The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.
Undergraduates, graduate students, and research mathematicians interested in mathematics.
The authors manage to breathe new life into topics that at first glance appear to be old hat.
This book results from a unique and innovative program at Pennsylvania State
University. Under the program, the “best of the best” students
nationwide are chosen to study challenging mathematical areas under the
guidance of experienced mathematicians. This program, Mathematics Advanced
Study Semesters (MASS), offers an unparalleled opportunity for talented
undergraduate students who are serious in the pursuit of mathematical
knowledge.
This volume represents various aspects of the MASS program over its six-year
existence, including core courses, summer courses, students' research, and
colloquium talks. The book is most appropriate for college professors of
mathematics who work with bright and eager undergraduate and beginning graduate
students, for such students who want to expand their mathematical horizons, and
for everyone who loves mathematics and wants to learn more interesting and
unusual material.
The first half of the book contains lecture notes of nonstandard courses. A
text for a semester-long course on \(p\)-adic analysis is centered
around contrasts and similarities with its real counterpart. A shorter text
focuses on a classical area of interplay between geometry, algebra and number
theory (continued fractions, hyperbolic geometry and quadratic forms). Also
provided are detailed descriptions of two innovative courses, one on geometry
and the other on classical mechanics. These notes constitute what one may call
the skeleton of a course, leaving the instructor ample room for innovation and
improvisation.
The second half of the book contains a large collection of essays on a broad
spectrum of exciting topics from Hilbert's Fourth Problem to geometric
inequalities and minimal surfaces, from mathematical billiards to fractals and
tilings, from unprovable theorems to the classification of finite simple groups
and lexicographic codes.
Professors of mathematics; general mathematical audience.
There is a tradition in Russia that holds that mathematics
can be both challenging and fun. One fine outgrowth of that tradition
is the magazine, Kvant, which has been enjoyed by many of the
best students since its founding in 1970. The articles in
Kvant assume only a minimal background, that of a good high
school student, yet are capable of entertaining mathematicians of
almost any level. Sometimes the articles require careful thought or a
moment's work with a pencil and paper. However, the industrious reader
will be generously rewarded by the elegance and beauty of the
subjects.
This book is the third collection of articles from
Kvant to be published by the AMS. The volume is devoted
mainly to combinatorics and discrete mathematics. Several of the
topics are well known: nonrepeating sequences, detecting a counterfeit
coin, and linear inequalities in economics, but they are discussed
here with the entertaining and engaging style typical of the
magazine. The two previous collections treat aspects of algebra and
analysis, including connections to number theory and other
topics. They were published as Volumes 14 and 15 in the Mathematical
World series.
The articles are written so as to present genuine
mathematics in a conceptual, entertaining, and accessible way. The
books are designed to be used by students and teachers who love
mathematics and want to study its various aspects, deepening and
expanding upon the school curriculum.
Graduate students and research mathematicians interested in combinatorics and discrete mathematics.
Advanced high school and undergraduate students interested in mathematics; mathematics teachers in high schools and colleges.
This volume and
Articles selected for these two volumes are written by leading Russian mathematicians and expositors. Some articles contain classical mathematical gems still used in university curricula today. Others feature cutting-edge research from the twentieth century.
The articles in these books are written so as to present genuine mathematics in a conceptual, entertaining, and accessible way. The volumes are designed to be used by students and teachers who love mathematics and want to study its various aspects, thus deepening and expanding the school curriculum.
The first volume is mainly devoted to various topics in number theory, whereas the second volume treats diverse aspects of analysis and algebra.
Published in the very popular and relatively basic-level Mathematical World series. Features a low price and a colorful cover. This is kind of a “What's Happening in Russian Mathematics” (Russian mathematics are still considered to be among the best in the world). Second of two volumes on the topic.
MAWRLD/14
This volume and
Articles selected for these two volumes are written by leading
Russian mathematicians and expositors. Some articles contain classical
mathematical gems still used in university curricula today. Others
feature cutting-edge research from the twentieth century.
