Series: Springer Monographs in Mathematics
Hardcover
ISBN 978-1-4939-3245-0
*Presents thorough treatment of hypersurfaces in real, complex, and
quaternionic space forms with connections to symmetric spaces,
homogeneous spaces, and Riemannian geometry
* Treats Dupin hypersurfaces using both standard and Lie sphere
geometric techniques
* Discusses the comprehensive treatment of the theory of
isoparametric hypersurfaces due to Cartan and Munzner that are
necessary for understanding the subject
This exposition provides the state-of-the art on the differential geometry of hypersurfaces
in real, complex, and quaternionic space forms. Special emphasis is placed on
isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces
in complex space forms. The book is accessible to a reader who has completed a oneyear
graduate course in differential geometry. The text, including open problems and an
extensive list of references, is an excellent resource for researchers in this area.
Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space
forms. Topics include shape operators, principal curvatures and foliations, tubes and
parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then
turns to the theory of isoparametric hypersurfaces in spheres. Important examples
and classification results are given, including the construction of isoparametric
hypersurfaces based on representations of Clifford algebras. An in-depth treatment of
Dupin hypersurfaces follows with results that are proved in the context of Lie sphere
geometry as well as those that are obtained using standard methods of submanifold
theory. Next comes a thorough treatment of the theory of real hypersurfaces in complex
space forms. A central focus is a complete proof of the classification of Hopf hypersurfaces
with constant principal curvatures due to Kimura and Berndt. The book concludes with the
basic theory of real hypersurfaces in quaternionic space forms, including statements of the
major classification results and directions for further research.
Series: Progress in Mathematics, Vol. 312
Hardcover
ISBN 978-3-319-23442-7
* A unique and comprehensive tribute for David A. Vogan, Jr., a
mathematician whose work has had a lasting impact on the field
* Highlights the depth and breadth of Vogan's influence on
representation theory, number theory, automorphic forms, algebraic
geometry, and combinatorics
* Explores many exciting new directions that remain to be explored in
the representation theory of reductive groups
Over the last forty years, David Vogan has left an indelible imprint on the representation
theory of reductive groups. His groundbreaking ideas have lead to deep advances in the
theory of real and p-adic groups, and have forged lasting connections with other subjects,
including number theory, automorphic forms, algebraic geometry, and combinatorics.
Representations of Reductive Groups is an outgrowth of the conference of the same name,
dedicated to David Vogan on his 60th birthday, which took place at MIT on May 19-23,
2014. This volume highlights the depth and breadth of Vogan's influence over the
subjects mentioned above, and point to many exciting new directions that remain to be
explored. Notably, the first article by McGovern and Trapa offers an overview of Vogan's
body of work, placing his ideas in a historical context.
Contributors: Pramod N. Achar, Jeffrey D. Adams, Dan Barbasch, Manjul Bhargava, Cedric
Bonnafe, Dan Ciubotaru, Meinolf Geck, William Graham, Benedict H. Gross, Xuhua He,
Jing-Song Huang, Toshiyuki Kobayashi, Bertram Kostant, Wenjing Li, George Lusztig, Eric
Marberg, William M. McGovern, Wilfried Schmid, Kari Vilonen, Diana Shelstad, Peter E.
Trapa, David A. Vogan, Jr., Nolan R. Wallach, Xiaoheng Wang, Geordie Williamson
Series: Levy Matters, Vol. 2149
Softcover
ISBN 978-3-319-23137-2
This three-chapter volume concerns the distributions of certain functionals of Levy
processes. The first chapter, by Makoto Maejima, surveys representations of the main
sub-classes of infinitesimal distributions in terms of mappings of certain Levy processes
via stochastic integration. The second chapter, by Lars Norvang Andersen, Soren
Asmussen, Peter W. Glynn and Mats Pihlsgard, concerns Levy processes reflected at two
barriers, where reflection is formulated a la Skorokhod. These processes can be used to
model systems with a finite capacity, which is crucial in many real life situations, a most
important quantity being the overflow or the loss occurring at the upper barrier. If a
process is killed when crossing the boundary, a natural question concerns its lifetime.
Deep formulas from fluctuation theory are the key to many classical results, which are
reviewed in the third chapter by Frank Aurzada and Thomas Simon. The main part,
however, discusses recent advances and developments in the setting where the process is
given either by the partial sum of a random walk or the integral of a Levy process.
Series: Graduate Texts in Mathematics, Vol. 273
Hardcover
ISBN 978-3-319-23487-8
* Updates a popular textbook from the golden era of the Moscow
school of I. M. Gelfand
* Presents material concisely but rigorously
* Illuminates the subject matter with a range of technical and artistic
illustrations, along with a wealth of examples and computations
meant to provide a treatment of the topic that is both deep and
broad
* Contains an entirely new chapter on K-theory and the Riemann-Roch
theorem
This classic text of the renowned Moscow mathematical school equips the aspiring
mathematician with a solid grounding in the core of topology, from the homotopical
perspective. Its comprehensiveness and depth of treatment are unmatched among
topology textbooks: in addition to covering the basics the fundamental notions and
constructions of homotopy theory, covering spaces and the fundamental group, CW
complexes, homology and cohomology, and homological algebra the book treats
essential advanced topics, such as obstruction theory, characteristic classes, Steenrod
squares, K-theory and cobordism theory, and, with distinctive thoroughness and lucidity,
spectral sequences. The organization of the material around the major achievements of
the golden era of topology the Adams conjecture, Bott periodicity, the Hirzebruch?
Riemann-Roch theorem, and the Atiyah-Singer index theorem, to name a few conveys
a clear picture of the canon of the subject. Grassmannians, loop spaces, and classical
groups play a central role in mathematics, and therefore in the presentation of this book,
as well.
A judicious focus on the key ideas, and an appropriate magnification of detail, enable
the reader to navigate the breadth of material, confidently, without the disorientation
of algebraic minutiae. Many exercises are integrated throughout the text to build up
the readerfs mastery of concepts and techniques. Numerous technical illustrations
elucidate both geometric constructions and the mechanics of spectral sequences and
other sophisticated techniques. Over fifty hauntingly captivating images by A. T. Fomenko
artistically render the wondrous beauty, and mystery, of the subject.