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This book grew out of the notes for a one-semester basic graduate
course in probability. As the title suggests, it is meant to be an
introduction to probability and could serve as textbook for a year
long text for a basic graduate course. It assumes some familiarity
with measure theory and integration so in this book we emphasize
only those aspects of measure theory that have special
probabilistic uses.The book covers the topics that are part of the
culture of an aspiring probabilist and it is guided by the author's
personal belief that probability was and is a theory driven by
examples. The examples form the main attraction of this subject.
For this reason, a large book is devoted to an eclectic collection
of examples, from classical to modern, from mainstream to 'exotic'.
The text is complemented by nearly 200 exercises, quite a few
nontrivial, but all meant to enhance comprehension and enlarge the
reader's horizons.While teaching probability both at undergraduate
and graduate level the author discovered the revealing power of
simulations. For this reason, the book contains a veiled invitation
to the reader to familiarize with the programing language R. In the
appendix, there are a few of the most frequently used operations
and the text is sprinkled with (less than optimal) R codes.
Nowadays one can do on a laptop simulations and computations we
could only dream as an undergraduate in the past. This is a book
written by a probability outsider. That brings along a bit of
freshness together with certain 'naiveties'.
The goal of this book is to introduce the reader to some of the
main techniques, ideas and concepts frequently used in modern
geometry. It starts from scratch and it covers basic topics such as
differential and integral calculus on manifolds, connections on
vector bundles and their curvatures, basic Riemannian geometry,
calculus of variations, DeRham cohomology, integral geometry (tube
and Crofton formulas), characteristic classes, elliptic equations
on manifolds and Dirac operators. The new edition contains a new
chapter on spectral geometry presenting recent results which appear
here for the first time in printed form.
The goal of this book is to introduce the reader to some of the
main techniques, ideas and concepts frequently used in modern
geometry. It starts from scratch and it covers basic topics such as
differential and integral calculus on manifolds, connections on
vector bundles and their curvatures, basic Riemannian geometry,
calculus of variations, DeRham cohomology, integral geometry (tube
and Crofton formulas), characteristic classes, elliptic equations
on manifolds and Dirac operators. The new edition contains a new
chapter on spectral geometry presenting recent results which appear
here for the first time in printed form.
The goal of this book is to introduce the reader to some of the
most frequently used techniques in modern global geometry. Suited
to the beginning graduate student willing to specialize in this
very challenging field, the necessary prerequisite is a good
knowledge of several variables calculus, linear algebra and
point-set topology.The book's guiding philosophy is, in the words
of Newton, that "in learning the sciences examples are of more use
than precepts". We support all the new concepts by examples and,
whenever possible, we tried to present several facets of the same
issue.While we present most of the local aspects of classical
differential geometry, the book has a "global and analytical bias".
We develop many algebraic-topological techniques in the special
context of smooth manifolds such as Poincare duality, Thom
isomorphism, intersection theory, characteristic classes and the
Gauss-Bonnet theorem.We devoted quite a substantial part of the
book to describing the analytic techniques which have played an
increasingly important role during the past decades. Thus, the last
part of the book discusses elliptic equations, including elliptic
Lpand Hoelder estimates, Fredholm theory, spectral theory, Hodge
theory, and applications of these. The last chapter is an in-depth
investigation of a very special, but fundamental class of elliptic
operators, namely, the Dirac type operators.The second edition has
many new examples and exercises, and an entirely new chapter on
classical integral geometry where we describe some mathematical
gems which, undeservedly, seem to have disappeared from the
contemporary mathematical limelight.
The goal of this book is to introduce the reader to some of the
most frequently used techniques in modern global geometry. Suited
to the beginning graduate student willing to specialize in this
very challenging field, the necessary prerequisite is a good
knowledge of several variables calculus, linear algebra and
point-set topology.The book's guiding philosophy is, in the words
of Newton, that "in learning the sciences examples are of more use
than precepts". We support all the new concepts by examples and,
whenever possible, we tried to present several facets of the same
issue.While we present most of the local aspects of classical
differential geometry, the book has a "global and analytical bias".
We develop many algebraic-topological techniques in the special
context of smooth manifolds such as Poincare duality, Thom
isomorphism, intersection theory, characteristic classes and the
Gauss-Bonnet theorem.We devoted quite a substantial part of the
book to describing the analytic techniques which have played an
increasingly important role during the past decades. Thus, the last
part of the book discusses elliptic equations, including elliptic
Lpand Hoelder estimates, Fredholm theory, spectral theory, Hodge
theory, and applications of these. The last chapter is an in-depth
investigation of a very special, but fundamental class of elliptic
operators, namely, the Dirac type operators.The second edition has
many new examples and exercises, and an entirely new chapter on
classical integral geometry where we describe some mathematical
gems which, undeservedly, seem to have disappeared from the
contemporary mathematical limelight.
This is a text that develops calculus 'from scratch', with complete
rigorous arguments. Its aim is to introduce the reader not only to
the basic facts about calculus but, as importantly, to mathematical
reasoning. It covers in great detail calculus of one variable and
multivariable calculus. Additionally it offers a basic introduction
to the topology of Euclidean space. It is intended to more advanced
or highly motivated undergraduates.
This is a text that develops calculus 'from scratch', with complete
rigorous arguments. Its aim is to introduce the reader not only to
the basic facts about calculus but, as importantly, to mathematical
reasoning. It covers in great detail calculus of one variable and
multivariable calculus. Additionally it offers a basic introduction
to the topology of Euclidean space. It is intended to more advanced
or highly motivated undergraduates.
This is a state-of-the-art introduction to the work of Franz
Reidemeister, Meng Taubes, Turaev, and the author on the concept of
torsion and its generalizations. Torsion is the oldest topological
(but not with respect to homotopy) invariant that in its almost
eight decades of existence has been at the center of many important
and surprising discoveries. During the past decade, in the work of
Vladimir Turaev, new points of view have emerged, which turned out
to be the "right ones" as far as gauge theory is concerned. The
book features mostly the new aspects of this venerable concept. The
theoretical foundations of this subject are presented in a style
accessible to those, who wish to learn and understand the main
ideas of the theory. Particular emphasis is upon the many and
rather diverse concrete examples and techniques which capture the
subleties of the theory better than any abstract general result.
Many of these examples and techniques never appeared in print
before, and their choice is often justified by ongoing current
research on the topology of surface singularities. The text is
addressed to mathematicians with geometric interests who want to
become comfortable users of this versatile invariant.
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