Format: Hardback, 241 pages, height x width: 235x155 mm, XX, 230 p. 10 illus.
Series: KIAS Springer Series in Mathematics 4
Pub. Date: 28-Jan-2025
ISBN-13: 9789819782482
Enumerative geometry is a core area of algebraic geometry that dates back to Apollonius in the second century BCE. It asks for the number of geometric figures with desired properties and has many applications from classical geometry to modern physics. Typically, an enumerative geometry problem is solved by first constructing the space of all geometric figures of fixed type, called the moduli space, and then finding the subspace of objects satisfying the desired properties. Unfortunately, many moduli spaces from nature are highly singular, and an intersection theory is difficult to make sense of. However, they come with deeper structures, such as perfect obstruction theories, which enable us to define nice subsets, called virtual fundamental classes. Now, enumerative numbers, called virtual invariants, are defined as integrals against the virtual fundamental classes.
Derived algebraic geometry is a relatively new area of algebraic geometry that is a natural generalization of Serres intersection theory in the 1950s and Grothendiecks scheme theory in the 1960s. Many moduli spaces in enumerative geometry admit natural derived structures as well as shifted symplectic structures.
The book covers foundations on derived algebraic and symplectic geometry. Then, it covers foundations on virtual fundamental classes and moduli spaces from a classical algebraic geometry point of view. Finally, it fuses derived algebraic geometry with enumerative geometry and covers the cutting-edge research topics about DonaldsonThomas invariants in dimensions three and four.
An Introduction to Derived Algebraic Geometry.- An Introduction to
Shifted Symplectic Structures.- An Introduction to Virtual Cycles via
Classical Algebraic Geometry.- An Introduction to Virtual Cycles via Derived
Algebraic Geometry.- An Introduction to Cohomological Donaldson Thomas
Theory.- Moduli Spaces of Sheaves: An Overview, Curves and Surfaces.- Sheaf
Counting Theory in Dimension Three and Four.
Format: Hardback, 493 pages, height x width: 235x155 mm, XIV, 493 p.,
Series: Springer Series in Computational Mathematics 63
Pub. Date: 27-Dec-2024
ISBN-13: 9783031743788
This book offers the first comprehensive account of how the logarithmic norm is used for matrices, nonlinear maps and linear differential operators, with a focus on initial and boundary value problems.
Complementing the usual operator norm, the logarithmic norm is a versatile tool which provides unique additional information on the magnitude of an operator. It is instrumental in the stability theory of dynamical systems and in the theory of elliptic operator equations.
The text adopts a unified approach to address a wide range of themes in applied mathematics. It explores the role of the logarithmic norm in scientific computing, compares the operator bounds with those of spectral theory, and illustrates the theory with classical models from science and engineering. Many previously unpublished results are presented alongside established material, supporting researchers in applied mathematics and computational engineering who seek a systematic approach to stability and perturbation bounds in initial value problems, boundary value problems and partial differential equations.
Primarily intended as a reference text, the book can also serve as a graduate text for PhD students.
Part I Introduction to logarithmic norms.- Part II Matrix Theory.- Part
III Nonlinear maps.- Part IV Differential Operators.- Further
reading.-Bibliography.- Index.
Format: Paperback 333 pages, height x width: 235x155 mm, 55 Illustrations, color; 12 Illustrations, black and white; XI, 322 p. 60 illus. in color.,
Series: Springer Undergraduate Mathematics Series
Pub. Date: 18-Jan-2025
ISBN-13: 9783031737268
This textbook provides a second course in complex analysis with a focus on geometric aspects. It covers topics such as the spherical geometry of the extended complex plane, the hyperbolic geometry of the Poincare disk, conformal mappings, the Riemann Mapping Theorem and uniformisation of planar domains, characterisations of simply connected domains, the convergence of Riemann maps in terms of Caratheodory convergence of the image domains, normal families and Picard's theorems on value distribution, as well as the fundamentals of univalent function theory. Throughout the text, the synergy between analysis and geometry is emphasised, with proofs chosen for their directness.
The textbook is self-contained, requiring only a first undergraduate course in complex analysis. The minimal topology needed is introduced as necessary. While primarily aimed at upper-level undergraduates, the book also serves as a concise reference for graduates working in complex analysis.
1 Introduction.- 2 The Complex Plane - Preparatory Topics.- 3 The
Riemann Sphere.- 4 The Hyperbolic Disk.- 5 Normal Families and Value
Distribution.- 6 Simply Connected Domains and the Riemann Mapping Theorem.- 7
Runge's Theorem and Further Characterisations of Simply Connected Domains.- 8
Univalent Functions - the Basics.- 9 Caratheodory Convergence of Domains and
Hyperbolic Geodesics.- 10 Uniformisation of Planar Domains.
Format: Hardback, 338 pages, height x width: 235x155 mm, 40 Illustrations, color; 10 Illustrations, black and white; X, 340 p.,
Series: Forum for Interdisciplinary Mathematics
Pub. Date: 30-Jan-2025
ISBN-13: 9789819787142
This book contains a comprehensive collection of chapters on recent and original research, along with review articles, on mathematical modeling of dynamical systems described by various types of differential equations. Structured into 18 chapters dedicated to exploring different aspects of differential equations and their applications in modeling both discrete and continuous systems, it highlights theoretical advancements in mathematics and their practical applications in modeling dynamic systems. Readers will find contributions by renowned scholars who delve into the intricacies of nonlinear dynamics, stochastic processes, and partial differential equations. This book is an essential resource for researchers, academicians, and practitioners in the field of mathematical modeling.
