Due 2018-11-29 1st ed. 2018, VI, 154 p. 33 illus., 15 illus. in color.
Hardcover ISBN 978-3-030-00640-2
Mathematics : Structures and Proofs
Includes both solved and unsolved exercises Includes informal discussions
of aspects of philosophy of mathematics, and of the relation between certain
mathematical notions and thought processes Helps engage students in a reflexion
on the nature of mathematics and periodically breaks away from technicalities
Provides a more extensive treatment of relations, including equivalence
relations and order relations, than most comparable books This is a textbook
for an undergraduate mathematics major transition course from techniquebased
mathematics (such as Algebra and Calculus) to proof-based mathematics.
It motivates the introduction of the formal language of logic and set theory
and develops the basics with examples, exercises with solutions and exercises
without. It then moves to a discussion of proof structure and basic proof
techniques, including proofs by induction with extensive examples.
An in-depth treatment of relations, particularly equivalence and order
relations completes the exposition of the basic language of mathematics.
The last chapter treats infinite cardinalities. An appendix gives some
complement on induction and order, and another provides full solutions
of the in-text exercises.
The primary audience is undergraduate mathematics major, but independent readers interested in mathematics can also use the book for self-study.
Due 2018-11-26 2nd ed. 2018, X, 244 p.
Hardcover ISBN 978-4-431-56851-3
Presents quite recent research works, all of very high standard, in the
field of several complex variables Selects only extremely important materials
from the conventional basic theory of complex analysis and manifold theory
Requires no more than a one-semester introductory course in complex analysis
as a prerequisite for understanding Makes the content more informative
with the addition of new materials and sections to each chapter Proves
Andreotti-Grauert's finiteness theorems by the method of Andreotti ? Vesentini
This monograph presents the current status of a rapidly developing part
of several
complex variables, motivated by the applicability of effective results
to algebraic geometry and differential geometry. Special emphasis is put
on the new precise results on the L2 extension of holomorphic functions
in the past 5 years. In Chapter 1, the classical questions of several complex
variables motivating the development of this field are reviewed after necessary
preparations from the basic notions of those variables and of complex manifolds
such as holomorphic functions, pseudoconvexity, differential forms, and
cohomology. In Chapter 2, the L2 method of solving the d-bar equation is
presented
emphasizing its differential geometric aspect. In Chapter 3, a refinement
of the Oka?Cartan theory is given by this method. The L2 extension theorem
with an optimal constant is included, obtained recently by Z. Bocki and
separately by Q.-A. Guan and X.-Y. Zhou. In Chapter 4, various results
on the Bergman kernel are presented, including recent works of Maitani-Yamaguchi,
Berndtsson, Guan?Zhou, and Berndtsson?Lempert. Most of these results are
obtained by the L2 method. In the last chapter, rather specific results
are discussed on the existence and classification of certain holomorphic
foliations and Levi flathypersurfaces as their stables sets. These are
also applications of the L2 method obtained during the past 15 years.
Due 2018-12-18 1st ed. 2018, X, 538 p. 4 illus.
Hardcover ISBN 978-3-030-02190-0
Celebrates one of the 20th century’s most significant mathematicians Explores
recent advances in topics that Kostant fundamentally influenced Equips
readers with an understanding of various subjects within Lie algebra and
representation theory This volume, dedicated to the memory of the great
American mathematician Bertram Kostant (May 24, 1928 ? February 2, 2017),
is a collection of 19 invited papers by leading mathematicians working
in Lie theory, representation theory, algebra, geometry, and mathematical
physics. Kostant’s fundamental work in all of these areas has provided
deep new insights and
connections, and has created new fields of research. This volume features
the only published articles of important recent results of the contributors
with full details of their proofs. Key topics include: Poisson structures
and potentials (A. Alekseev, A. Berenstein, B. Hoffman) Vertex algebras
(T. Arakawa, K. Kawasetsu) Modular irreducible representations of semisimple
Lie algebras (R. Bezrukavnikov, I. Losev) Asymptotic Hecke algebras (A.
