TEXTBOOK
To Be Published: 11 June 2021
Publisher: International Press of Boston, Inc.
Paperback
382 pages
Hodge theory?one of the pillars of modern algebraic geometry?is a deep theory with many applications and open problems, the most enigmatic of which is the Hodge conjecture, one of the Clay Institute’s seven Millennium Prize Problems. Hodge theory is also famously difficult to learn, requiring training in many different branches of mathematics.
The present volume begins with an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann, among many others; and then the studies of two-dimensional multiple integrals by Poincare and Picard. Thenceforth, the focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials, for which many tools from singularity theory, such as Brieskorn modules and Milnor fibrations, are used.
Another aspect of this volume is its computational presentation of many well-known theoretical concepts, such as the Gauss?Manin connection, homology of varieties in terms of vanishing cycles, Hodge cycles, Noether?Lefschetz, and Hodge loci. All are explained for the generalized Fermat variety, which for Hodge theory boils down to a heavy linear algebra. Most of the algorithms introduced here are implemented in Singular, a computer algebra system for polynomial computations.
Finally, the author introduces a few problems, mainly for talented undergraduate students, which can be understood with a basic knowledge of linear algebra. The origins of these problems may be seen in the discussions of advanced topics presented throughout this volume.
Brings under one roof results previously scattered in many research papers
published during the past 50 years since the origin of the three-dimensional
theory of quasiconformal and quasiregular mappings
Contains an extensive set of exercises, including solutions
Can be used as learning material/collateral reading for several courses
This book is an introduction to the theory of quasiconformal and quasiregular mappings in the
euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop
this theory as the literature shows. The authors' approach is based on the use of metrics, in
particular conformally invariant metrics, which will have a key role throughout the whole book.
The intended readership consists of mathematicians from beginning graduate students to
researchers. The prerequisite requirements are modest: only some familiarity with basic ideas
of real and complex analysis is expected.
1st ed. 2020, XIX, 502 p. 56 illus.
Hardcover
ISBN 978-3-030-32067-6
Product category : Monograph
Series : Springer Monographs in Mathematics
Mathematics : Potential Theory
Explores Euclidean and non-Euclidean geometries, culminating in a
mathematical model for special relativity
Introduces students familiar with calculus to the rigorous foundations of
plane geometry: Euclidean, spherical, hyperbolic, and relativistic
Offers a pathway from classical to abstract geometries by focusing on
isometries
This textbook offers a geometric perspective on special relativity, bridging Euclidean space,
hyperbolic space, and Einstein’s spacetime in one accessible, self-contained volume. Using tools
tailored to undergraduates, the author explores Euclidean and non-Euclidean geometries,
gradually building from intuitive to abstract spaces. By the end, readers will have encountered
a range of topics, from isometries to the Lorentz?Minkowski plane, building an understanding
of how geometry can be used to model special relativity. Beginning with intuitive spaces, such
as the Euclidean plane and the sphere, a structure theorem for isometries is introduced that
serves as a foundation for increasingly sophisticated topics, such as the hyperbolic plane and
the Lorentz?Minkowski plane. By gradually introducing tools throughout, the author offers
readers an accessible pathway to visualizing increasingly abstract geometric concepts.
Numerous exercises are also included with selected solutions provided. Geometry: from
Isometries to Special Relativity offers a unique approach to non-Euclidean geometries,
culminating in a mathematical model for special relativity. The focus on isometries offers
undergraduates an accessible progression from the intuitive to abstract; instructors will
appreciate the complete instructor solutions manual available online. A background in
elementary calculus is assumed.
1st ed. 2020, XIII, 258 p. 92 illus., 18 illus. in color.
Hardcover
ISBN 978-3-030-42100-7
Product category : Undergraduate textbook
Series : Undergraduate Texts in Mathematics
Mathematics : Hyperbolic Geometry
Introduces the almost simple groups together with their maximal subgroups
and automorphisms
Provides a very well-written, comprehensive account of Shintani descent for
applications in group theory in a useful context
Classifies the finite 3/2-generated groups in the important and substantial
case of almost simple classical groups
This monograph studies generating sets of almost simple classical groups, by bounding the
spread of these groups. Guralnick and Kantor resolved a 1962 question of Steinberg by
proving that in a finite simple group, every nontrivial element belongs to a generating pair.
Groups with this property are said to be 3/2-generated. Breuer, Guralnick and Kantor
conjectured that a finite group is 3/2-generated if and only if every proper quotient is cyclic.
We prove a strong version of this conjecture for almost simple classical groups, by bounding
the spread of these groups. This involves analysing the automorphisms, fixed point ratios and
subgroup structure of almost simple classical groups, so the first half of this monograph is
dedicated to these general topics. In particular, we give a general exposition of Shintani
descent. This monograph will interest researchers in group generation, but the opening
chapters also serve as a general introduction to the almost simple classical groups.
