The Moduli Space of Curves
Title | The Moduli Space of Curves PDF eBook |
Author | Robert H. Dijkgraaf |
Publisher | Springer Science & Business Media |
Total Pages | 570 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461242649 |
The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.
Moduli Spaces of Riemann Surfaces
Title | Moduli Spaces of Riemann Surfaces PDF eBook |
Author | Benson Farb |
Publisher | American Mathematical Soc. |
Total Pages | 371 |
Release | 2013-08-16 |
Genre | Mathematics |
ISBN | 0821898876 |
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
The Geometry of Moduli Spaces of Sheaves
Title | The Geometry of Moduli Spaces of Sheaves PDF eBook |
Author | Daniel Huybrechts |
Publisher | Cambridge University Press |
Total Pages | 345 |
Release | 2010-05-27 |
Genre | Mathematics |
ISBN | 1139485822 |
This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.
Algebraic Curves
Title | Algebraic Curves PDF eBook |
Author | Maxim E. Kazaryan |
Publisher | Springer |
Total Pages | 231 |
Release | 2019-01-21 |
Genre | Mathematics |
ISBN | 3030029433 |
This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces. The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion. Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework
Moduli Spaces
Title | Moduli Spaces PDF eBook |
Author | Leticia Brambila |
Publisher | Cambridge University Press |
Total Pages | 347 |
Release | 2014-03-13 |
Genre | Mathematics |
ISBN | 1107636388 |
A graduate-level introduction to some of the important contemporary ideas and problems in the theory of moduli spaces.
Birational Geometry and Moduli Spaces
Title | Birational Geometry and Moduli Spaces PDF eBook |
Author | Elisabetta Colombo |
Publisher | Springer Nature |
Total Pages | 200 |
Release | 2020-02-25 |
Genre | Mathematics |
ISBN | 303037114X |
This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. In particular, the book includes both surveys and original papers on irreducible holomorphic symplectic manifolds, Severi varieties, degenerations of Calabi-Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformations, and deformations of diagrams of algebras. The intention is to disseminate the knowledge of advanced results and key techniques used to solve open problems. The book is intended for all advanced graduate students and researchers interested in the new research frontiers of birational geometry and moduli spaces.
Geometry of Moduli Spaces and Representation Theory
Title | Geometry of Moduli Spaces and Representation Theory PDF eBook |
Author | Roman Bezrukavnikov |
Publisher | American Mathematical Soc. |
Total Pages | 436 |
Release | 2017-12-15 |
Genre | Algebraic varieties |
ISBN | 1470435748 |
This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.