Sheldon Katz, University of Illinois at Urbana-Champaign, IL
Enumerative Geometry and String Theory
Student Mathematical Library, Volume: 32
2006; approx. 214 pp; softcover
ISBN: 0-8218-3687-0
Perhaps the most famous example of how ideas from modern physics
have revolutionized mathematics is the way string theory has led
to an overhaul of enumerative geometry, an area of mathematics
that started in the eighteen hundreds. Century-old problems of
enumerating geometric configurations have now been solved using
new and deep mathematical techniques inspired by physics!
The book begins with an insightful introduction to enumerative
geometry. From there, the goal becomes explaining the more
advanced elements of enumerative algebraic geometry. Along the
way, there are some crash courses on intermediate topics which
are essential tools for the student of modern mathematics, such
as cohomology and other topics in geometry.
The physics content assumes nothing beyond a first undergraduate
course. The focus is on explaining the action principle in
physics, the idea of string theory, and how these directly lead
to questions in geometry. Once these topics are in place, the
connection between physics and enumerative geometry is made with
the introduction of topological quantum field theory and quantum
cohomology.
Readership
Undergraduate and graduate students interested in algebraic
geometry or in mathematical physics.
Table of Contents
Warming up to enumerative geometry
Enumerative geometry in the projective plane
Stable maps and enumerative geometry
Crash course in topology and manifolds
Crash course in $C^\infty$ manifolds and cohomology
Cellular decompositions and line bundles
Enumerative geometry of lines
Excess intersection
Rational curves on the quintic threefold
Mechanics
Introduction to supersymmetry
Introduction to string theory
Topological quantum field theory
Quantum cohomology and enumerative geometry
Bibliography
Index
Edited by: Katrin Becker and Melanie Becker, Texas A&M University, College Station, TX, Aaron Bertram,
University of Utah, Salt Lake City, UT, Paul S. Green, University of Maryland,
College Park, MD, and Benjamin McKay, University College, Cork, Ireland
Snowbird Lectures on String Geometry
Contemporary Mathematics Volume: 401
2006; 104 pp; softcover
ISBN: 0-8218-3663-3
The interaction and cross-fertilization of mathematics and
physics is ubiquitous in the history of both disciplines. In
particular, the recent developments of string theory have led to
some relatively new areas of common interest among mathematicians
and physicists, some of which are explored in the papers in this
volume. These papers provide a reasonably comprehensive sampling
of the potential for fruitful interaction between mathematicians
and physicists that exists as a result of string theory.
Readership
Graduate students and research mathematicians interested in
mathematical physics.
Table of Contents
Burkard Polster, Monash University, Clayton, Victoria, Australia
The Shoelace Book
A Mathematical Guide to the Best (and Worst) Ways to Lace Your
Shoes
Mathematical World Volume: 24
2006; 125 pp; softcover
ISBN: 0-8218-3933-0
Crisscross, zigzag, bowtie, devil, angel, or star: which are the
longest, the shortest, the strongest, and the weakest lacings?
Pondering the mathematics of shoelaces, the author paints a vivid
picture of the simple, beautiful, and surprising
characterizations of the most common shoelace patterns. The
mathematics involved is an attractive mix of combinatorics and
elementary calculus. This book will be enjoyed by mathematically
minded people for as long as there are shoes to lace.
Burkard Polster is a well-known mathematical juggler, magician,
origami expert, bubble-master, shoelace charmer, and "Count
von Count" impersonator. His previous books include A
Geometrical Picture Book, The Mathematics of Juggling, and QED:
Beauty in Mathematical Proof.
Readership
General mathematical audience interested in the mathematics of
lacing.
