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Lectures on the topological recursion for Higgs bundles and quantum curves
by Olivia Dumitrescu, Motohico Mulase - arXiv , 2015
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the discovery of the relation between the topological recursion and the quantization of Hitchin spectral curves associated with Higgs bundles.
Current Developments in Algebraic Geometry
by Lucia Caporaso, et al. - Cambridge University Press , 2012
An introductory panorama of current progress in the field, addressed to both beginners and experts. This volume offers expository overviews of the state of the art in many areas of algebraic geometry. Prerequisites are kept to a minimum ...
Brief Introduction to Tropical Geometry
by E. Brugallé, I. Itenberg, G. Mikhalkin, K. Shaw - arXiv , 2015
Lecture notes for a mini-course. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.
Convex Bodies and Algebraic Geometry
by Tadao Oda - Springer , 1988
The theory of toric varieties describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications ...
Ample Subvarieties of Algebraic Varieties
by Robin Hartshorne - Springer , 1970
These notes are an enlarged version of a three-month course of lectures. Their style is informal. I hope they will serve as an introduction to some current research topics, for students who have had a one year course in modern algebraic geometry.
Abel's Theorem and the Allied Theory
by H.F. Baker - Cambridge University Press , 1897
This classic book covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today.
Geometric Complexity Theory: An Introduction for Geometers
by J.M. Landsberg - arXiv , 2013
This is survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory. The article is written to be accessible to graduate students. Numerous open questions are presented.
Introduction to Algebraic Geometry
by Sudhir R. Ghorpade - Indian Institute of Technology Bombay , 2007
This text is a brief introduction to algebraic geometry. We will focus mainly on two basic results in algebraic geometry, known as Bezout's Theorem and Hilbert's Nullstellensatz, as generalizations of the Fundamental Theorem of Algebra.
From D-modules to Deformation Quantization Modules
by Pierre Schapira - UPMC , 2012
The aim of these lecture notes is first to introduce the reader to the theory of D-modules in the analytical setting and also to make a link with the theory of deformation quantization (DQ for short) in the complex setting.
Classical Algebraic Geometry: A Modern View
by Igor V. Dolgachev - Cambridge University Press , 2012
The main purpose of the present treatise is to give an account of some of the topics in algebraic geometry which while having occupied the minds of many mathematicians in previous generations have fallen out of fashion in modern times.
Lectures on Birational Geometry
by Caucher Birkar - arXiv , 2012
Topics covered: introduction into the subject, contractions and extremal rays, pairs and singularities, Kodaira dimension, minimal model program, cone and contraction, vanishing, base point freeness, flips and local finite generation, etc.
Lectures on Moduli of Curves
by D. Gieseker - Tata Institute of Fundamental Research , 1982
These lecture notes are based on some lectures given in 1980. The object of the lectures was to construct a projective moduli space for stable curves of genus greater than or equal two using Mumford's geometric invariant theory.
Lectures on Curves on Rational and Unirational Surfaces
by Masayoshi Miyanishi - Tata Institute of Fundamental Research , 1978
From the table of contents: Introduction; Geometry of the affine line (Locally nilpotent derivations, Algebraic pencils of affine lines, Flat fibrations by the affine line); Curves on an affine rational surface; Unirational surfaces; etc.
Lectures on Torus Embeddings and Applications
by Tadao Oda - Tata Institute of Fundamental Research , 1978
Theory of torus embeddings has find many applications. The point of the theory lies in its ability of translating meaningful algebra-geometric phenomena into very simple statements about the combinatorics of cones in affine space over the reals.
Lectures on Expansion Techniques In Algebraic Geometry
by S.S. Abhyankar - Tata Institute Of Fundamental Research , 1977
From the table of contents: Meromorphic Curves; G-Adic Expansion and Approximate Roots; Characteristic Sequences of a Meromorphic Curve; The Fundamental Theorem and applications; Irreducibility, Newton's Polygon; The Jacobian Problem.
