IAS/Park City Mathematics Series, Volume: 11
2006; 285 pp; hardcover
ISBN-10: 0-8218-3431-2
ISBN-13: 978-0-8218-3431-2
Each summer the IAS/Park City Mathematics Institute Graduate
Summer School gathers some of the best researchers and educators
in a particular field to present diverse sets of lectures. This
volume presents three weeks of lectures given at the Summer
School on Quantum Field Theory, Supersymmetry, and Enumerative
Geometry, three very active research areas in mathematics and
theoretical physics.
With this volume, the Park City Mathematics Institute returns to
the general topic of the first institute: the interplay between
quantum field theory and mathematics. Two major themes at this
institute were supersymmetry and algebraic geometry, particularly
enumerative geometry. The volume contains two lecture series on
methods of enumerative geometry that have their roots in QFT. The
first series covers the Schubert calculus and quantum cohomology.
The second discusses methods from algebraic geometry for
computing Gromov-Witten invariants. There are also three sets of
lectures of a more introductory nature: an overview of classical
field theory and supersymmetry, an introduction to
supermanifolds, and an introduction to general relativity.
Readership
Graduate students and research mathematicians interested in
enumerative geometry and mathematical physics.
Table of Contents
W. Fulton, Enumerative geometry (with notes by Alastair Craw)
Enumerative geometry (with notes by Alastair Craw)
Bibliography
A. Bertram, Computing Gromov-Witten invariants with algebraic
geometry
Introduction and motivation
Localization
$J$-functions
An alternative to WDVV
Bibliography
D. S. Freed, Classical field theory and supersymmetry
Introduction
Classical mechanics
Lagrangian field theory and symmetries
Classical bosonic theories on Minkowski spacetime
Fermions and the supersymmetric particle
Free theories, quantization, and approximation
Supersymmetric field theories
Supersymmetric $\sigma$-models
Bibliography
J. W. Morgan, Introduction to supermanifolds
Introduction to supermanifolds
Bibliography
C. V. Johnson, Notes on introductory general relativity
Notes on introductory general relativity
Bibliography
SMF/AMS Texts and Monographs, Volume: 12
2006; 104 pp; softcover
ISBN-10: 0-8218-3164-X
ISBN-13: 978-0-8218-3164-9
In this text, the author presents a general framework for
applying the standard methods from homotopy theory to the
category of smooth schemes over a reasonable base scheme $k$. He
defines the homotopy category $h(\mathcal{E}_k)$ of smooth $k$-schemes
and shows that it plays the same role for smooth $k$-schemes as
the classical homotopy category plays for differentiable
varieties. It is shown that certain expected properties are
satisfied, for example, concerning the algebraic $K$-theory of
those schemes. In this way, advanced methods of algebraic
topology become available in modern algebraic geometry.
Readership
Graduate students and research mathematicians interested in
algebraic geometry and algebraic topology.
Table of Contents
Introduction
The homotopic category
Homotopic excision, homotopic purity and projective blow-ups
Homotopic classification of vector bundles
Review of homotopic algebra
Ample families of invertible bundles on a scheme
References
American Mathematical Society Translations--Series 2, Volume:
217
Advances in the Mathematical Sciences
2006; 246 pp; hardcover
ISBN-10: 0-8218-4208-0
ISBN-13: 978-0-8218-4208-9
This volume, devoted to the 70th birthday of the well-known St.
Petersburg mathematician A. M. Vershik, contains a collection of
articles by participants in the conference "Representation
Theory, Dynamical Systems, and Asymptotic Combinatorics",
held in St. Petersburg in June of 2004. The book is suitable for
graduate students and researchers interested in combinatorial and
dynamical aspects of group representation theory.
Readership
Graduate students and research mathematicians interested in
various aspects of group representation theory.
Table of Contents
V. Arnold -- Statistics of the symmetric group representations as
a natural science question on asymptotics of Young diagrams
A. Borodin and G. Olshanski -- Stochastic dynamics related to
Plancherel measure on partitions
A. Bufetov, Y. G. Sinai, and C. Ulcigrai -- A condition for
continuous spectrum of an interval exchange transformation
I. Fesenko -- Several nonstandard remarks
G. A. Freiman and A. A. Yudin -- The interface between
probability theory and additive number theory (local limit
theorems and structure theory of set addition)
V. Ivanov -- Plancherel measure on shifted Young diagrams
A. Joseph -- Results and problems in enveloping algebras arising
from quantum groups
V. Malyshev and A. Manita -- Asymptotic behaviour in the time
synchronization model
Y. A. Neretin -- Stable densities and operators of fractional
differentiation
W. Parry and M. Pollicott -- Skew products and Livsic theory
M. Pollicott and R. Sharp -- Distribution of ergodic sums for
hyperbolic maps
J. Renault -- Transverse properties of dynamical systems
B.-Z. Rubshtein -- Minimal one-sided Markov shifts and their
cofiltrations
K. Schmidt -- Quotients of $\ell^{\infty}(\mathbb{Z},\mathbb{Z})$
and symbolic covers of toral automorphisms
2006; approx. 305 pp; hardcover
ISBN-10: 0-8218-3750-8
ISBN-13: 978-0-8218-3750-4
Is there always a prime number between $n$ and $2n$? Where,
approximately, is the millionth prime? And just what does
calculus have to do with answering either of these questions? It
turns out that calculus has a lot to do with both questions, as
this book can show you.
The theme of the book is approximations. Calculus is a powerful
tool because it allows us to approximate complicated functions
with simpler ones. Indeed, replacing a function locally with a
linear--or higher order--approximation is at the heart of
calculus. The real star of the book, though, is the task of
approximating the number of primes up to a number $x$. This leads
to the famous Prime Number Theorem--and to the answers to the two
questions about primes.
While emphasizing the role of approximations in calculus, most
major topics are addressed, such as derivatives, integrals, the
Fundamental Theorem of Calculus, sequences, series, and so on.
However, our particular point of view also leads us to many
unusual topics: curvature, Pade approximations, public key
cryptography, and an analysis of the logistic equation, to name a
few.
The reader takes an active role in developing the material by
solving problems. Most topics are broken down into a series of
manageable problems, which guide you to an understanding of the
important ideas. There is also ample exposition to fill in
background material and to get you thinking appropriately about
the concepts.
Approximately Calculus is intended for the reader who has already
had an introduction to calculus, but wants to engage the concepts
and ideas at a deeper level. It is suitable as a text for an
honors or alternative second semester calculus course.
Readership
Undergraduate students interested in calculus and number theory.
Table of Contents
Patterns and induction
Divisibility
Primes
Derivatives and approximations of functions
Antiderivatives and integration
Distribution of primes
Log, exponential, and the inverse trigonometric functions
The mean value theorem and approximations
Linearization topics
Defining integrals, areas, and arclengths
Improper integrals and techniques of integration
The prime number theorem
Local approximation of functions and integral estimations
Sequences and series
Power series and Taylor series
More on series
Limits of functions
Differential equations
Logical arguments
Hints for selected problems
Bibliography
Index