Part of Elements in Non-local Data Interactions: Foundations and Applications
DATE PUBLISHED: May 2023
AVAILABILITY: Not yet published - available from June 2023
FORMAT: Paperback ISBN: 9781009327855
In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.
1. Introduction
2. Mathematical Background
3. Nonlocal p-Laplacian evolution problem on graphs
4. Nonlocal p-Laplacian variational problem on graphs
5. Nonlocal p-Laplacian Dirichlet problem on graphs
6. Algorithmic framework based on proximal splitting
References.
Part of London Mathematical Society Student Texts
PUBLICATION PLANNED FOR: July 2023
AVAILABILITY: Not yet published - available from July 2023
FORMAT: HardbackISBN: 9781009372626
FORMAT: PaperbackISBN: 9781009372633
Symington's almost toric fibrations have played a central role in symplectic geometry over the past decade, from Vianna's discovery of exotic Lagrangian tori to recent work on Fibonacci staircases. Four-dimensional spaces are of relevance in Hamiltonian dynamics, algebraic geometry, and mathematical string theory, and these fibrations encode the geometry of a symplectic 4-manifold in a simple 2-dimensional diagram. This text is a guide to interpreting these diagrams, aimed at graduate students and researchers in geometry and topology. First the theory is developed, and then studied in many examples, including fillings of lens spaces, resolutions of cusp singularities, non-toric blow-ups, and Vianna tori. In addition to the many examples, students will appreciate the exercises with full solutions throughout the text. The appendices explore select topics in more depth, including tropical Lagrangians and Markov triples, with a final appendix listing open problems. Prerequisites include familiarity with algebraic topology and differential geometry.
Contains many worked examples and figures to help the reader interpret the general theory
Includes a range of exercises, with complete solutions, to develop understanding
Features a list of open-ended open research problems, with accompanying discussion, to stimulate research in the area
Accessible to beginning graduate students, with extensive appendices providing background material
1. The Arnold?Liouville theorem
2. Lagrangian fibrations
3. Global action-angle coordinates and torus actions
4. Symplectic reduction
5. Visible Lagrangian submanifolds
6. Focus-focus singularities
7. Examples of focus-focus systems
8. Almost toric manifolds
9. Surgery
10. Elliptic and cusp singularities
A. Symplectic linear algebra
B. Lie derivatives
C. Complex projective spaces
D. Cotangent bundles
E. Moser's argument
F. Toric varieties revisited
G. Visible contact hypersurfaces and Reeb flows
H. Tropical Lagrangian submanifolds
I. Markov triples
J. Open problems
References
Index.
Surveys in Differential Geometry Volume 24 (2019)
Published: 31 May 2022
Hardcover
504 pages
In 2019, mathematicians around the world celebrated the 70th birthday of Professor Shing-Tung Yau and expressed their appreciation and admiration. A number of conferences were held throughout the year for this special occasion: at Harvard University (May), at the University of Rome (May/June), at the Chinese University of Hong Kong (June), at the 8th ICCM Congress at Tsinghua University (June), at Lehigh University (November), and elsewhere.
This is the second of the two Surveys in Differential Geometryvolumes in honor of Professor Yau, consisting of articles by some of the speakers at those special conferences, and several additional contributions.
The articles in this volume include: Tristan C. Collins and Yun Shi, “Stability and the deformed Hermitian Yang?Mills equation”; Simon Donaldson and Christopher Scaduto, “Associative submanifolds and gradient cycles”; Davide Gaiotto and Edward Witten, “Probing quantization via branes”; L. Gottsche and M. Kool, “Sheaves on surfaces and virtual invariants”; Goo Ishikawa and Stanislaw Janeczko, “Symplectic singularities of differentiable mappings”; Jun Li and Chiu-Chu Melissa Liu, “Counting curves in quintic Calabi?Yau threefolds and Landau-Ginzberg models”; Nikolai Nadirashvili et al., “Conformally maximal metrics for Laplace eigenvalues on surfaces”; Kieran G. O’Grady, “Moduli of sheaves on K3
’s and higher dimensional HK varieties”; Richard Schoen and Shing-Tung Yau “Positive scalar curvature and minimal hypersurface singularities”; Penny Smith and Karen Uhlenbeck, “Removeability of a codimension four singular set for solutions of a Yang?Mills?Higgs equation with small energy”; and Mu?Tao Wang, “Limits of quasi-local angular momentum on an isolated gravitating system”.
