By Alexander I. Bobenko and Yuri B. Suris
An rising box of discrete differential geometry goals on the improvement of discrete equivalents of notions and strategies of classical differential geometry. The latter seems as a restrict of a refinement of the discretization. present curiosity in discrete differential geometry derives not just from its value in natural arithmetic but additionally from its purposes in special effects, theoretical physics, structure, and numerics. fairly abruptly, the very uncomplicated buildings of discrete differential geometry change into on the topic of the speculation of integrable structures. one of many major ambitions of this e-book is to bare this integrable constitution of discrete differential geometry. For a given delicate geometry you possibly can recommend many various discretizations. Which one is the easiest? This booklet solutions this question by means of supplying basic discretization ideas and utilising them to various concrete difficulties. It seems that clever theoretical discretizations are extraordinary additionally by way of their strong functionality in purposes. The meant viewers of this booklet is threefold. it's a textbook on discrete differential geometry and integrable platforms compatible for a one semester graduate direction. however, it's addressed to experts in geometry and mathematical physics. It displays the new development in discrete differential geometry and comprises many unique effects. The 3rd team of readers at which this publication is concentrated is shaped by means of experts in geometry processing, special effects, architectural layout, numerical simulations, and animation. they could locate right here solutions to the query ``How can we discretize differential geometry?'' bobbing up of their particular box. must haves for studying this publication comprise commonplace undergraduate heritage (calculus and linear algebra). No wisdom of differential geometry is anticipated, even though a few familiarity with curves and surfaces will be useful.
By Jing-Song Huang
This monograph provides a entire remedy of significant new principles on Dirac operators and Dirac cohomology. Dirac operators are commonly used in physics, differential geometry, and group-theoretic settings (particularly, the geometric development of discrete sequence representations). The comparable notion of Dirac cohomology, that is outlined utilizing Dirac operators, is a far-reaching generalization that connects index idea in differential geometry to illustration idea. utilizing Dirac operators as a unifying topic, the authors reveal how one of the most vital leads to illustration thought healthy jointly whilst considered from this angle.
Key issues lined include:
* evidence of Vogan's conjecture on Dirac cohomology
* basic proofs of many classical theorems, corresponding to the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem
* Dirac cohomology, outlined through Kostant's cubic Dirac operator, in addition to different heavily similar types of cohomology, akin to n-cohomology and (g,K)-cohomology
* Cohomological parabolic induction and $A_q(\lambda)$ modules
* Discrete sequence thought, characters, life and exhaustion
* sprucing of the Langlands formulation on multiplicity of automorphic varieties, with applications
* Dirac cohomology for Lie superalgebras
An very good contribution to the mathematical literature of illustration thought, this self-contained exposition bargains a scientific exam and panoramic view of the topic. the cloth may be of curiosity to researchers and graduate scholars in illustration concept, differential geometry, and physics.
By Stephan C. Carlson
Topology of Surfaces, Knots, and Manifolds bargains an intuition-based and example-driven method of the elemental rules and difficulties related to manifolds, rather one- and two-dimensional manifolds. a mix of examples and routines leads the reader to expect common definitions and theorems touching on curves, surfaces, knots, and links--the items of curiosity within the attractive set of mathematical principles often called "rubber sheet geometry." the result's a e-book that gives strong insurance of the maths underlying those issues.
By M. M. Postnikov
By Steven G. Krantz
The implicit functionality theorem is a part of the bedrock of mathematical research and geometry. discovering its genesis in eighteenth century reports of actual analytic services and mechanics, the implicit and inverse functionality theorems have now blossomed into strong instruments within the theories of partial differential equations, differential geometry, and geometric research. there are various assorted different types of the implicit functionality theorem, together with (i) the classical formula for C^k capabilities, (ii) formulations in different functionality areas, (iii) formulations for non- soft services, (iv) formulations for services with degenerate Jacobian. quite robust implicit functionality theorems, resembling the Nash--Moser theorem, were built for particular purposes (e.g., the imbedding of Riemannian manifolds). All of those subject matters, and plenty of extra, are handled within the current quantity. The background of the implicit functionality theorem is a full of life and complicated tale, and is in detail certain up with the advance of basic principles in research and geometry. this complete improvement, including mathematical examples and proofs, is mentioned for the first time right here. it really is a thrilling story, and it keeps to conform. "The Implicit functionality Theorem" is an available and thorough remedy of implicit and inverse functionality theorems and their purposes. will probably be of curiosity to mathematicians, graduate/advanced undergraduate scholars, and to those that practice arithmetic. The booklet unifies disparate rules that experience performed an very important function in glossy arithmetic. It serves to record and position in context a considerable physique of mathematical rules.
