Softcover ISBN: 978-1-4704-7429-4
Contemporary Mathematics Volume: 807;
2024; Estimated: 229 pp
MSC: Primary 15; 33; 30; 34; 37; 42
This volume contains a collection of papers that focus on recent research in the broad field of special functions.
The articles cover topics related to differential equations, dynamic systems, integrable systems, billiards, and random matrix theory. Linear classical special functions, such as hypergeometric functions, Heun functions, and various orthogonal polynomials and nonlinear special functions (e.g., the Painleve transcendents and their generalizations), are studied from different perspectives.
This volume serves as a useful reference for a large audience of mathematicians and mathematical physicists interested in modern theory of special functions. It is suitable for both graduate students and specialists in the field.
Graduate students and research mathematicians interested in special functions, integrable systems, and differential equations.
Amilcar Branquinho, Ana Foulquie-Moreno and Manuel Manas ? Banded matrices and their orthogonality
M. M. Castro, A. Foulquie-Moreno and A. Fradi ? Time-and-band limiting for matrix-valued orthogonal polynomials related with 2×2 hypergeometric operators
Yang Chen and Shulin Lyu ? Gaussian unitary ensembles with jump discontinuities, PDEs, and the coupled Painleve IV system
Ewa Ciechanowicz ? Properties of meromorphic solutions of certain equations via Nevanlinna theory
Nicholas Ercolani, Joceline Lega and Brandon Tippings ? Map enumeration from a dynamical perspective
Sean Gasiorek and Milena Radnovi? ? Periodic trajectories and topology of the integrable Boltzmann system
P. R. Gordoa and A. Pickering ? Higher-order nonlinear special functions: Painleve hierarchies, a survey
Xing Li and Da-jun Zhang ? The Lame functions and elliptic soliton solutions: Bilinear approach
Egmont Porten and Cornelia Schiebold ? From the operator KP equation to scalar and matrix-valued solutions
Softcover ISBN: 978-1-4704-7180-4
Contemporary Mathematics Volume: 808;
2024; 276 pp
MSC: Primary 55; 57; 58; 19; 14
This volume contains the proceedings of the virtual AMS Special Session on Equivariant Cohomology, held March 19?20, 2022.
Equivariant topology is the algebraic topology of spaces with symmetries. At the meeting, “equivariant cohomology” was broadly interpreted to include related topics in equivariant topology and geometry such as Bredon cohomology, equivariant cobordism, GKM (Goresky, Kottwitz, and MacPherson) theory, equivariant K
-theory, symplectic geometry, and equivariant Schubert calculus.
This volume offers a view of the exciting progress made in these fields in the last twenty years. Several of the articles are surveys suitable for a general audience of topologists and geometers. To be broadly accessible, all the authors were instructed to make their presentations somewhat expository. This collection should be of interest and useful to graduate students and researchers alike.
Graduate students and research mathematicians interested in algebraic topology, group actions, and equivariant cohomology.
Table of Contents
Noe Barcenas ? A survey of computations of Bredon cohomology
Jack Carlisle ? Cobordism of G -manifolds
Jeffrey D. Carlson ? The cohomology of homogeneous spaces in historical context
Chi-Kwong Fok ? A stroll in equivariant K -theory
Matthias Franz ? The Chang?Skjelbred lemma and generalizations
Oliver Goertsches, Panagiotis Konstantis and Leopold Zoller ? Low-dimensional GKM theory
Rebecca Goldin ? On positivity for the Peterson variety
Chen He ? Localization of equivariant cohomology rings of real and oriented Grassmannians
Matvei Libine ? Localization of integrals of equivariant forms for non-compact group actions
Andres Pedroza ? Induced Hamiltonian function on the symplectic one-point blowup
Loring W. Tu ? Gysin formulas and equivariant cohomology
Julianna Tymoczko ? A concise introduction to GKM theory, 25 years on
Hardcover ISBN: 978-1-4704-7499-7
Graduate Studies in Mathematics Volume: 248;
2024; Estimated: 514 pp
MSC: Primary 17
Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entry-level graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinite-dimensional algebras are treated extensively along with the finite-dimensional ones.
After some motivation, the text gives a detailed and concise treatment of the Killing?Cartan classification of finite-dimensional semisimple algebras over algebraically closed fields of characteristic 0. Important constructions such as Chevalley bases follow. The second half of the book serves as a broad introduction to algebras of arbitrary dimension, including Kac?Moody (KM), loop, and affine KM algebras. Finite-dimensional semisimple algebras are viewed as KM algebras of finite dimension, their representation and character theory developed in terms of integrable representations. The text also covers triangular decomposition (after Moody and Pianzola) and the BGG category O
. A lengthy chapter discusses the Virasoro algebra and its representations. Several applications to physics are touched on via differential equations, Lie groups, superalgebras, and vertex operator algebras.
