Lectures on Hilbert Schemes of Points on Surfaces

Author: Hiraku Nakajima

Publisher: American Mathematical Soc.

ISBN: 0821819569

Category: Mathematics

Page: 132

View: 1268

This beautifully written book deals with one shining example: the Hilbert schemes of points on algebraic surfaces ... The topics are carefully and tastefully chosen ... The young person will profit from reading this book. --Mathematical Reviews The Hilbert scheme of a surface $X$ describes collections of $n$ (not necessarily distinct) points on $X$. More precisely, it is the moduli space for 0-dimensional subschemes of $X$ of length $n$. Recently it was realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory--even theoretical physics. The discussion in the book reflects this feature of Hilbert schemes. One example of the modern, broader interest in the subject is a construction of the representation of the infinite-dimensional Heisenberg algebra, i.e., Fock space. This representation has been studied extensively in the literature in connection with affine Lie algebras, conformal field theory, etc. However, the construction presented in this volume is completely unique and provides an unexplored link between geometry and representation theory. The book offers an attractive survey of current developments in this rapidly growing subject. It is suitable as a text at the advanced graduate level.

Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

Author: Zhenbo Qin

Publisher: American Mathematical Soc.

ISBN: 1470441888

Category: Hilbert schemes

Page: 336

View: 5857

Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes of collections of points (zero-dimensional subschemes) in a smooth algebraic surface . Schemes turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of , including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of and the Gromov–Witten correspondence. The last part of the book presents results about quantum cohomology of and related questions. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.

Algebraic Geometry and Number Theory

Summer School, Galatasaray University, Istanbul, 2014

Author: Hussein Mourtada,Celal Cem Sarıoğlu,Christophe Soulé,Ayberk Zeytin

Publisher: Birkhäuser

ISBN: 331947779X

Category: Mathematics

Page: 232

View: 4390

This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.

Donaldson Type Invariants for Algebraic Surfaces

Transition of Moduli Stacks

Author: Takuro Mochizuki

Publisher: Springer

ISBN: 354093913X

Category: Mathematics

Page: 383

View: 1636

In this monograph, we de?ne and investigate an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We may expect the existence of interesting “universal relations among invariants”, which would be a natural generalization of the “wall-crossing formula” and the “Witten conjecture” for classical Donaldson invariants. Our goal is to obtain a weaker version of such relations, in other brief words, to describe a relation as the sum of integrals over the products of m- uli spaces of objects with lower ranks. Fortunately, according to a recent excellent work of L. Gottsche, ̈ H. Nakajima and K. Yoshioka, [53], a wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case. We hope that our work in this monograph would, at least tentatively, provides a part of foundation for the further study on such universal relations. In the rest of this preface, we would like to explain our motivation and some of important ingredients of this study. See Introduction for our actual problems and results. Donaldson Invariants Let us brie?y recall Donaldson invariants. We refer to [22] for more details and precise. We also refer to [37], [39], [51] and [53]. LetX be a compact simply con- ? nected oriented real 4-dimensional C -manifold with a Riemannian metric g. Let P be a principalSO(3)-bundle on X.

Orbifolds in Mathematics and Physics

Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory, May 4-8, 2001, University of Wisconsin, Madison, Wisconsin

Author: Alejandro Adem,Jack Morava,Yongbin Ruan

Publisher: American Mathematical Soc.

ISBN: 0821829904

Category: Mathematics

Page: 358

View: 3667

This book publishes papers originally presented at a conference on the Mathematical Aspects of Orbifold String Theory, hosted by the University of Wisconsin-Madison. The recent explosion of activity on the topic has been powered by applications of orbifolds to moduli problems and quantum field theory. The present volume contains information not fully covered in the published literature, and presents an interdisciplinary look at orbifold problems. Topics such as stacks, vertex operator algebras, branes, groupoids, K-theory and quantum cohomology are discussed. The book reflects the thinking of distinguished investigators working in the areas of mathematical physics, algebraic geometry, algebraic topology, symplectic geometry and representation theory. Advanced graduate students and researchers working in these areas, as well as those interested in connections between mathematical subject areas, will find this book of interest.

International Conference on Theoretical Physics

Th-2002, Paris, July 22-27, 2002

Author: Daniel Iagolnitzer,Vincent Rivasseau,Jean Zinn-Justin

Publisher: Springer Science & Business Media

ISBN: 9783764324339

Category: Computers

Page: 984

View: 6775

Theoretical physics is a vast set of subjects, ideas and methods, with wide and unexpected applications to many interdisciplinary problems. But no general international conference had tried to review in depth this huge and burgeoning field since the Trieste conference in 1968. The International Conference on Theoretical Physics, TH-2002, which took place at the Unesco building, Paris, from July 22 to 27, 2002, addressed this challenge.The reader will find in this book all invited and received contributions to the conference. After the general lectures of Nobel prize winners Anderson and Yang, the contributions by experts cover all aspects of modern theoretical physics ranging from particle physics, string theory, cosmology, statistical and condensed matter physics to dynamical systems and quantum chaos, the physics/biology interface, information theory and quantum computing.