By Helmut Hofer
The discoveries of the final a long time have opened new views for the previous box of Hamiltonian platforms and resulted in the production of a brand new box: symplectic topology. mind-blowing tension phenomena display that the character of symplectic mappings is especially diversified from that of quantity retaining mappings. This increases new questions, lots of them nonetheless unanswered. nonetheless, research of an previous variational precept in classical mechanics has verified international periodic phenomena in Hamiltonian structures. because it seems, those possible diverse phenomena are mysteriously similar. one of many hyperlinks is a category of symplectic invariants, known as symplectic capacities. those invariants are the most subject matter of this publication, consisting of such subject matters as easy symplectic geometry, symplectic capacities and tension, periodic orbits for Hamiltonian structures and the motion precept, a bi-invariant metric at the symplectic diffeomorphism crew and its geometry, symplectic fastened aspect thought, the Arnold conjectures and primary order elliptic platforms, and at last a survey on Floer homology and symplectic homology.
The exposition is self-contained and addressed to researchers and scholars from the graduate point onwards.
All the chapters have a pleasant advent with the ancient improvement of the topic and with an ideal description of the state-of-the-art. the most rules are brightly uncovered through the e-book. (…) This booklet, written by way of skilled researchers, will surely fill in a spot within the idea of symplectic topology. The authors have taken half within the improvement of this sort of conception by means of themselves or via their collaboration with different awesome humans within the area.
This publication is a gorgeous creation to at least one outlook at the intriguing new advancements of the final ten to 15 years in symplectic geometry, or symplectic topology, as definite features of the topic are in recent times known as. (…) The authors are visible masters of the sphere, and their reflections right here and there during the booklet at the ambient literature and open difficulties are might be the main attention-grabbing components of the volume.
By Alex Lubotzky, Jonathan D. Rogawski
The ebook offers the ideas to 2 difficulties: the 1st is the development of increasing graphs – graphs that are of primary value for verbal exchange networks and computing device technology; the second one is the Ruziewicz challenge about the finitely additive invariant measures on spheres. either difficulties have been partly solved utilizing the Kazhdan estate (T) from illustration thought of semi-simple Lie teams. Later, whole soultions have been received for either difficulties utilizing the Ramanujan conjecture from analytic quantity conception. the writer, who performed a big position in those advancements, explains the 2 difficulties and their ideas from a standpoint which unearths why these types of doubtless unrelated themes are so interconnected. The unified strategy exhibits interrelations among varied branches of arithmetic corresponding to graph thought, degree idea, Riemannian geometry, discrete subgroups of Lie teams, illustration idea and analytic quantity thought. distinct efforts have been made to make the publication available to graduate scholars in arithmetic and computing device technology. a couple of difficulties and proposals for additional examine are awarded. stories: "This interesting booklet marks the genesis of a brand new box. it's a box during which one passes from side to side at will throughout the taking a look glass dividing the discrete from the continual. (...) The ebook is an enthralling mix of issues from team conception (finite and infinite), combinatorics, quantity conception, harmonic analysis." - Zentralblatt MATH "The Appendix, written via J. Rogawski, explains the Jacquet-Langlands concept and shows Deligne’s facts of the Petersson-Ramanujan conjecture. it's going to advantage its personal evaluation. (...) In end, this can be a incredible manner of transmitting contemporary mathematical study without delay "from the manufacturer to the consumer." - MathSciNet "The e-book is available to mature graduate scholars in arithmetic and theoretical computing device technology. it's a great presentation of a gem on the border of research, geometry, algebra and combinatorics. those that take the hassle to look what occurs at the back of the scene won’t remorse it." - Acta Scientiarum Mathematicarum
By Maks A. Akivis, Vladislav V. Goldberg
During this publication the authors research the differential geometry of types with degenerate Gauss maps. They use the most tools of differential geometry, particularly, the tools of relocating frames and external differential varieties in addition to tensor tools. by way of those equipment, the authors notice the constitution of types with degenerate Gauss maps, be sure the singular issues and singular types, locate focal pictures and build a class of the types with degenerate Gauss maps.
The authors introduce the above pointed out equipment and follow them to a chain of concrete difficulties coming up within the thought of types with degenerate Gauss maps. What makes this ebook certain is the authors' use of a scientific software of equipment of projective differential geometry in addition to equipment of the classical algebraic geometry for learning types with degenerate Gauss maps.
This e-book is meant for researchers and graduate scholars attracted to projective differential geometry and algebraic geometry and their purposes. it may be used as a textual content for complicated undergraduate and graduate scholars.
Both authors have released over a hundred papers each one. each one has written a few books, together with Conformal Differential Geometry and Its Generalizations (Wiley 1996), Projective Differential Geometry of Submanifolds (North-Holland 1993), and Introductory Linear Algebra (Prentice-Hall 1972), that have been written via them together.
By Helmut Eschrig
A concise yet self-contained advent of the valuable recommendations of contemporary topology and differential geometry on a mathematical point is given in particular with functions in physics in brain. All easy suggestions are systematically supplied together with sketches of the proofs of so much statements. tender finite-dimensional manifolds, tensor and external calculus working on them, homotopy, (co)homology concept together with Morse thought of serious issues, in addition to the idea of fiber bundles and Riemannian geometry, are taken care of. Examples from physics include topological fees, the topology of periodic boundary stipulations for solids, gauge fields, geometric stages in quantum physics and gravitation.