Lectures on Hilbert Schemes of Points on Surfaces

Author: Hiraku Nakajima
Publisher: American Mathematical Soc.
ISBN: 9780821819562
Release Date: 1999
Genre: Mathematics

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: 9781470441883
Release Date: 2018-02-26
Genre: Hilbert schemes

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

Author: Hussein Mourtada
Publisher: Birkhäuser
ISBN: 9783319477794
Release Date: 2017-05-07
Genre: Mathematics

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

Author: Takuro Mochizuki
Publisher: Springer
ISBN: 9783540939139
Release Date: 2009-04-20
Genre: Mathematics

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

Author: Alejandro Adem
Publisher: American Mathematical Soc.
ISBN: 9780821829905
Release Date: 2002
Genre: Mathematics

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

Author: Daniel Iagolnitzer
Publisher: Springer Science & Business Media
ISBN: 3764324333
Release Date: 2004
Genre: Computers

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.

Geometry Topology

Author:
Publisher:
ISBN: UOM:39015072617262
Release Date: 2007
Genre: Geometry

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
ISBN: UVA:X030255316
Release Date: 2008
Genre: Mathematics

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.