Many classical problems in pure and applied mathematics remain unsolved or partially solved. This book studies some of these questions by presenting new and important results that should motivate future research. Strong bookstore candidate.

This is the second edition of a popular work offering a unique introduction to Clifford algebras and spinors. The beginning chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This edition has three new chapters, including material on conformal invariance and a history of Clifford algebras.

This book collects original peer-reviewed contributions to the conferences organised by the international research network “Minimal surfaces: Integrable Systems and Visualization” financed by the Leverhulme Trust. The conferences took place in Cork, Granada, Munich and Leicester between 2016 and 2019. Within the theme of the network, the presented articles cover a broad range of topics and explore exciting links between problems related to the mean curvature of surfaces in homogeneous 3-manifolds, like minimal surfaces, CMC surfaces and mean curvature flows, integrable systems and visualisation. Combining research and overview articles by prominent international researchers, the book offers a valuable resource for both researchers and students who are interested in this research area.

Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence. The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica®, MatlabTM, and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress. The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

Topological geometrodynamics (TGD) is a modification of the theory of general relativity inspired by the problems related to the definition of inertial and gravitational energies in the earlier hypotheses. TGD is also a generalization of super string models. TGD brings forth an elegant theoretical projection of reality and builds upon the work by renowned scientists (Wheeler, Feynman, Penrose, Einstein, Josephson to name a few). In TGD, Physical space-time planes are visualized as four-dimensional surfaces in a certain 8-dimensional space (H). The choice of H is fixed by symmetries of standard model and leads to a geometric mapping of known classical fields and elementary particle numbers. TGD differs from Einstein’s geometrodynamics in the way space-time planes or ‘sheets’ are lumped together. Extending the theory based on fusing number concepts implies a further generalisation of the space-time concept allowing the identification of space-time correlates of cognition and intentionality. Additionally, zero energy ontology forces an extension of quantum measurement theory to a theory of consciousness and a hierarchy of phases is identified. Dark matter is thus predicted with far reaching implications for the understanding of consciousness and living systems. Therefore, it sets a solid foundation for modeling our universe in geometric terms. Topological Geometrodynamics: An Overview explains basic and advanced concepts about TGD. The book covers introductory information and classical TGD concepts before delving into twistor-space theory, particle physics, infinite-dimensional spinor geometry, generalized number theory, Planck constants, and the applications of TGD theory in research. The book is a valuable guide to TDG theory for researchers and advanced graduates in theoretical physics and cosmology.

This book brings together several advanced topics in computer graphics that are important in the areas of game development, three-dimensional animation and real-time rendering. The book is designed for final-year undergraduate or first-year graduate students, who are already familiar with the basic concepts in computer graphics and programming. It aims to provide a good foundation of advanced methods such as skeletal animation, quaternions, mesh processing and collision detection. These and other methods covered in the book are fundamental to the development of algorithms used in commercial applications as well as research.

In the present work, a coordinate-free way is suggested to handle conformal maps of a Rie­mannian sur­face into a space of constant curvature of maximum dimension 4, modeled on the non-commutative field of quaternions. This setup for the target space and the idea to treat dif­fe­rential 2-forms on Rie­mannian surfaces as quadratic functions on the tangent space, are the starting points for the development of the theory of conformal maps and in particular of con­formal immersions. As a first result, very nice condi­tions for the conformality of immersions into 3- and 4-dimensional space-forms are deduced and a sim­ple way to write the second fun­damental form is found. If the target space is euclidean 3-space, an alternative approach is proposed by fixing a spin structure on the Rie­mannian surface. The problem of finding a local immersion is then reduced to that of solving a linear Dirac equation with a potential whose square is the Willmore in­tegrand. This allows to make statements about the structure of the moduli space of conformal immersions and to derive a very nice criterion for a conformal immersion to be con­strained Willmore. As an application the Dirac equation with constant potential over spheres and tori is solved. This yields explicit immersion formulae out of which there were produced pictures, the Dirac-spheres and -tori. These immersions have the property that their Willmore integrand generates a metric of vanishing and constant curvature, respectively. As a next step an affine immersion theory is developped. This means, one starts with a given conformal immersion into euclidean 3-space and looks for new ones in the same conformal class. This is called a spin-transformation and it leads one to solve an affine Dirac equation. Also, it is shown how the coordi­nate-dependent generalized Weierstrass representation fits into the present framework. In particular, it is now natural to consider the class of conformal im­mersions that admit new conformal immersions having the same potential. It turns out, that all geometri­cally interesting immersions admit such an isopotential spin-transformation and that this property of an immersion is even a conformal invariant of the ambient space. It is shown that conformal isothermal immersions generate both via their dual and via Darboux trans­formations non-trivial families of new isopotential conformal immersions. Similarly to this, conformal (constrained) Willmore immersions produce non-trivial families of isopotential im­mer­sions of which subfamilies are (constrained) Willmore again having even the same Will­more integral. Another obser­vation is, that the Euler-Lagrange equation for the Willmore pro­blem is the integrability condition for a quaternionic 1-form, which generates a conformal mi­nimal im­mersions into hyperbolic 4-space. Vice versa, any such immersion determines a con­formal Willmore immersion. As a conse­quence, there is a one-to-one correspondence between con­formal minimal immersions into Lorentzian space and those into hyperbolic space, which gene­ralizes to any dimension. There is also induced an action on conformal minimal immersi­ons into hyperbolic 4-space. Another fact is, that conformal con­stant mean curvature (cmc) immersions into some 3-dimensional space form unveil to be isothermal and constrained Will­more. The reverse statement is true at least for tori. Finally a very simple proof of a theorem by R.Bryant concer­ning Willmore spheres is given. In the last part, time-dependent conformal immersions are considered. Their deformation for­mulae are computed and it is investigated under what conditions the flow commutes with Moe­bius transforma­tions. The modified Novikov-Veselov flow is written down in a conformal in­variant way and explicit deformation formulae for the immersion function itself and all of its invariants are given. This flow commutes with Moebius transformations. Its definition is cou­pled with a delta-bar problem, for which a so­lution is presented under special conditions. These are fulfilled at least by cmc immersions and by sur­faces of revolution and the general flow for­mulae reduce to very nice formulae in these cases.