Lectures on Deformations of Singularities
by Michael Artin - Tata Institute of Fundamental Research , 1976
These notes are based on a series of lectures given in 1973. The lectures are centered about the work of M. Scahlessinger and R. Elkik on infinitesimal deformations. Contents: Formal Theory and Computations; Elkik's Theorems on Algebraization.
Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group
by J.P. Murre - Tata Institute of Fundamental Research , 1967
The purpose of this text is to give an introduction to Grothendieck's theory of the fundamental group in algebraic geometry with the study of the fundamental group of an algebraic curve over an algebraically closed field of arbitrary characteristic.
Lectures On Old And New Results On Algebraic Curves
by P. Samuel - Tata Institute Of Fundamental Research , 1966
The aim of this text is to give a proof, due to Hans Grauert, of an analogue of Mordell's conjecture. Contents: Introduction; Algebro-Geometric Background; Algebraic Curves; The Theorem of Grauert (Mordell's conjecture for function fields).
by Johan de Jong, et al. , 2012
The stacks project aims to build up enough basic algebraic geometry as foundations for algebraic stacks. This implies a good deal of theory on commutative algebra, schemes, varieties, algebraic spaces, has to be developed en route.
by Winfried Bruns, Udo Vetter - Springer , 1988
Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. The book gives a coherent treatment of the structure of determinantal rings. The approach is via the theory of algebras with straightening law.
Introduction to Algebraic Topology and Algebraic Geometry
by U. Bruzzo , 2008
Introduction to algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tools necessary for algebraically integrable systems and the geometry of quantum field and string theory.
Lectures on Siegel's Modular Functions
by H. Maass - Tata Institute of Fundamental Research , 1955
Contents: Modular Group of Degree n; Symplectic group of degree n; Reduction Theory of Positive Definite Quadratic Forms; Fundamental Domain of the Modular Group of Degree n; Modular Forms of Degree n; Algebraic dependence of modular forms; etc.
Analysis on Homogeneous Spaces
by Ralph Howard - Royal Institute of Technology Stockholm , 1994
The main goal of these notes is to give a proof of the basic facts of harmonic analysis on compact symmetric spaces and then to apply these to concrete problems involving things such as the Radon and related transforms on these spaces.
Strings and Geometry
by M. Douglas, J. Gauntlett, M. Gross - American Mathematical Society , 2004
This volume highlights the interface between string theory and algebraic geometry. The topics covered include manifolds of special holonomy, supergravity, supersymmetry, D-branes, the McKay correspondence and the Fourier-Mukai transform.
by Cumrun Vafa, Eric Zaslow - American Mathematical Society , 2003
The book provides an introduction to the field of mirror symmetry from both a mathematical and physical perspective. After covering the relevant background material, the monograph is devoted to the proof of mirror symmetry from various viewpoints.
Lectures on Algebraic Groups
by Alexander Kleshchev - University of Oregon , 2005
Contents: General Algebra; Commutative Algebra; Affine and Projective Algebraic Sets; Varieties; Morphisms; Tangent spaces; Complete Varieties; Basic Concepts; Lie algebra of an algebraic group; Quotients; Semisimple and unipotent elements; etc.
Algorithms in Real Algebraic Geometry
by S. Basu, R. Pollack, M. Roy - Springer , 2009
The monograph gives a detailed exposition of the algorithmic real algebraic geometry. It is well written and will be useful both for beginners and for advanced readers, who work in real algebraic geometry or apply its methods in other fields.
Introduction to Stokes Structures
by Claude Sabbah - arXiv , 2010
The purpose of these lectures is to introduce the notion of a Stokes-perverse sheaf as a receptacle for the Riemann-Hilbert correspondence for holonomic D-modules. They develop the original idea of P. Deligne in dimension one.
Linear Systems Theory and Introductory Algebraic Geometry
by Robert Hermann - Math Sci Press , 1974
Systems theory offers a unified mathematical framework to solve problems in a wide variety of fields. This mathematics is not of the traditional sort involved in engineering education, but involves virtually every field of modern mathematics.
