Expected publication date is August 29, 2003
Description
The volume is from the proceedings of the
international
conference held in celebration of Stanley
Osher's sixtieth
birthday. It presents recent developments
and exciting new
directions in scientific computing and partial
differential
equations for time dependent problems and
its interplay with
other fields, such as image processing, computer
vision and
graphics. Over the past decade, there have
been very rapid
developments in the field. This volume emphasizes
the strong
interaction of advanced mathematics with
real-world applications
and algorithms.
The book is suitable for graduate students
and research
mathematicians interested in scientific computing
and partial
differential equations.
Contents
Y. Brenier -- Systems of particles involving
permutations and
their continuous limits
R. E. Caflisch and D. G. Meyer -- A reduced
order model for
epitaxial growth
T. F. Chan and J. Shen -- On the role of
the BV image model in
image restoration
L.-T. Cheng and W. E -- The heterogeneous
multi-scale method for
interface dynamics
A. Ditkowski and D. Gottlieb -- On the Engquist
Majda absorbing
boundary conditions for hyperbolic systems
M.-H. Giga and Y. Giga -- Minimal vertical
singular diffusion
preventing diffusion preventing overturning
for the Burgers
equation
K. Harriman, P. Houston, B. Senior, and E.
Suli -- $hp$-version
discontinuous Galerkin methods with interior
penalty for partial
differential equations with nonnegative characteristic
form
D. Li and F. J. Hickernell -- Trigonometric
spectral collocation
methods on lattices
L. Ju, M. D. Gunzburger, and L. S. Hou --
Approximation of exact
boundary controllability problems for the
1-D wave equation by
optimization-based methods
P. Ming and Z.-c. Shi -- Some low order quadrilateral
Reissner-Mindlin
plate elements
H. Tang and T. Tang -- Multi-dimensional
moving mesh methods for
shock computations
D. D. Vvedensky, C. Baggio, A. Chua, C. Haselwandter,
and R.
Vardavas -- Stochastic differential equations
for driven lattice
systems
T. Yabe, K. Takizawa, F. Xiao, and A. Ikehata
-- Universal solver
CIP for all phases of matter
Details:
Series: Contemporary Mathematics, Volume:
330
Publication Year: 2003
ISBN: 0-8218-3155-0
Paging: 222 pp.
Binding: Softcover
Expected publication date is September 18,
2003
Description
This thorough and detailed exposition is
the result of an
intensive month-long course on mirror symmetry
sponsored by the
Clay Mathematics Institute. It develops mirror
symmetry from both
mathematical and physical perspectives with
the aim of furthering
interaction between the two fields. The material
will be
particularly useful for mathematicians and
physicists who wish to
advance their understanding across both disciplines.
Mirror symmetry is a phenomenon arising in
string theory in which
two very different manifolds give rise to
equivalent physics.
Such a correspondence has significant mathematical
consequences,
the most familiar of which involves the enumeration
of
holomorphic curves inside complex manifolds
by solving
differential equations obtained from a "mirror"
geometry. The inclusion of D-brane states
in the equivalence has
led to further conjectures involving calibrated
submanifolds of
the mirror pairs and new (conjectural) invariants
of complex
manifolds: the Gopakumar-Vafa invariants.
This book gives a single, cohesive treatment
of mirror symmetry.
Parts 1 and 2 develop the necessary mathematical
and physical
background from "scratch". The
treatment is focused,
developing only the material most necessary
for the task. In
Parts 3 and 4 the physical and mathematical
proofs of mirror
symmetry are given. From the physics side,
this means
demonstrating that two different physical
theories give
isomorphic physics. Each physical theory
can be described
geometrically, and thus mirror symmetry gives
rise to a "pairing"
of geometries. The proof involves applying
$R\leftrightarrow 1/R$
circle duality to the phases of the fields
in the gauged linear
sigma model. The mathematics proof develops
Gromov-Witten theory
in the algebraic setting, beginning with
the moduli spaces of
curves and maps, and uses localization techniques
to show that
certain hypergeometric functions encode the
Gromov-Witten
invariants in genus zero, as is predicted
by mirror symmetry.
