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Pluripotential theory is a very powerful tool in geometry, complex
analysis and dynamics. This volume brings together the lectures
held at the 2011 CIME session on "pluripotential theory" in
Cetraro, Italy. This CIME course focused on complex Monge-Ampére
equations, applications of pluripotential theory to Kahler geometry
and algebraic geometry and to holomorphic dynamics. The
contributions provide an extensive description of the theory and
its very... more...

Since Benoit Mandelbrot's pioneering work in the late 1970s,
scores of research articles and books have
been published on the topic of fractals. Despite
the volume of literature in the field, the general
level of theoretical understanding has remained low; most work is
aimed either at too mainstream an audience to achieve any depth
or at too specialized a community to achieve widespread
use. Written... more...

This book offers a unique opportunity to understand the essence
of one of the great thinkers of western civilization. A guided
reading of Euclid's Elements leads to a critical discussion and
rigorous modern treatment of Euclid's geometry and its more
recent descendants, with complete proofs. Topics include the
introduction of coordinates, the theory of area, history of the
parallel postulate, the various non-Euclidean geometries, and... more...

Harmonic maps between Riemannian manifolds were first established
by James Eells and Joseph H. Sampson in 1964. Wave maps are
harmonic maps on Minkowski spaces and have been studied since the
1990s. Yang-Mills fields, the critical points of Yang-Mills
functionals of connections whose curvature tensors are harmonic,
were explored by a few physicists in the 1950s, and biharmonic maps
(generalizing harmonic maps) were introduced by Guoying Jiang in
1986.... more...

Seki was a Japanese mathematician in the seventeenth century
known for his outstanding achievements, including the elimination
theory of systems of algebraic equations, which preceded the
works of Étienne Bézout and Leonhard Euler by 80 years. Seki was
a contemporary of Isaac Newton and Gottfried Wilhelm Leibniz,
although there was apparently no direct interaction between them.
The Mathematical Society of Japan and the History... more...

This book contains a detailed account of the result of the author's
recent Annals paper and JAMS paper on arithmetic invariant,
including μ-invariant, L-invariant, and similar
topics. This book can be regarded as an introductory
text to the author's previous book p-Adic Automorphic Forms on
Shimura Varieties. Written as a down-to-earth
introduction to Shimura varieties, this text includes many examples
and applications of the theory... more...

In the spring of 1976, George Andrews of Pennsylvania State
University visited the library at Trinity College, Cambridge, to
examine the papers of the late G.N. Watson. Among these papers,
Andrews discovered a sheaf of 138 pages in the handwriting of
Srinivasa Ramanujan. This manuscript was soon designated,
"Ramanujan's lost notebook." Its discovery has frequently been
deemed the mathematical equivalent of finding Beethoven's... more...

In recent years, research in K3 surfaces and Calabi–Yau varieties
has seen spectacular progress from both arithmetic and geometric
points of view, which in turn continues to have a huge influence
and impact in theoretical physics—in particular, in string
theory. The workshop on Arithmetic and Geometry of K3
surfaces and Calabi–Yau threefolds, held at the Fields Institute
(August 16-25, 2011), aimed to give a... more...

Diophantine geometry has been studied by number theorists for
thousands of years, since the time of Pythagoras, and has continued
to be a rich area of ideas such as Fermat's Last Theorem, and most
recently the ABC conjecture. This monograph is a bridge between the
classical theory and modern approach via arithmetic geometry. The
authors provide a clear path through the subject for graduate
students and researchers. They have re-examined many results... more...

This book provides a tour of the principal areas and methods of
modern differential geometry. Beginning at the introductory level
with curves in Euclidian space, the sections become more
challenging, arriving finally at the advanced topics that form
the greatest part of the book: transformation groups, the
geometry of differential equations, geometric structures, the
equivalence problem, the geometry of elliptic operators.

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