1st ed. 2021, VIII, 154 p. 35 illus.
Softcover
ISBN 978-3-030-74099-3
Product category : Monograph
Series : Lecture Notes in Mathematics
Mathematics : Group Theory and Generalizations
Provides a comprehensive overview of cutting-edge statistical approaches for
microbiome research
Explores the intersection of big data and next generation sequencing
technologies
Discusses innovative methodology and applications that will be of interest to
statisticians as well as non-statistical scientists and advanced students in
microbiome research
Microbiome research has focused on microorganisms that live within the human body and their
effects on health. During the last few years, the quantification of microbiome composition in
different environments has been facilitated by the advent of high throughput sequencing
technologies. The statistical challenges include computational difficulties due to the high
volume of data; normalization and quantification of metabolic abundances, relative taxa and
bacterial genes; high-dimensionality; multivariate analysis; the inherently compositional nature
of the data; and the proper utilization of complementary phylogenetic information. This has
resulted in an explosion of statistical approaches aimed at tackling the unique opportunities
and challenges presented by microbiome data. This book provides a comprehensive overview of
the state of the art in statistical and informatics technologies for microbiome research. In
addition to reviewing demonstrably successful cutting-edge methods, particular emphasis is
placed on examples in R that rely on available statistical packages for microbiome data. With
its wide-ranging approach, the book benefits not only trained statisticians in academia and
industry involved in microbiome research, but also other scientists working in microbiomics and
in related fields.
Due 2021-07-28
1st ed. 2021, VIII, 242 p. 64 illus., 49 illus. in color.
Hardcover
ISBN 978-3-030-73350-6
Product category : Contributed volume
Series : Frontiers in Probability and the Statistical Sciences
Statistics : Statistics for Life Sciences, Medicine, Health Sciences
Shows the SOT problem to be partly the generalization of the OT problem and
partly Schrodinger's problem
Explains fundamental results of the stochastic optimal transportation
problem, including duality theorem
Encompasses the zero-noise limit, the Lipschitz continuity, and the
semiconcavity of Schrodinger's problem
In this book, the optimal transportation problem (OT) is described as a variational problem for
absolutely continuous stochastic processes with fixed initial and terminal distributions. Also
described is Schrodinger’s problem, which is originally a variational problem for one-step
random walks with fixed initial and terminal distributions. The stochastic optimal transportation
problem (SOT) is then introduced as a generalization of the OT, i.e., as a variational problem for
semimartingales with fixed initial and terminal distributions. An interpretation of the SOT is also
stated as a generalization of Schrodinger’s problem. After the brief introduction above, the
fundamental results on the SOT are described: duality theorem, a sufficient condition for the
problem to be finite, forward?backward stochastic differential equations (SDE) for the
minimizer, and so on. The recent development of the superposition principle plays a crucial
role in the SOT. A systematic method is introduced to consider two problems: one with fixed
initial and terminal distributions and one with fixed marginal distributions for all times. By the
zero-noise limit of the SOT, the probabilistic proofs to Monge’s problem with a quadratic cost
and the duality theorem for the OT are described. Also described are the Lipschitz continuity
and the semiconcavity of Schrodinger’s problem in marginal distributions and random variables
with given marginals, respectively. As well, there is an explanation of the regularity result for
the solution to Schrodinger’s functional equation when the space of Borel probability measures
is endowed with a strong or a weak topology, and it is shown that Schrodinger’s problem can
be considered a class of mean field games.
Due 2021-07-03
1st ed. 2021, XI, 121 p. 11 illus.
Softcover
ISBN 978-981-16-1753-9
Product category : Brief
Series : SpringerBriefs in Mathematics
Mathematics : Probability Theory and Stochastic Processes
Volume II of Ernst Kummer's Collected Papers can be divided into four different parts. The first
part covers his work on the hypergeometric function, and on repeated integrals of rational
functions. In the second part (Algebraic Geometry), we see how the discovery of Kummer
surfaces seems to have been an outgrowth of the authors interest in optical properties of
biaxial crystals, and in the "cyclides" of Dupin. The relation between Kummer surfaces and
quotients of abelian surfaces was discovered only much later. In this section, one also finds a
number of papers describing actual plaster models of particular Kummer surfaces with
symmetries. The third part is devoted to his work on aerodynamics and ballistics. Finally, the
fourth part (Speeches and Reviews) spans a broad range of topics, including a long
retrospective on the life and work of Dirichlet.
Due 2021-07-19
1st ed. 2021, Approx. 950 p.
Softcover
ISBN 978-3-662-63345-8
Product ＣＡＴＥＧＯＲＹ : Collected works
Series : Springer Collected Works in Mathematics
Mathematics : Mathematics (general)