Format: Paperback / softback, 134 pages, height x width: 235x155 mm, 3 Illustrations, color;
35 Illustrations, black and white; VIII, 134 p. 38 illus., 3 illus. in color.,
Series: SpringerBriefs in Mathematics
Pub. Date: 10-May-2023
ISBN-13: 9783031277030
The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings).
The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology.
The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds.
The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality.
Chapter.
1. IntroductionPart I: Leitfaden
Chapter.
2. Manifold and orbifold constructionsChapter.
3. Spectra, group
representations and twisted LaplaciansChapter.
4. Detecting representation
isomorphism through twisted spectraChapter.
5. Representations with a unique
monomial structureChapter.
6. Construction of suitable covers and proof of
the main theoremChapter.
7. Geometric construction of the covering
manifoldChapter.
8. Homological widenessChapter.
9. Examples of homologically
wide actionsChapter.
10. Homological wideness, "class field theory" for
covers, and a number theoretical analogueChapter.
11. Examples concerning the
main resultChapter.
12. Length spectrumReferencesIndex
Format: Hardback, 312 pages, height x width: 235x155 mm, 3 Tables, color; VIII, 312 p., 1 Hardback
Pub. Date: 13-May-2023
ISBN-13: 9783031254239
Over the course of his distinguished career, Vladimir Maz'ya has made a number of groundbreaking contributions to numerous areas of mathematics, including partial differential equations, function theory, and harmonic analysis. The chapters in this volume - compiled on the occasion of his 80th birthday - are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.
A. Cialdea, The scientific work of Vladimir Maz'ya.- E. Afanas'eva and
A. Golberg, Topological mappings of finite area distortion.- A. Alberico, A.
Cianchi, L. Pick, and L. Slavikova, On fractional Orlicz-Sobolev spaces.- C.
De Filippis and G. Mingione, Interpolative gap bounds for nonautonomous
integrals.- R. Kr. Giri and Y. Pinchover, Positive Liouville theorem and
asymptotic behaviour for (p, A)-Laplacian type elliptic equations with
Fuchsian potentials in Morrey space.- V. Gol'dshtein, R. Hurri-Syrjanen, V.
Pchelintsev, and A. Ukhlov, Space quasiconformal composition operators with
applications to Neumann eigenvalues.- S. L. Krushkai, Teichmuller spaces and
coefficient problems for univalent holomorphic functions.- N. V. Krylov, A
review of some new results in the theory of linear elliptic equations with
drift in L_d.- F. Lanzara, V. Maz'ya, and G. Schmidt, Fast computation of
elastic and hydrodynamic potentials using approximate approximations.- A.
Laptev and T. Weth, Spectral Properties of the logarithmic Laplacian.- E.
Liflyand, L^1 Convergence of Fourier transforms.- D. Mitrea, I. Mitrea, and
M. Mitrea, Failure of Fredholm solvability for the Dirichlet problem
corresponding to weakly elliptic systems.- G. Seregin, Local regularity of
axisymmetric solutions to the Navier-Stokes equations.- D. Shoikhet,
Nonlinear resolvent and rigidity of holomorphic mappings.- Y. Yomdin, "Smooth
rigidity" and Remez-type inequalities.
Format: Hardback, 266 pages, height x width: 240x168 mm, 20 Tables, color;
31 Illustrations, color; XVIII, 266 p. 31 illus. in color.,
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 08-May-2023
ISBN-13: 9783031264573
This book examines the most fundamental parts of convex analysis and its applications to optimization and location problems. Accessible techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and to build a theory of generalized differentiation for convex functions and sets in finite dimensions. The book serves as a bridge for the readers who have just started using convex analysis to reach deeper topics in the field. Detailed proofs are presented for most of the results in the book and also included are many figures and exercises for better understanding the material. Applications provided include both the classical topics of convex optimization and important problems of modern convex optimization, convex geometry, and facility location.
