Hardcover ISBN: 978-1-4704-7637-3
Graduate Studies in Mathematics Volume: 250
2024; 413 pp
MSC: Primary 14; 13
NOTE: For this title, the eBook is available for free as part of a pilot program to promote access to key mathematical texts. We are collecting information about the readers reached through the AMS Bookstore. These data will only be reported and analyzed at the aggregate level in order to measure the success of the pilot.
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 Greenfs 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.
Chapters
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
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.
Softcover ISBN: 978-1-4704-7959-6
2024; 154 pp
MSC: Primary 57; Secondary 00; 54
Topology is the field of mathematics that studies those properties of a shape which persist when the shape gently evolves and changes its specific form. This book gives a playful and intuitive introduction to topology, favoring simple and crisp illustrations over long explanations and formal definitions. The material includes some of the classic results one would see in undergraduate courses on topology and graph theory, but presents these results in a way that readers without any knowledge of mathematics will be able to understand.
Loops
Paths
Trees
Trees and Polygons
Cell Divisions
Eulerfs Formula
Connecting Five Points
Connecting Three Points to Three Points
The Cylinder
The Mobius Band
The Torus
Connecting Seven Points on a Torus
The Klein Bottle and the Projective Plane
The Genus-2 Surface
Mix and Match
Undergraduate students interested in finding out what topology is all about.
Softcover ISBN: 978-1-4704-7301-3
Expected availability date: February 19, 2025
Contemporary Mathematics Volume: 809
2025; Estimated: 146 pp
MSC: Primary 53; 22; 47; 58; 81; 35
This volume contains the proceedings of the AMS-EMS-SMF Special Session on Sub-Riemannian Geometry and Interactions, held from July 18?20, 2022, at the Universite de Grenoble-Alpes, Grenoble, France.
Sub-Riemannian geometry is a generalization of Riemannian one, where a smooth metric is defined only on a preferred subset of tangent directions. Under the so-called Hormander condition, all points are connected by finite-length curves, giving rise to a well-defined metric space. Sub-Riemannian geometry is nowadays a lively branch of mathematics, connected with probability, harmonic and complex analysis, subelliptic PDEs, geometric measure theory, optimal transport, calculus of variations, and potential analysis.
The articles in this volume present some developments of a broad range of topics in sub-Riemannian geometry, including the theory of sub-elliptic operators, holonomy, spectral theory, and the geometry of the exponential map.
Articles
I. Beschastnyi ? Lie groupoids for sub-elliptic operators
Samuel Borza ? Normal forms for the sub-Riemannian exponential map of G¿
, SU(2)
, and SL(2)
Fabrice Baudoin and Sylvie Vega-Molino ? Holonomy of H-type Foliations
Marco Carfagnini and Maria Gordina ? Spectral gap bounds on H-type groups
Ivan Beschastnyi, Ugo Boscain, Daniele Cannarsa and Eugenio Pozzoli ? Embedding the Grushin cylinder in R3
and Schroedinger evolution
Jeremy T. Tyson ? Polar coordinates in Carnot groups II
Fabrice Baudoin, Michel Bonnefont and Li Chen ? Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck-type operators
Marco Inversi and Giorgio Stefani ? Lagrangian stability for a system of non-local continuity equations under Osgood condition
Graduate students and research mathematicians interested in differential geometry.
Softcover ISBN: 978-1-4704-7338-9
Expected availability date: February 19, 2025
Contemporary Mathematics Volume: 810
2025; 293 pp
MSC: Primary 32; 14; 52; 53
This volume presents a collection of research articles arising from the conference on gConvex and Complex: Perspectives on Positivity in Geometry,h held in Cetraro, Italy, from October 31?November 4, 2022. The conference celebrated the 70th birthday of Bo Berndtsson and the vitality of current research across complex and convex geometry, as well as interactions between the two areas, all united by the overarching concept of positivity.