Chapter 1 Fractional Differential Equations in Engineering Sciences.-
Chapter 2 A New Operator Approach for Solving Time-Fractional Nonlinear
Burgess Equation.
Chapter 3 Explicit Formulas for the Solutions of a Solvable.
Chapter 4 Stochastic Equations with Deviating Argument Driven by Poisson
Jumps.
Chapter 5 A Review on Fractional Calculus in Modelling of Cancer.
Format: Hardback, 429 pages, height x width: 235x155 mm, 43 Illustrations, color; 62 Illustrations, black and white; XIII, 424 p. 32 illus.,
Series: Undergraduate Texts in Mathematics
Pub. Date: 24-Jan-2025
ISBN-13: 9783031734335
The aim of this text is to introduce discrete mathematics to beginning students of mathematics or computer science. It does this by bringing some coherency into the seemingly incongruent subjects that compose discrete math, such as logic, set theory, algebra, and combinatorics. It emphasizes their theoretical foundations and illustrates proofs along the way. The book prepares readers for the analysis of algorithms by discussing asymptotic analysis and a discrete calculus for sums. The book also deduces combinatorial methods from the foundations that are laid out. Unlike other texts on this subject, there is a greater emphasis on foundational material that leads to a better understanding. To further assist the reader in grasping and practicing concepts, roughly 690 exercises are provided at various levels of difficulty. Readers are encouraged to study the examples in the text and solve as many of the exercises as possible.
The text is intended for freshman or sophomore undergraduate students in mathematics, computer science, or similar majors. The assumed background is precalculus. The chapter dependency chart included is designed to help students, independent readers, and instructors follow a systematic path for learning and teaching the material, with the option to explore material in later chapters.
Preface.- Notation.-
I. Discrete Structures.-
1. Introduction.-2. Mathematical Arguments.-3. Sets.-
4. Proof by Induction.-5. Equivalence Relations.-
6. Partial Orders and Lattices.-
7. Floor and Ceiling Functions.-
8. Number Theory.-
II. Summation and Asymptotics.-
10. Asymptotic Analysis.-
III. Combinatorics.-
11. Counting.-
12. Generating Functions.-
13. Recurrence Relations.-
14. Graphs.-
15. Probability.- Bibliography.- Index.
Format: Hardback, 480 pages, height x width: 235x155 mm, 38 Illustrations, black and white; X, 490 p. 39 illus.,
Series: Springer Monographs in Mathematics
Pub. Date: 22-Jan-2025
ISBN-13: 9789819781621
This books area is special functions of one or several complex variables. Special functions have been applied to dynamics and physics. Special functions such as elliptic or automorphic functions have an algebro-geometric nature. These attributes permeate the book. The Kleinian sigma function, or higher-genus Weierstrass sigma function generalizes the elliptic sigma function. It appears for the first time in the work of Weierstrass. Klein gave an explicit definition for hyperelliptic or genus-three curves, as a modular invariant analogue of the Riemann theta function on the Jacobian (the two functions are equivalent). H.F. Baker later used generalized Legendre relations for meromorphic differentials, and brought out the two principles of the theory: on the one hand, sigma uniformizes the Jacobian so that its (logarithmic) derivatives in one direction generate the field of meromorphic functions on the Jacobian, therefore algebraic relations among them generate the ideal of the Jacobian as a projective variety; on the other hand, a set of nonlinear PDEs (which turns out to include the integrable hierarchies of KdV type), characterize sigma. We follow Bakers approach.
There is no book where the theory of the sigma function is taken from its origins up to the latest most general results achieved, which cover large classes of curves. The authors propose to produce such a book, and cover applications to integrable PDEs, and the inclusion of related al functions, which have not yet received comparable attention but have applications to defining specific subvarieties of the degenerating family of curves. One reason for the attention given to sigma is its relationship to Sato's tau function and the heat equations for deformation from monomial curves.
The book is based on classical literature and contemporary research, in particular our contribution which covers a class of curves whose sigma had not been found explicitly before.
Overview of Work on Sigma Function from Historical Viewpoint.- Curves in
Weierstrass Canonical Form (W-curves).- Theory of Sigma Function.-
Application of the Sigma Function Theory to Integrable Systems.
Format: Hardback, 468 pages, height x width: 235x155 mm, 56 Illustrations, color; 108 Illustrations, black and white; VI, 469 p.
Series: Simons Symposia
Pub. Date: 30-Jan-2025
ISBN-13: 9783031741333
This book is a tribute to the memory of Yuri Ivanovich Manin, who passed away on January 7, 2023. Manin was one of the giants of modern mathematics. His work covered a wide range of fields, including logic, number theory, geometry, mathematical physics, theoretical computer science, and linguistics. The contributions collected here are on topics close to his life-long passion: arithmetic and algebraic geometry.
Table of Contents
Preface.- Projecting lattice polytopes according to the Minimal Model
Program.- Zeta-polynomials, superpolynomials, DAHA and plane curve
singularities.- Rational points over C1 fields.- On isomorphisms of
ind-varieties of generalized flags.- Spectral description of non-commutative
local systems on surfaces and non-commutative cluster varieties.- Semi-stable
reduction of foliations.- The Hasse principle for 9-nodal cubic 3-folds.-
Manin's work in birational geometry.- Endomorphism Algebras and Automorphism
Groups of certain Complex Tori.