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Recent books
P. G. Hjorth, C. L. Petersen, Eds.: Dynamics on the Riemann
Sphere: A Bodil Branner Festschrift, European Mathematical
Society, Zürich, 2006, 222 pp., EUR 68, ISBN 3-03719-011-6
The papers collected in this volume were written to celebrate
Bodil Branner’s 60th birthday. Most of them were presented at
the ‘Bodil Fest’, a symposium on holomorphic dynamics held
in June 2003 in Holbæk, Denmark. The main research theme of
Bodil Branner is the iteration of cubic polynomials. Together
with John H. Hubbard, she described the global topology of the
parameter space C 2 of polynomials of the form P a,b (z) = z 3 – a 2 z
+ b. Several decompositions of the parameter space have been
considered. The fi rst splitting is to separate the connectedness
locus (where both critical points have bounded orbit and the
Julia set is therefore connected) from the escape locus (where
at least one critical point escapes to infi nity). The second splitting
is to foliate the escape locus into different hyper-surfaces,
each one corresponding to a fi xed maximal escape rate of the
critical points. A particular way of constructing Teichmüller almost
complex structures, which are invariant under Pa,b, was
introduced as wringing and stretching of the complex structure;
this technique is now referred to as Branner-Hubbard motion.
In the volume, Branner-Hubbard motion is described in the papers
of C. L. Petersen, Tan Lei, and A. Douady. A survey paper
of J. Milnor treats Lattès maps. A. Avila and M. Lyubich give
examples of infi nitely renormalizable quadratic maps whose
Julia sets have Hausdorff dimension arbitrarily close to one. A.
Chéritat studies the linearizability of the family P θ (z) = e2πiθz
+ z 2 . Two papers (written by P. Blanchard et al. and P. Roesch)
treat the parameter space of the family f λ (z) = z 2 + λ/z 2 . T. Kawahira
studies small perturbations of geometrically fi nite maps
into other geometrically fi nite maps that are (semi)-conjugate
on the Julia set to the original map. W. Jung constructs by quasiconformal
surgery a class of homeomorphisms of subsets of the
Mandelbrot set. N. Fagella and Ch. Henriksen study Arnold
tongues in the complexifi cation of analytic diffeomorphisms
f a,t (x) = x + t + (a/2π) sin(2πx) of the circle. (pku)
A. Kanel-Belov, L. H. Rowen: Computational Aspects of Polynomial
Identities, Research Notes in Mathematics, vol. 9, A.K.
Peters, Wellesley, 2005, 378 pp., USD 69, ISBN 1-56881-163-2
In 1950 Wilhelm Specht asked whether every T-ideal of a free
F-algebra on the set of letters X is fi nitely based. An affi rmative
answer implies that every identity of a PI-algebra is a consequence
of a fi nite set of identities. Various aspects of the Specht
problem have been studied by many researchers. The main purpose
of this monograph is to present a complete and concise
answer to the Specht question.
The fi rst six chapters of the book develop tools needed for
the Kemer proof of the Specht conjecture in characteristic zero.
The choice and exposition of the topics of this part of the book
(such as the Schirschov height theorem, the Razmyslov-Kemer-Braun
theorem for the nilpotency of the Jacobson radical
of fi nitely generated PI-algebras, the development of Kemer
polynomials, and the theory of Grassmann algebras and their
connection to superalgebras) is guided by this aim. Chapter 7
contains counterexamples to the Specht question in positive
characteristics and related theory. The fi nal fi ve chapters of the
monograph are devoted to further development of themes that
were established on the way to the Kemer proof. This part of the
book deals with notions of Hilbert series and Gelfand-Kirillov
dimension as well as the theory of cocharacters, trace identities,
and the general theory of identities. The book ends with exercises,
the list of main theorems, examples and counterexamples,
and a list of open problems. All topics of the monograph are
well-arranged and developed in a clear way. The book is suitable
not only as a useful reference for researchers but also as
part of a course on PI-algebras for graduate students. (jzem)
W. Kocay, D.L. Kreher: Graphs, Algorithms, and Optimization,
Discrete Mathematics and its Applications, Chapman
& Hall/CRC, Boca Raton, 2004, 482 pp., USD 89,95, ISBN 1-
58488-396-0
The authors decided to write a textbook that would introduce
the realm of graph theory to computer science students.
In many ways they have undoubtedly succeeded. Rigorously
stated mathematical theorems and proofs continuously interlace
with tables of programming codes formally describing the
algorithms presented. Spice is added to the menu now and then
by slight detours to the theory of NP-completeness and a few
samples of polynomial reductions, a few more advanced techniques
of proof exploiting linear algebra, and a slightly informal
but illustrative exposition of embeddings of graphs on surfaces
of higher genus. The topics covered include basic notions
of degree sequences, trees, cycles and connectivity, all illustrated
from an algorithmic point of view. Further topics include
matchings and Hamilton cycles, network fl ows and Menger
theorems, and graph colorings. The relatively slow pace of the
introductory chapters becomes a bit faster towards the end of
the textbook and in the end the fi nal chapters face an ambitious
task of introducing linear programming to the reader. It is hard
to imagine that all the material could be covered in a single
one-semester course but, if the lecturer invests just a little bit of
effort in careful selection, this textbook can be used to teach a
very nice course on computer science. (jkrat)
S. G. Krantz: Geometric Function Theory – Explorations in
Complex Analysis, Cornerstones, Birkhäuser, Boston, 2006, 314
pp., EUR 58, ISBN 0-8176-4339-7
Standard courses in complex analysis do not guide students to
choose this seemingly closed subject as their research topic.
However, this book presents one part of modern complex analysis:
geometric function theory, not as a closed subject but instead
as a fruitful synthesis of many different areas: functional
analysis, invariant geometry, harmonic analysis, differential geometry,
partial differential equations, the automorphism groups
of domains and abstract algebra. It will capture the imagination
of advanced undergraduate and graduate students and
other mathematicians and it shows how to formulate problems.
Moreover, the book is nicely-written, readable and a methodologically
designed showroom of modern mathematics. It contains
many exercises, examples and an extensive bibliography.
The statement, “There is hardly any other analysis text that offers
such a variety and synthesis of mathematical topics”, (cited
from the preface) is not exaggerated. The book will be useful
for a wide range of students and mathematicians. (zv)
S. B. Kuksin: Randomly Forced Nonlinear PDE’s and Statistical
Hydrodynamics in 2 Space Dimensions, Zurich Lectures
in Advanced Mathematics, European Mathematical Society,
Zürich, 2006, 92 pp., EUR 28, ISBN 3-03719-021-3
52 EMS Newsletter December 2006