Each chapter concludes with a problem section and a section on context and history. There is an extensive bibliography, and appendices present some algebraic results used in the book.
Undergraduate and graduate students and researchers interested in learning and teaching representations of finite-dimensional and infinite-dimensional Lie algebras.
Part I. Preliminaries
Algebras
Examples of Lie algebras
Lie groups
Part II. Classification
Lie algebra basics
The Cartan decomposition
Semisimple Lie algebras: Basic structure
Classification of root systems
Semisimple Lie algebras: Classification
Part III. Important constructions
Finite degree representations of sl2(K)
PBW and free Lie algebras
Casimir operators and Weyl’s Theorem II
Chevalley bases and integration
Kac?Moody Lie algebras
Part IV. Representation
Integrable representations
The spherical case and Serre’s Theorem
Irreducible weight modules for sl2(K)
Part V. Infinite dimension
Some infinite-dimensional Lie algebras
Triangular decomposition and category O
Character theory
Representation of the Virasoro algebra
Part VI. Appendices
Appendix A. Algebra basics
Appendix B. Bilinear forms
Appendix C. Finite groups generated by reflections
Bibliography
Index
Hardcover ISBN: 978-1-4704-6503-2
Graduate Studies in Mathematics Volume: 249;
2024; Estimated: 627 pp
MSC: Primary 35; 47; 52; 58; 82
The theory of one-dimensional ergodic operators involves a beautiful synthesis of ideas from dynamical systems, topology, and analysis. Additionally, this setting includes many models of physical interest, including those operators that model crystals, disordered media, or quasicrystals. This field has seen substantial progress in recent decades, much of which has yet to be discussed in textbooks. The current volume addresses specific classes of operators, including the important examples of random and almost-periodic operators. The text serves as a self-contained introduction to the field for junior researchers and beginning graduate students, as well as a reference text for people already working in this area.
The general theory of one-dimensional ergodic operators was presented in the book by the same authors as volume 221 in the Graduate Studies in Mathematics series.
Graduate students and researchers interested in differential operators with ergodic coefficients.
Highlights from Part I
Part II: Specific classes
Random potentials
Almost-periodic potentials
Periodic potentials
Limit-periodic potentials
Quasi-periodic potentials
Subshift potentials
Appendices
Continued fractions
Topological groups
A crash course in combinatorial word theory
List of open problems
Glossary of notation
Bibliography
Index
Hardcover ISBN: 978-1-4704-7637-3
Graduate Studies in Mathematics
Volume: 250; 2024; 413 pp
MSC: Primary 14; 13
This textbook provides readers with a working knowledge of the modern theory of complex projective algebraic curves. Also known as compact Riemann surfaces, such curves shaped the development of algebraic geometry itself, making this theory essential background for anyone working in or using this discipline. Examples underpin the presentation throughout, illustrating techniques that range across classical geometric theory, modern commutative algebra, and moduli theory.
The book begins with two chapters covering basic ideas, including maps to projective space, invertible sheaves, and the Riemann?Roch theorem. Subsequent chapters alternate between a detailed study of curves up to genus six and more advanced topics such as Jacobians, Hilbert schemes, moduli spaces of curves, Severi varieties, dualizing sheaves, and linkage of curves in 3-space. Three chapters treat the refinements of the Brill?Noether theorem, including applications and a complete proof of the basic result. Two chapters on free resolutions, rational normal scrolls, and canonical curves build context for Green’s conjecture. The book culminates in a study of Hilbert schemes of curves through examples. A historical appendix by Jeremy Gray captures the early development of the theory of algebraic curves. Exercises, illustrations, and open problems accompany the text throughout.
The Practice of Algebraic Curves offers a masterclass in theory that has become essential in areas ranging from algebraic geometry itself to mathematical physics and other applications. Suitable for students and researchers alike, the text bridges the gap from a first course in algebraic geometry to advanced literature and active research.
Graduate students considering working in the field of algebraic curves and researchers in a related field whose work has led them to questions about algebraic curves.
Introduction
Linear series and morphisms to projective space
The Riemann-Roch theorem
Curves of genus 0
Smooth plane curves and curves of genus 1
Jacobians
Hyperelliptic curves and curves of genus 2 and 3
Fine moduli spaces
Moduli of curves
Curves of genus 4 and 5
Hyperplane sections of a curve
Monodromy of hyperplane sections
Brill-Noether theory and applications to genus 6
Inflection points
Proof of the Brill-Noether theorem
Using a singular plane model
Linkage and the canonical sheave of a singular curves
Scrolls and the curves they contain
Free resolutions and canonical curves
Hilbert schemes
Appendix A: A historical essay on some topics in algebraic geometry (by Jeremy Gray)
Hints to marked exercises
Bibliography
Index