The articles in these books are written so as to present genuine
mathematics in a conceptual, entertaining, and accessible way. The
volumes are designed to be used by students and teachers who love
mathematics and want to study its various aspects, thus deepening and
expanding the school curriculum.
The first volume is mainly devoted to various
topics in number theory, whereas the second volume treats diverse
aspects of analysis and algebra.
Advanced high school and undergraduate students interested in mathematics; mathematics teachers in high schools and colleges.
The slate of the authors is most impressive … It becomes even more impressive in view of the fact that these ‘serious’ mathematicians went out of their way to make their sophisticated material understandable by a broad readership … a welcome edition to mathematics literature where rigor co-exists with fun and accessibility.
-- Zentralblatt MATH
Mathematical billiards describe the motion of a mass point in
a domain with elastic reflections off the boundary or, equivalently,
the behavior of rays of light in a domain with ideally reflecting
boundary. From the point of view of differential geometry, the
billiard flow is the geodesic flow on a manifold with boundary. This
book is devoted to billiards in their relation with differential
geometry, classical mechanics, and geometrical optics.
Topics covered include variational principles of billiard motion,
symplectic geometry of rays of light and integral geometry, existence
and nonexistence of caustics, optical properties of conics and
quadrics and completely integrable billiards, periodic billiard
trajectories, polygonal billiards, mechanisms of chaos in billiard
dynamics, and the lesser-known subject of dual (or outer)
billiards.
The book is based on an advanced undergraduate topics
course. Minimum prerequisites are the standard material covered in the
first two years of college mathematics (the entire calculus sequence,
linear algebra). However, readers should show some mathematical
maturity and rely on their mathematical common sense.
A unique feature of the book is the coverage of many diverse topics
related to billiards, for example, evolutes and involutes of plane
curves, the four-vertex theorem, a mathematical theory of rainbows,
distribution of first digits in various sequences, Morse theory, the
Poincaré recurrence theorem, Hilbert's fourth problem, Poncelet
porism, and many others. There are approximately 100
illustrations.
The book is suitable for advanced undergraduates, graduate
students, and researchers interested in ergodic theory and
geometry.
This book is published in cooperation with Mathematics Advanced Study Semesters
Advanced undergraduates, graduate students, and research mathematicians interested in ergodic theory and geometry.
(This book) is very well written, with nice illustrations. The author presents the results very clearly, with interesting digressions and he mentions applications of billiards to various fields.
-- Zentralblatt MATH
This volume presents contributions by leading experts in the
field. The articles are dedicated to D. B. Fuchs on the occasion of
his 60th birthday. Contributors to the book were directly influenced
by Professor Fuchs and include his students, friends, and professional
colleagues. In addition to their research, they offer personal
reminicences about Professor Fuchs, giving insight into the history of
Russian mathematics.
The main topics addressed in this unique work are infinite-dimensional Lie
algebras with applications (vertex operator algebras, conformal field theory,
quantum integrable systems, etc.) and differential topology. The volume
provides an excellent introduction to current research in the field.
Graduate students, research mathematicians, and physicists interested in algebraic geometry; theoretical physicists.
Vladimir Arnold, an eminent mathematician of our time, is known both
for his mathematical results, which are many and prominent, and for
his strong opinions, often expressed in an uncompromising and
provoking manner. His dictum that “Mathematics is a part of physics
where experiments are cheap” is well known.
This book consists of two parts: selected articles by and an
interview with Vladimir Arnold, and a collection of articles about him
written by his friends, colleagues, and students. The book is
generously illustrated by a large collection of photographs, some
never before published. The book presents many a facet of this
extraordinary mathematician and man, from his mathematical discoveries
to his daredevil outdoor adventures.
Mathematicians of all levels: teachers, researchers, graduate and undergraduate students, and other scientists interested in the recent history and ideology of mathematics.