Braverman, D. Kazhdan) Tensor categories and quantum groups (A. Davydov,
P. Etingof, D. Nikshych) NilHecke algebras and Whittaker D-modules (V.
Ginzburg) Toeplitz operators (V. Guillemin, A. Uribe, Z. Wang) Kashiwara
crystals (A. Joseph) Characters of highest weight modules (V. Kac, M. Wakimoto)
Alcove polytopes (T.Lam, A. Postnikov) Representation theory of quantized
Gieseker varieties (I. Losev) Generalized Bruhat cells and integrable systems
(J.-H. Liu, Y. Mi) Almost characters (G. Lusztig) Verlinde formulas (E.
Meinrenken) Dirac operator and equivariant index (P.-E. Paradan, M. Vergne)
Modality of representations and geometry of-groups (V. L. Popov) Distributions
on homogeneous spaces (N. Ressayre) Reduction of orthogonal representations
(J.P. Serre)
Due 2018-12-20 1st ed. 2018, Approx. 200 p.
Hardcover ISBN 978-981-13-2894-7
Elucidates automorphisms of groups as a fundamental topic of study in group
theory Explores various developments on the relationship between orders
of finite groups and their automorphism groups Provides a unified account
of important group-theoretic advances arising from this study Includes
open problems for future work The book describes developments on some well-known
problems regarding the relationship between orders of finite groups and
that of their automorphism groups. It is broadly divided into three parts:
the first part offers an exposition of the fundamental exact sequence of
Wells that relates automorphisms, derivations and cohomology of groups,
along with some interesting applications of the sequence. The second part
offers an account of important developments on a conjecture that a finite
group has at least a prescribed number of automorphisms if the order of
the group is sufficiently large. A non-abelian group of primepower order
is said to have divisibility property if its order divides that of its
automorphism group. The final part of the book discusses the literature
on divisibility property of groups culminating in the existence of groups
without this property. Unifying various ideas developed over the years,
this largely self-contained book includes results that are either proved
or with complete references provided. It is aimed at researchers working
in group theory, in particular, graduate students in algebra.
Due 2018-12-24 1st ed. 2018, XVI, 287 p. 24 illus., 9 illus. in color.
Hardcover ISBN 978-3-030-02603-5
Right Triangles, Sums of Squares, and Arithmetic Offers an innovative approach
to elementary number theory motivated by right triangles Inspires students
to explore number theory through investigation of concrete examples Provides
historical context throughout, showing how ideas developed in the field
Features numerous engaging exercises, including many designed for SageMath
Right triangles are at the heart of this textbook’s vibrant new approach
to elementary number theory. Inspired by the familiar Pythagorean theorem,
the author invites the reader to ask natural arithmetic questions about
right triangles, then proceeds to develop the theory needed to respond.
Throughout, students are encouraged to engage with the material by posing
questions, working through exercises, using technology, and learning about
the broader context in which ideas developed. Progressing from the fundamentals
of number theory through to Gauss sums and quadratic reciprocity, the first
part of this text presents an innovative first course in elementary number
theory.
The advanced topics that follow, such as counting lattice points and the
four squares theorem, offer a variety of options for extension, or a higher-level
course; the breadth and modularity of the later material is ideal for creating
a senior capstone course. Numerous exercises are included throughout, many
of which are designed for SageMath. By involving students in the active
process of inquiry and investigation, this textbook imbues the foundations
of number theory with insights into the lively mathematical process that
continues to advance the field today. Experience writing proofs is the
only formal prerequisite
for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.
Due 2018-12-30 1st ed. 2018, XI, 302 p. 1 illus.
Hardcover ISBN 978-3-030-02124-5
Presents core material on elliptic operators as well as advanced topics
Provides detailed information about the function spaces related to elliptic
operators Includes results on extensions of adjoint operators and applications
to elliptic boundary value problems This book deals with elliptic differential
equations, providing the analytic background necessary for the treatment
of associated spectral questions, and covering important topics previously
scattered throughout the literature. Starting with the basics of elliptic
operators and their naturally associated function spaces, the authors then
proceed to cover
various related topics of current and continuing importance. Particular
attention is given to the characterisation of self-adjoint extensions ofsymmetric
operators acting in a Hilbert space and, for elliptic operators, the realisation
of such extensions in terms of boundary conditions. A good deal of material
not previously available in book form, such as the treatment of the Schauder
estimates, is included. Requiring only basic knowledge of measure theory
and functional analysis, the book is accessible to graduate students and
will be of interest to all researchers in partial differential equations.
The reader will value
its self-contained, thorough and unified presentation of the modern theory of elliptic operators.