Table of Contents
Setting the stage
One-column lacings
Counting lacings
The shortest lacings
Variations on the shortest lacing problem
The longest lacings
The strongest lacings
The weakest lacings
Related mathematics
Loose ends
References
Index
Edited by: Zvi Arad, Bar Ilan University, Ramat-Gan, Israel, Mariagrazia Bianchi, Universita degli Studi
di Milano, Italy, Wolfgang Herfort, University of Technology, Vienna, Austria,
Patrizia Longobardi and Mercede Maj, Universita di Salerno, Fisciano, (SA),
Italy, and Carlo Scoppola
Ischia Group Theory 2004
Proceedings of a Conference in honour of Marcel Herzog
Contemporary Mathematics Volume: 402
2006; approx. 274 pp; softcover
ISBN: 0-8218-3711-7
Experts in the theory of finite groups and in representation
theory provide insight into various aspects of group theory, such
as the classification of finite simple groups, character theory,
groups with special properties, table algebras, etc.
Readership
Graduate students and research mathematicians interested in group
theory.
Table of Contents
Z. Arad and W. Herfort -- The history of the classification of
finite groups with a CC-subgroup
Y. Berkovich and Z. Janko -- Structure of finite $p$-groups with
given subgroups
E. A. Bertram -- Lower bounds for the number of conjugacy classes
in finite groups
M. Bianchi, A. Gillio, and L. Verardi -- Monounary simple
algebras
D. Chillag -- Algebras with positive bases, commutators and
covering numbers
C. Delizia and C. Nicotera -- On certain group theoretical
properties generalizing commutativity
E. Detomi and A. Lucchini -- Probabilistic non-generators in
profinite groups
M. Giudici, C. H. Li, C. E. Praeger, A. Seress, and V. Trofimov
-- Limits of vertex-transitive graphs
R. Gobel and O. H. Kegel -- Group rings with simple augmentation
ideals
M. Herzog, P. Longobardi, and M. Maj -- On the number of
commutators in groups
Z. Janko -- New results in the theory of finite 2-groups
G. Kaplan -- The existence of normal and characteristic subgroups
in finite groups
A. Lev -- On the covering numbers of finite groups: Some old and
new results
M. Mainardis -- Normal subgroups in the subgroup lattices of
finite $p$-groups
A. Mann -- On characters-classes duality and orders of
centralizers
A. Regev -- Bijections for identities of multisets of hook
numbers
D. O. Revin and E. P. Vdovin -- Hall subgroups of finite groups
Lars V. Ahlfors
Lectures on Quasiconformal Mappings: Second edition
University Lecture Series Volume: 38
2006; approx. 156 pp; softcover
ISBN: 0-8218-3644-7
Lars Ahlfors's Lectures on Quasiconformal Mappings, based on a
course he gave at Harvard University in the spring term of 1964,
was first published in 1966 and was soon recognized as the
classic it was shortly destined to become. These lectures develop
the theory of quasiconformal mappings from scratch, give a self-contained
treatment of the Beltrami equation, and cover the basic
properties of Teichmuller spaces, including the Bers embedding
and the Teichmuller curve. It is remarkable how Ahlfors goes
straight to the heart of the matter, presenting major results
with a minimum set of prerequisites. Many graduate students and
other mathematicians have learned the foundations of the theories
of quasiconformal mappings and Teichmuller spaces from these
lecture notes.
This edition includes three new chapters. The first, written by
Earle and Kra, describes further developments in the theory of
Teichmuller spaces and provides many references to the vast
literature on Teichmuller spaces and quasiconformal mappings. The
second, by Shishikura, describes how quasiconformal mappings have
revitalized the subject of complex dynamics. The third, by
Hubbard, illustrates the role of these mappings in Thurston's
theory of hyperbolic structures on 3-manifolds. Together, these
three new chapters exhibit the continuing vitality and importance
of the theory of quasiconformal mappings.
Readership
Graduate students and research mathematicians interested in
complex analysis.
Table of Contents
Differentiable quasiconformal mappings
The general definition
Extremal geometric properties
Boundary correspondence
The mapping theorem
Teichmuller spaces
A supplement to Ahlfors's lectures
Complex dynamics and quasiconformal mappings
Hyperbolic structures on three-manifolds that fiber over the
circle