Sūgaku Expositions

A Translation of Sūgaku

Author: N.A

Publisher: N.A


Category: Mathematics

Page: N.A

View: 2778


Geometry & Topology

Author: N.A

Publisher: N.A


Category: Geometry

Page: N.A

View: 4398

Fully refereed international journal dealing with all aspects of geometry and topology and their applications.

Noncommutative geometry and Cayley-smooth orders

Author: Lieven le Bruyn

Publisher: Chapman & Hall


Category: Mathematics

Page: 524

View: 8052

Noncommutative Geometry and Cayley-smooth Orders explains the theory of Cayley-smooth orders in central simple algebras over function fields of varieties. In particular, the book describes the étale local structure of such orders as well as their central singularities and finite dimensional representations. After an introduction to partial desingularizations of commutative singularities from noncommutative algebras, the book presents the invariant theoretic description of orders and their centers. It proceeds to introduce étale topology and its use in noncommutative algebra as well as to collect the necessary material on representations of quivers. The subsequent chapters explain the étale local structure of a Cayley-smooth order in a semisimple representation, classify the associated central singularity to smooth equivalence, describe the nullcone of these marked quiver representations, and relate them to the study of all isomorphism classes of n-dimensional representations of a Cayley-smooth order. The final chapters study Quillen-smooth algebras via their finite dimensional representations. Noncommutative Geometry and Cayley-smooth Orders provides a gentle introduction to one of mathematics' and physics' hottest topics.

The Geometry of Moduli Spaces of Sheaves

A Publication of the Max-Planck-Institut für Mathematik, Bonn

Author: Daniel Huybrechts,Manfred Lehn

Publisher: Vieweg+Teubner Verlag

ISBN: 9783663116257

Category: Technology & Engineering

Page: 270

View: 3227

This book is intended to serve as an introduction to the theory of semistable sheaves and at the same time to provide a survey of recent research results on the geometry of moduli spaces. The first part introduces the basic concepts in the theory: Hilbert polynomial, slope, stability, Harder-Narasimhan filtration, Grothendieck's Quot-scheme. It presents detailed proofs of the Grauert-Mülich Theorem, the Bogomolov Inequality, the semistability of tensor products, and the boundedness of the family of semistable sheaves. It also gives a self-contained account of the construction of moduli spaces of semistable sheaves on a projective variety à la Gieseker, Maruyama, and Simpson. The second part presents some of the recent results of the geometry of moduli spaces of sheaves on an algebraic surface, following work of Mukai, O'Grady, Gieseker, Li and many others. In particular, moduli spaces of sheaves on K3 surfaces and determinant line bundles on the moduli spaces are treated in some detail. Other topics include the Serre correspondence, restriction of stable bundles to curves, symplectic structures, irreducibility and Kodaira-dimension of moduli spaces.

Surveys in Combinatorics

Invited Papers for the ... British Combinatorial Conference

Author: Christopher D. Wensley

Publisher: N.A


Category: Combinatorial analysis

Page: N.A

View: 9689


Surveys in Combinatorics 2003

Author: C. D. Wensley

Publisher: Cambridge University Press

ISBN: 9780521540124

Category: Mathematics

Page: 378

View: 6228

The British Combinatorial Conference attracts a large following from the U.K. and international research community. Held at the University of Wales, Bangor, in 2003, the speakers included renowned experts on topics currently attracting significant research interest, as well as less traditional areas such as the combinatorics of protecting digital content. All the contributions are survey papers presenting an overview of the state of the art in a particular area.

Commutative Algebra, Algebraic Geometry, and Computational Methods

Author: David Eisenbud

Publisher: Springer Verlag

ISBN: 9789814021500

Category: Mathematics

Page: 320

View: 4122

This volume contains papers presented at the International Conference on Commutative Algebra, Algebraic geometry, and Computational methods held in Hanoi in 1996, as well as papers written subsequently. It features both expository articles as well as research papers on a range of currently active areas in commutative algebra, algebraic geometry (particularly surveys on intersection theory) and combinatorics. In addition, a special feature is a section on the life and work of Wolfgang Vogel, who was an organiser of the conference.


Author: Constance Reid,Hermann Weyl

Publisher: Springer-Verlag

ISBN: 3662286157

Category: Mathematics

Page: 290

View: 8242