Part 5 is devoted to advanced topics in mirror
symmetry,
including the role of D-branes in the context
of mirror symmetry,
and some of their applications in physics
and mathematics:
topological strings and large $N$ Chern-Simons
theory; geometric
engineering; mirror symmetry at higher genus;
Gopakumar-Vafa
invariants; and Kontsevich's formulation
of the mirror phenomenon
as an equivalence of categories.
This one-of-a-kind book is suitable for graduate
students and
research mathematicians interested in mathematics
and
mathematical and theoretical physics.
Titles in this series are published by the
AMS for the Clay
Mathematics Institute (Cambridge, MA).
Contents
Part 1. Mathematical Preliminaries
Differential geometry
Algebraic geometry
Differential and algebraic topology
Equivariant cohomology and fixed-point theorems
Complex and Kahler geometry
Calabi-Yau manifolds and their moduli
Toric geometry for string theory
Part 2. Physics Preliminaries
What is a QFT?
QFT in $d=0$
QFT in dimension 1: Quantum mechanics
Free quantum field theories 1 + 1 dimensions
$\mathcal{N} = (2,2)$ supersymmetry
Non-linear sigma models and Landau-Ginzburg
models
Renormalization group flow
Linear sigma models
Chiral rings and topological field theory
Chiral rings and the geometry of the vacuum
bundle
BPS solitons in $\mathcal{N}=2$ Landau-Ginzburg
theories
D-branes
Part 3. Mirror Symmetry: Physics Proof
Proof of mirror symmetry
Part 4. Mirror Symmetry: Mathematics Proof
Introduction and overview
Complex curves (non-singular and nodal)
Moduli spaces of curves
Moduli spaces $\bar{\mathcal M}_{g,n}(X,\beta)$
of stable maps
Cohomology classes on $\bar{\mathcal M}_{g,n}$
and ($\bar{\mathcal
M})_{g,n}(X,\beta)$
The virtual fundamental class, Gromov-Witten
invariants, and
descendant invariants
Localization on the moduli space of maps
The fundamental solution of the quantum differential
equation
The mirror conjecture for hypersurfaces I:
The Fano case
The mirror conjecture for hypersurfaces II:
The Calabi-Yau case
Part 5. Advanced Topics
Topological strings
Topological strings and target space physics
Mathematical formulation of Gopakumar-Vafa
invariants
Multiple covers, integrality, and Gopakumar-Vafa
invariants
Mirror symmetry at higher genus
Some applications of mirror symmetry
Aspects of mirror symmetry and D-branes
More on the mathematics of D-branes: Bundles,
derived categories
and Lagrangians
Boundary $\mathcal{N}=2$ theories
References
Bibliography
Index
Details:
Series: Clay Mathematics Monographs,Volume:
1
Publication Year: 2003
ISBN: 0-8218-2955-6
Paging: 929 pp.
Binding: Hardcover
Expected publication date is September 28,
2003
Description
This book is the second of two proceedings
volumes stemming from
the International Conference and Workshop
on Valuation Theory
held at the University of Saskatchewan (Saskatoon,
SK, Canada).
It contains the most recent applications
of valuation theory to a
broad range of mathematical ideas. Valuation
theory arose in the
early part of the twentieth century in connection
with number
theory and continues to have many important
applications to
algebra, geometry, and analysis.
The research and survey papers in this volume
cover a variety of
topics, including Galois theory, the Grunwald-Wang
Theorem,
algebraic geometry, resolution of singularities,
curves over
Prufer domains, model theory of valued fields
and the Frobenius,
Hardy fields, Hensel's Lemma, fixed point
theorems, and
computations in valued fields.
It is suitable for graduate students and
research mathematicians
interested in algebra, algebraic geometry,
number theory, and
mathematical logic.
