modelling economic behaviour based on optimal control by "representative-agents" are ill-suited to a world typified by concurrency, decentralized control, and interaction. To this end, it argues for the development of new, process-based approaches to analysis, modelling, and simulation. The book provides the context—both philosophical and mathematical—for the construction and application of new, rigorous, and meaningful analytical tools. In terms of social theory, it adopts a Post-Cognitivist approach, the elements of which include the nature philosophy of Schelling, Marx’s critique of political economy, Peircean Pragmatism, Whitehead’s process philosophy, and Merleau-Ponty’s phenomenology of the flesh, along with cognitive scientific notions of embodied cognition and neural Darwinism, as well as more questionable notions of artificial intelligence that are encompassed by the rubric of "perception-and-action-without-intelligence".

Galois Theories of Linear Difference Equations: An Introduction

Charlotte Hardouin,Jacques Sauloy,Michael F. Singer
Galois Theories of Linear Difference Equations An Introduction Book PDF

Author : Charlotte Hardouin,Jacques Sauloy,Michael F. Singer
Publisher : American Mathematical Soc.
Page : 171 pages
File Size : 45,7 Mb
Release : 2016-04-27
Category : Difference and functional equations -- Difference equations -- Linear equations
ISBN : 9781470426552

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Galois Theories of Linear Difference Equations: An Introduction by Charlotte Hardouin,Jacques Sauloy,Michael F. Singer Pdf

This book is a collection of three introductory tutorials coming out of three courses given at the CIMPA Research School “Galois Theory of Difference Equations” in Santa Marta, Columbia, July 23–August 1, 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations. The authors motivate and give elementary examples of the basic ideas and techniques, providing the reader with an entry to current research. In addition each article contains an extensive bibliography that includes recent papers; the authors have provided pointers to these articles allowing the interested reader to explore further.

Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

Jared Speck
Shock Formation in Small Data Solutions to 3D Quasilinear Wave Equations Book PDF

Author : Jared Speck
Publisher : American Mathematical Soc.
Page : 544 pages
File Size : 43,9 Mb
Release : 2016-12-07
Category : Differential equations, Nonlinear
ISBN : 9781470428570

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Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations by Jared Speck Pdf

In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation. In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he provides a sharp description of the blow-up. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that small-data solutions are global when the null condition is satisfied. Readers who master the material will have acquired tools on the cutting edge of PDEs, fluid mechanics, hyperbolic conservation laws, wave equations, and geometric analysis.

The Dynamical Mordell–Lang Conjecture

Jason P. Bell,Dragos Ghioca,Thomas J. Tucker
The Dynamical Mordell Lang Conjecture Book PDF

Author : Jason P. Bell,Dragos Ghioca,Thomas J. Tucker
Publisher : American Mathematical Soc.
Page : 280 pages
File Size : 55,9 Mb
Release : 2016-04-20
Category : Arithmetical algebraic geometry
ISBN : 9781470424084

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The Dynamical Mordell–Lang Conjecture by Jason P. Bell,Dragos Ghioca,Thomas J. Tucker Pdf

The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.

The Projective Heat Map

Richard Evan Schwartz
The Projective Heat Map Book PDF

Author : Richard Evan Schwartz
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 53,9 Mb
Release : 2017-04-20
Category : Dynamical systems and ergodic theory
ISBN : 9781470435141

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The Projective Heat Map by Richard Evan Schwartz Pdf

This book introduces a simple dynamical model for a planar heat map that is invariant under projective transformations. The map is defined by iterating a polygon map, where one starts with a finite planar -gon and produces a new -gon by a prescribed geometric construction. One of the appeals of the topic of this book is the simplicity of the construction that yet leads to deep and far reaching mathematics. To construct the projective heat map, the author modifies the classical affine invariant midpoint map, which takes a polygon to a new polygon whose vertices are the midpoints of the original. The author provides useful background which makes this book accessible to a beginning graduate student or advanced undergraduate as well as researchers approaching this subject from other fields of specialty. The book includes many illustrations, and there is also a companion computer program.