Positivity plays a central role in complex and convex geometry. It arises from a range of complementary perspectives, as illustrated by the breadth of the papers appearing in this volume, including existence Kahler?Einstein edge metrics, Santalo-type inequalities, curvature of direct images of bundles, extension theorems for holomorphic functions, optimal transport and Hessian manifolds, interpolation and Brunn?Minkowski theory, and non-Archimedean geometry. The format of the workshop was innovative compared to standard conferences in mathematics, with focused 30-minute talks, aimed at stimulating lively discussions and a gflipped classroomh where the audience becomes more engaged and the speaker is not expected to transmit more information than listeners can possibly absorb. Lengthy breaks between talks and a relatively small number of talks allowed for useful time blocks for collaboration.
This volume reflects the spirit of the conference, showcasing the vitality of current research in these areas as well as the profound impact of Bo Berndtsson's contributions to the field.
Articles
Chenzi Jin ? The Cheltsov?Rubinstein problem for strongly asymptotically log del Pezzo surfaces
Vlassis Mastrantonis ? A Santalo inequality for the Lp
-polar body
Frederic Campana, Junyan Cao and Mihai P?un ? Subharmonicity of direct images and applications
Tsz On Mario Chan and Young-Jun Choi ? An application of adjoint ideal sheaves to injectivity and extension theorems
Tai Terje Huu Nguyen and Xu Wang ? A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem
Dror Varolin ? Curvature of smooth proper direct images by way of a holomorphic Gauss formula
Sebastien Boucksom and Mattias Jonsson ? Measures of finite energy in pluripotential theory: A synthetic approach
Jakob Hultgren ? Duality of Hessian manifolds and optimal transport
Laszlo Lempert ? To the geometry of spaces of plurisubharmonic functions on a Kahler manifold
Julius Ross and David Witt Nystrom ? Harmonic interpolation and a Brunn-Minkowski theorem for random determinants
Mingchen Xia ? Operations on transcendental non-Archimedean metrics
Graduate students and research mathematicians interested in complex and convex geometry.
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN: 978-1-4704-7757-8
Expected availability date: March 14, 2025
Classroom Resource Materials Volume: 74;
2025; Estimated: 196 pp
MSC: Primary 05;
This book provides an active-learning approach to combinatorial reasoning and proof through a thoughtful sequence of low threshold, high ceiling activities. A novel feature is its narrative format, with much of the text written from the perspective of a student working through the material with peers. Furthermore, each chapter includes detailed notes for the instructor such as additional scaffolding, extensions, and notation for more advanced students. The exposition is complemented by over 300 colorful illustrations.
The main focus of the book is the study of integer compositions with forays into graph theory and recreational mathematics. Befitting the constructive nature of the book, compositions are represented by trains made up of cars. By physically constructing these objects, students become proficient in hands-on verifications of numerous identities.
Developed by a recipient of the MAA's Haimo Award for Distinguished Teaching and used in several teacher professional development workshops and college courses, the book has very modest prerequisites. In particular, no prior experience with symbolic formalism is presumed, allowing this material to be used in multiple classroom settings, from enrichment activities for secondary school students through undergraduate classes in discrete mathematics. The structure of the book also makes it conducive to self-study. Get ready to gbuild some trainsh and explore the enlightening world of combinatorial proofs!
Hands-on Combinatorics is a wonderful book, cleverly designed for readers of all mathematical levels. With eye-catching illustrations, Brian Hopkins creates beautiful bijections and clever combinatorial arguments with binomial coefficients, Fibonacci numbers, and beyond.
?Arthur T. Benjamin, Harvey Mudd College, co-author of Proofs That Really Count
Circular regions and colorful trains
Counting cars
Restricted trains
Circular regions redux
A fleet of restricted trains
Barrycades
Other triangles of trains
A culminating connection
Additional train topics
Generating functions
Bibliography
Index
Undergraduate students interested in teaching discrete mathematics, particularly in an IBL styl