Convex Sets and Functions.- Convex Separation and Some Consequences.- Convex Generalized Differentiation.- Fenchel Conjugate and Further Topics In Subdifferentiation.- Remarkable Consequences of Convexity.- Minimal Time Functions and Related Issues.- Applications To Problems of Optimization and Equilibrium.- Applications To Location Problems.
Format: Paperback / softback, 155 pages, height x width: 235x155 mm, 19 Tables, color;
29 Illustrations, color; 5 Illustrations, black and white; VIII, 155 p. 34 illus., 29 illus. in color.,
Series: Compact Textbooks in Mathematics
Pub. Date: 05-May-2023
ISBN-13: 9783031264542
This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated as spectrahedra and their shadows (projections). Several general results obtained in the last 15 years by a variety of different methods are presented in the book, along with the necessary background from algebra and geometry.
1. Introduction and Preliminaries.-
2. Linear Matrix Inequalities and
Spectrahedra.-
3. Spectrahedral Shadows.- Appendix A: Real Algebraic
Geometry.- Appendix B: Convexity.
Format: Hardback, 275 pages, height x width: 235x155 mm, 1 Illustrations, color;
49 Illustrations, black and white; XIX, 275 p. 50 illus., 1 illus. in color.
Series: Graduate Texts in Physics
Pub. Date: 28-Apr-2023
ISBN-13: 9783031161384
Description
This book systematically develops the mathematical foundations of the theory of relativity and links them to physical relations. For this purpose, differential geometry on manifolds is introduced first, including differentiation and integration, and special relativity is presented as tensor calculus on tangential spaces. Using Einstein's field equations relating curvature to matter, the relativistic effects in the solar system including black holes are discussed in detail.
The text is aimed at students of physics and mathematics and assumes only basic knowledge of classical differential and integral calculus and linear algebra.
Table of Contents
Differentiable manifolds.- Tangent vectors.- Tensors.- Semi-Riemann manifolds.- Special relativity.- Differential forms.- Covariant derivation of vector fields.- Curvature.- Matter.- Geodesy.- Covariant differentiation of tensor fields.- Lie derivation.- Integration on manifolds.- Non-rotating black holes.- Cosmology.- Rotating black holes.- An overview of string theory.
Format: Hardback, 264 pages, height x width: 235x155 mm, 5 Tables, color;
6 Illustrations, color; 23 Illustrations, black and white; VIII, 264 p. 29 illus., 6 illus. in color.,
Series: Tutorials, Schools, and Workshops in the Mathematical Sciences
Pub. Date: 08-May-2023
ISBN-13: 9783031256653
This volume presents lectures given at the Wisla 20-21 Winter School and Workshop: Groups, Invariants, Integrals, and Mathematical Physics, organized by the Baltic Institute of Mathematics. The lectures were dedicated to differential invariants - with a focus on Lie groups, pseudogroups, and their orbit spaces - and Poisson structures in algebra and geometry and are included here as lecture notes comprising the first two chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and category theory. Specific topics covered include:
The multisymplectic and variational nature of Monge-Ampere equations in dimension four Integrability of fifth-order equations admitting a Lie symmetry algebra Applications of the van Kampen theorem for groupoids to computation of homotopy types of striped surfaces A geometric framework to compare classical systems of PDEs in the category of smooth manifolds
Groups, Invariants, Integrals, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry and category theory is assumed.
Lychagin, V., Roop, M., Differential Invariants in Algebra.- Rubtsov,
V., Suchanek, R., Lectures on Poisson Algebras.- Suchanek,R., Some Remarks on
Multisymplectic and Variational Nature of Monge-Ampere Equations in Dimension
Four.- Ruiz, A., Muriel, C., Generalized Solvable Structures Associated to
Symmetry Algebras Isomorphic to $\mathfrak{gl}(2,\mathbb{R}) \ltimes
\mathbb{R}$.- Maksymenko, S., Nikitchenko, O., Fundamental Groupoids and
Homotopy Types of Non-Compact Surfaces.- Barth, L. S., A Geometric Framework
to Compare Classical Field Theories.