Contents
K. Aghigh and S. K. Khanduja -- A note on
tame fields
M. Aschenbrenner -- Some remarks about asymptotic
couples
H. H. Brungs, H. Marubayashi, and E. Osmanagic
-- Prime segments
for cones and rings
V. Cossart and G. Moreno-Socias -- Irreducibility
criterion: A
geometric point of view
J. Denef and H. Schoutens -- On the decidability
of the
existential theory of ${\mathbb F_p}[[t]]$
W. Gao, D. B. Leep, J. Minac, and T. L. Smith
-- Galois groups
over nonrigid fields
B. Green -- Automorphisms of formal power
series rings over a
valuation ring
H. Knaf -- Regular curves over Prufer domains
J. Koenigsmann -- Encoding valuations in
absolute Galois groups
F.-V. Kuhlmann, H. Lombardi, and H. Perdry
-- Dynamic
computations inside the algebraic closure
of a valued field
G. Leloup -- Preorders, rings, lattice-ordered
groups and formal
power series
F. Lorenz and P. Roquette -- The theorem
of Grunwald-Wang in the
setting of valuation theory
R. I. Michler -- Invariants of singular plane
curves
J. Ohm -- $\mathcal V$-rational fields
H. Perdry -- A generalization of Hensel's
lemma
F. Pop -- Classically projective groups and
pseudo classically
closed fields
P. Popescu-Pampu -- Approximate roots
T. Scanlon -- Quantifier elimination for
the relative Frobenius
E. Schorner -- Ultrametric fixed point theorems
and applications
B. Teissier -- Valuations, deformations,
and toric geometry
Details:
Series: Fields Institute Communications,
Volume: 33
Publication Year: 2003
ISBN: 0-8218-3206-9
Paging: 459 pp.
Binding: Hardcover
Expected publication date is October 12,
2003
Description
The book contains survey and research articles
devoted mainly to
geometry and harmonic analysis of symmetric
spaces and to
corresponding aspects of group representation
theory. The volume
is dedicated to the memory of Russian mathematician,
F. I.
Karpelevich (1927-2000).
Of particular interest are the survey articles
by Sawyer on the
Abel transform on noncompact Riemannian symmetric
spaces, and by
Anker and Ostellari on estimates for heat
kernels on such spaces,
as well as the article by Bernstein and Gindikin
on integral
geometry for families of curves. There are
also many research
papers on topics of current interest.
The book is suitable for graduate students
and research
mathematicians interested in harmonic analysis
and representation
theory.
Contents
D. N. Akhiezer -- Asymptotic distribution
of eigenvalues for
certain elements of the group ring of a compact
Lie group
D. V. Alekseevsky and A. J. Di Scala -- Minimal
homogeneous
submanifolds of symmetric spaces
J.-P. Anker and P. Ostellari -- The heat
kernel on noncompact
symmetric spaces
E. M. Baruch, I. Piatetski-Shapiro, and S.
Rallis -- On the
uniqueness of Fourier Jacobi models for representations
of $U(2,1)$
J. Bernstein and S. Gindikin -- Notes on
integral geometry for
manifolds of curves
B. Enriquez and P. Etingof -- Quantization
of Alekseev-Meinrenken
dynamical $r$-matrices
J. Faraut -- Analysis on the crown of a Riemannian
symmetric
space
I. Gelfand, V. Retakh, and R. L. Wilson --
Quaternionic
quasideterminants and determinants
S. Gindikin -- Product-formula for $c$-functions
and inverse
horospherical transform
J. Hilgert, A. Pasquale, and E. B. Vinberg
-- The dual
horospherical Radon transform as a limit
of spherical Radon
transforms
A. W. Knapp -- The Gindikin-Karpelevic formula
and intertwining
operators
T. Kobayashi and S. Nasrin -- Multiplicity
one theorem in the
orbit method
B. Krotz and G. Olafsson -- The c-function
for non-compactly
causal symmetric spaces and its relations
to harmonic analysis
and representation theory
I. G. Macdonald -- A formal identity for
affine root systems
V. F. Molchanov -- Canonical representations
and overgroups
Y. A. Neretin -- Pencils of geodesics in
symmetric spaces,
Karpelevich boundary, and associahedron-like
polyhedra
M. A. Olshanetsky and V.-B. K. Rogov -- Poisson
formula for a
family of non-commutative Lobachevsky spaces
A. L. Onishchik -- Real semisimple Lie algebras
and their
representations
T. Oshima -- A calculation of $c$-function
for semisimple
symmetric spaces
P. Sawyer -- The Abel transform on symmetric
spaces of noncompact
type
Details:
Series: American Mathematical Society Translations--Series
2,
Volume: 210
Publication Year: 2003
ISBN: 0-8218-3472-X
Paging: 355 pp.
Binding: Hardcover