A publication of Higher Education Press (Beijing)
Felix Klein's famous Erlangen program made the theory of group actions
into a central part of mathematics. In the spirit of this program, Klein
set out to write a grand series of books which unified many different subjects
of mathematics, including number theory, geometry, complex analysis, and
discrete subgroups.
The first book on icosahedron and the solution of equations of the fifth
degree showed closed relations between three seemingly different subjects:
the symmetries of the icosahedron, the solution to fifth degree algebraic
equations, and the differential equation of hypergeometric functions. It
was translated into English in 1888, four years after its original German
version was published in 1884. It was followed by two volumes on elliptic
modular functions by Klein
and Fricke and two more volumes on automorphic functions also by Klein and Fricke.
These four classic books are vast generalizations of the first volume and contain the highly original works of Poincare and Klein on automorphic forms.
They have been very influential in the development of mathematics and are
now available in English for the first time. These books contain many original
ideas, striking examples, explicit computations, and details which are
not available anywhere else. They will be very valuable references for
people at all levels and allow the reader to see the unity of mathematics
through the eyes of one of the most influential mathematicians with vision,
Felix Klein.
Classical Topics in Mathematics Volume: 1
2017; 639 pp; Hardcover
MSC: Primary 11;
Print ISBN: 978-7-04-047872-3
Graduate students and research mathematicians interested in the theory elliptic modular functions.
Classical Topics in Mathematics, Volume: 2
2017; 589 pp; Hardcover
Print ISBN: 978-7-04-047837-2
Felix Klein's famous Erlangen program made the theory of group actions
into a central part of mathematics. In the spirit of this program, Klein
set out to write a grand series of books which unified many different subjects
of mathematics, including number theory, geometry, complex analysis, and
discrete subgroups.
The first book on icosahedron and the solution of equations of the fifth
degree showed closed relations between three seemingly different subjects:
the symmetries of the icosahedron, the solution to fifth degree algebraic
equations, and the differential equation of hypergeometric functions. It
was translated into English in 1888, four years after its original German
version was published in 1884. It was followed by two volumes on elliptic
modular functions by Klein and Fricke and two more volumes on automorphic
functions also by Klein and Fricke.
These four classic books are vast generalizations of the first volume and
contain the highly original works of Poincare and Klein on automorphic
forms. They have been very influential in the development of mathematics
and are now available in English for the first time. These books contain
many original ideas, striking examples, explicit computations, and details
which are not available anywhere else. They will be very valuable references
for people at all levels and allow the reader to see the unity of mathematics
through the eyes of one of the most influential mathematicians with vision,
Felix Klein.
Graduate students and research mathematicians interested in the theory of elliptic modular functions.
Miscellaneous Books
2017; 243 pp; Softcover
Print ISBN: 978-1-4704-3658-2
This is the second edition of a book originally published in 1997. Today
the internet virtually consumes all of our lives (especially the lives
of writers). As both readers and writers, we are all aware of blogs, chat
rooms, and preprint servers. There are now electronic-only journals and
print-on-demand books, Open Access journals and joint research projects
such as MathOverflow?not to mention a host of other new realities. It truly
is a brave new world, one that can be overwhelming and confusing. The truly
new feature of this second edition is an extensive discussion of technological
developments. Similar to the first edition, Krantz's frank and straightforward
approach makes this book particularly suitable as a textbook for an undergraduate
course.
Undergraduate and graduate students and researchers interested in how to write about mathematics.
Krantz, a prolific and distinguished mathematical author, discourses engagingly
(yet seriously) on the art and etiquette of virtually all types of writing
an academic mathematician is likely to encounter … Grammatical points,
stylistic and typesetting issues, and the correct and effective use of
mathematical notation are handled deftly and with good humor … [Hopefully]
senior faculty will consider it mandatory reading for graduate students
and even upper-division undergraduates. An enjoyable way to learn some
fundamentals of good mathematical writing. Highly recommended.
-- CHOICE
This monograph is a detailed survey of an area of differential geometry
surrounding the Bochner technique. This is a technique that falls under
the general heading of “curvature and topology” and refers to a method
initiated by Salomon Bochner in the 1940s for proving that on compact Riemannian
manifolds, certain objects of geometric interest (e.g., harmonic forms,
harmonic spinor fields, etc.) must satisfy additional differential equations
when appropriate curvature conditions are imposed.
In 1953 K. Kodaira applied this method to prove the vanishing theorem for
harmonic forms with values in a holomorphic vector bundle. This theorem,
which bears his name, was the crucial step that allowed him to prove his
famous imbedding theorem. Subsequently, the Bochner technique has been
extended, on the one hand, to spinor fields and harmonic maps and, on the
other, to harmonic functions and harmonic maps on noncompact manifolds.
The last has led to the proof of rigidity properties of certain Kahler
manifolds and locally symmetric spaces.
This monograph gives a self-contained and coherent account of some of these
developments, assuming the basic facts about Riemannian and Kahler geometry
as well as the statement of the Hodge theorem. The brief introductions
to the elementary portions of spinor geometry and harmonic maps may be
especially useful to beginners.
Classical Topics in Mathematics, Volume: 6
2017; 214 pp Hardcover
Print ISBN: 978-7-04-047838-9
Graduate students and research mathematicians interested in differential geometry.
This book is motivated by the problem of determining the set of rational
points on a variety, but its true goal is to equip readers with a broad
range of tools essential for current research in algebraic geometry and
number theory. The book is unconventional in that it provides concise accounts
of many topics instead of a comprehensive account of just one?this is intentionally
designed to bring readers up to speed rapidly. Among the topics included
are Brauer groups, faithfully flat descent, algebraic groups, torsors,
etale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and
descent obstructions. A final chapter applies all these to study the arithmetic
of surfaces.
The down-to-earth explanations and the over 100 exercises make the book
suitable for use as a graduate-level textbook, but even experts will appreciate
having a single source covering many aspects of geometry over an unrestricted
ground field and containing some material that cannot be found elsewhere.
Graduate Studies in Mathematics, Volume: 186
2017; 337 pp Hardcover
Print ISBN: 978-1-4704-3773-2
Graduate students and researchers interested in arithmetic geometry.
The origins of arithmetic (or Diophantine) geometry can be traced back
to antiquity, and it remains a lively and wide research domain up to our
days. The book by Bjorn Poonen, a leading expert in the field, opens doors
to this vast field for many readers with different experiences and backgrounds.
It leads through various algebraic geometric constructions towards its
central subject: obstructions to existence of rational points.
-- Yuri Manin, Max-Planck-Institute, Bonn
It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student.
-- Hendrik Lenstra, University LeidenUnderstanding rational points on arbitrary
algebraic varieties is the ultimate challenge. We have conjectures but
few results. Poonen's book, with its mixture of basic constructions and
openings into current research, will attract new generations to the Queen
of Mathematics.
-- Jean-Louis Colliot-Thelene, Universite Paris-Sud
A beautiful subject, handled by a master.
-- Joseph Silverman, Brown University