The Many Facets Of Geometry

This title celebrates the academic career of Professor Nigel Hitchin - one of the most influential figures in the field of differential and algebraic geometry.

Few people have proved more influential in the field of differential and algebraic geometry, and in showing how this links with mathematical physics, than Nigel Hitchin. Oxford University's Savilian Professor of Geometry has made fundamental contributions in areas as diverse as: spin geometry, instanton and monopole equations, twistor theory, symplectic geometry of moduli spaces, integrables systems, Higgs bundles, Einstein metrics, hyperkähler geometry, Frobenius manifolds, Painlevé equations, special Lagrangian geometry and mirror symmetry, theory of grebes, and many more. He was previously Rouse Ball Professor of Mathematics at Cambridge University, as well as Professor of Mathematics at the University of Warwick, is a Fellow of the Royal Society and has been the President of the London Mathematical Society. The chapters in this fascinating volume, written by some of the greats in their fields (including four Fields Medalists), show how Hitchin's ideas have impacted on a wide variety of subjects. The book grew out of the Geometry Conference in Honour of Nigel Hitchin, held in Madrid, with some additional contributions, and should be required reading for anyone seeking insights into the overlap between geometry and physics.

The primary objective of this book is to provide you with the opportunity to gain insight into the nature of mathematics by exploring some of the many facets of mathematics.

This book presents a collection of essays that address various facets of the hydrogeology of Israel. Despite its small geographic size, Israel exhibits a variety of climates and is located between two regional fluctuating base levels. The respective chapters discuss the variety of hydrogeological configurations and hydrological processes produced by these geographical circumstances. In some cases, the interpretation of these aspects is deliberately left open to debate, because the authors were asked to provide, in addition to their own views, also alternative and even conflicting ones. Hydrogeological configurations similar to those in Israel can be found in other countries around the world. Therefore, researchers, scholars and professionals in this interdisciplinary field can benefit from and directly apply the considerable experience and expertise that has been gathered in Israel over the past few decades.

Over the past few years, long-duration gamma-ray bursts (GRBs), including the subclass of X-ray flashes (XRFs), have been revealed to be a rare variety of Type Ibc supernova (SN Ibc). While all these events result from the death of massive stars, the electromagnetic luminosities of GRBs and XRFs exceed those of ordinary Type Ibc SNe by many orders of magnitude. The observed diversity of stellar death corresponds to large variations in the energy, velocity, and geometry of the explosion ejecta. Using multi-wavelength (radio, optical, X-ray) observations of the nearest GRBs, XRFs, and SNe Ibc, I show that GRBs and XRFs couple at least 1048 erg to relativistic material while SNe Ibc typically couple less than 1048 erg to their fastest (albeit non-relativistic) outflows. Specifically, I find that less than 3 percent of local SNe Ibc show any evidence for association with a GRB or XRF. Interestingly, this dichotomy is not echoed by the properties of their optical SN emission, dominated by the radioactive decay of Nickel-56; I find that GRBs, XRFs, and SNe Ibc show significant overlap in their optical peak luminosity and photospheric velocities. Recently, I identified a new class of GRBs and XRFs that are under-luminous in comparison with the statistical sample of GRBs. Owing to their faint high-energy emission, these sub-energetic bursts are only detectable nearby (z

Nigel Hitchin is one of the world's foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin's 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics.

Articles in this volume are based on lectures given at three conferences on Geometry at the Frontier, held at the Universidad de la Frontera, Pucón, Chile in 2016, 2017, and 2018. The papers cover recent developments on the theory of algebraic varieties—in particular, of their automorphism groups and moduli spaces. They will be of interest to anyone working in the area, as well as young mathematicians and students interested in complex and algebraic geometry.

This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, held from January 6-7, 2012, in Boston, MA. The very wide range of topics represented in this volume illustrates

This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.

Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.