Published: 23 June 2022
Paperback
This volume presents lively and engaging articles from the lecturers and participants of the 27th Gokova Geometry-Topology Conference, held online as a series of virtual seminars in 2021.
The 27th Gokova Geometry-Topology Conference was sponsored by the National Science Foundation, by the Turkish Mathematical Society, and by the European Research Council.
Published: 9 September 2022
Hardcover
364 pages
In the last half-century, tremendous progress has been made in the study of 3-dimensional topology. Many revolutions in 3-manifold topology during this period have come from outside of the field, including Kleinian groups, minimal surfaces, foliations, von Neumann algebras, gauge theory, mathematical physics, 4-manifolds, symplectic topology, contact topology, Riemannian geometry and PDEs, number theory, dynamics, and geometric group theory. The influx of ideas from neighboring fields has made the subject of 3-manifolds (and more generally low-dimensional topology) a very rich subject, creating subfields such as quantum topology. But this also means that there is a tremendous amount of background material for a novitiate in the subject to learn and master.
This volume is a collection of surveys meant to bring certain subfields of 3-manifold topology up-to-date. These include: Richard Bamler on Ricci flow-with-surgery on 3-manifolds stemming from Perelman’s work on the geometrization theorem; Tobias Colding, David Gabai, and Daniel Ketover on minimal surface techniques applied to the study of Heegaard splittings of 3-manifolds, including the resolution of the Pitts?Rubinstein conjecture; Vincent Colin and Ko Honda on the theory of foliations and contact structures on sutured 3-manifolds; John Etnyre and Lenhard Ng on Legendrian contact homology of knots; Sang-Hyun Kim and Genevieve Walsh on hyperbolic groups with planar limit sets in relation to Kleinian groups; Marc Lackenby on algorithms in knot theory and 3-manifold topology, including results on computational complexity; Yi Liu and Hongbin Sun on the resolution of the virtual Haken conjecture, including subgroup separability, degree one maps between 3-manifolds, and torsion in the homology of covers; Mahan Mj on Cannon?Thurston maps following his resolution of Question 14 from Thurston’s problem list; and Jean-Marc Schlenker on renormalized volume of Kleinian groups and its relation to other notions of volume.
Published: 31 May 2023
Paperback
162 pages
Einstein’s theory of relativity is as beautiful as it is profound, and a brilliant stroke of genius. But for the non-expert, his theory of relativity seems quite hard to pin down: one doesn’t know where to begin in order to understand its wonderful implications.
The author, however, believes that understanding Einstein’s theory of relativity is not an insurmountable task.
This book starts with the simplest Newtonian theory of universal gravitation and uses easy-to-grasp language to explain the gulf of knowledge traversed by humanity over the course of 200 years: establishing that light travels at a finite speed, and that the propagation of light does not require a medium. After Einstein began his pursuits, he overturned Newton’s stubborn concept of absolute time with an easily understood thought experiment.
And it turns out that time can expand or contract according to how fast one travels and the strength of the gravitational field in which one is located. Einstein persistently worked to transform Newton’s conception of gravitational fields into his unique notion of a local curvature in the four-dimensional spacetime geometry. Here the author once again uses plain language and easily understood descriptions that allow the reader to fully appreciate the physical implications hidden behind the esoteric mathematics of Einstein’s theory of relativity. In this way, readers can establish a comprehensive understanding of Einstein’s theory of relativity?it’s like a stimulus check of knowledge.
Einstein predicted the existence of gravitational waves a hundred years ago. These are waves that come from the deepest reaches of the universe and are intimately related to the expanding dark universe, discovered at the onset of the 21st century. After more than fifty years of hard work, in 2015 humanity at last detected a gravitational wave?generated from a collision between two black holes 1.3 billion years ago. The author reports in detail this tremendous achievement (which won the 2017 Nobel Prize in Physics), and what we have gained by it.