Computations in Algebraic Geometry with Macaulay 2
by D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels - Springer , 2001
This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out.
by J. S. Milne , 2008
Introduction to both the geometry and the arithmetic of abelian varieties. It includes a discussion of the theorems of Honda and Tate concerning abelian varieties over finite fields and the paper of Faltings in which he proves Mordell's Conjecture.
Modular Functions and Modular Forms
by J. S. Milne , 2009
This is an introduction to the arithmetic theory of modular functions and modular forms, with an emphasis on the geometry. Prerequisites are the algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses.
Introduction to Projective Varieties
by Enrique Arrondo - Universidad Complutense de Madrid , 2007
The scope of these notes is to present a soft and practical introduction to algebraic geometry, i.e. with very few algebraic requirements but arriving soon to deep results and concrete examples that can be obtained 'by hand'.
An Introduction to Semialgebraic Geometry
by Michel Coste - Universite de Rennes , 2002
Semialgebraic geometry is the study of sets of real solutions of systems of polynomial equations and inequalities. These notes present the first results of semialgebraic geometry and related algorithmic issues. Their content is by no means original.
An Introduction to Complex Algebraic Geometry
by Chris Peters - Institut Fourier Grenoble , 2004
This is an advanced course in complex algebraic geometry presupposing only some familiarity with theory of algebraic curves or Riemann surfaces. The goal is to understand the Enriques classification of surfaces from the point of view of Mori-theory.
Lectures on Logarithmic Algebraic Geometry
by Arthur Ogus - University of California, Berkeley , 2006
Logarithmic geometry deals with two problems in algebraic geometry: compactification and degeneration. Contents: The geometry of monoids; Log structures and charts; Morphisms of log schemes; Differentials and smoothness; De Rham and Betti cohomology.
Algebraic Geometry over the Complex Numbers
by Donu Arapura - Purdue University , 2009
Algebraic geometry is the geometric study of sets of solutions to polynomial equations over a field (or ring). In this book the author maintains a reasonable balance between rigor and intuition; so it retains the informal quality of lecture notes.
Introduction to Algebraic Geometry
by Yuriy Drozd , 1999
From the table of contents: Affine Varieties; Ideals and varieties. Hilbert's Basis Theorem. Regular functions and regular mappings. Projective and Abstract Varieties; Dimension Theory; Regular and singular points; Intersection theory.
by Andreas Gathmann - University of Kaiserslautern , 2003
From the contents: Introduction; Affine varieties; Functions, morphisms, and varieties; Projective varieties; Dimension; Schemes; First applications of scheme theory; More about sheaves; Cohomology of sheaves; Intersection theory; Chern classes.
Current Topics in Complex Algebraic Geometry
by Herbert Clemens, János Kollár - Cambridge University Press , 1996
The 1992/93 year at the Mathematical Sciences Research Institute was devoted to Complex Algebraic Geometry. This volume collects articles that arose from this event, which took place at a time when algebraic geometry was undergoing a major change.
Algebraic Groups and Discontinuous Subgroups
by Armand Borel, George D. Mostow - American Mathematical Society , 1966
The book covers linear algebraic groups and arithmetic groups, adeles and arithmetic properties of algebraic groups, automorphic functions and spectral decomposition of L2-spaces, vector valued cohomology and deformation of discrete subgroups, etc.
by Marc Levine - American Mathematical Society , 1998
This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry.
Quasi-Projective Moduli for Polarized Manifolds
by Eckart Viehweg - Springer , 1995
This book discusses two subjects of quite different nature: Construction methods for quotients of quasi-projective schemes by group actions or by equivalence relations and properties of direct images of certain sheaves under smooth morphisms.
by J.S. Milne , 2008
These notes are an introduction to the theory of algebraic varieties. In contrast to most such accounts they study abstract algebraic varieties, not just subvarieties of affine and projective space. This approach leads naturally to scheme theory.
Algebraic geometry and projective differential geometry
by Joseph M. Landsberg - arXiv , 1998
Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, and more.
Complex Analytic and Differential Geometry
by Jean-Pierre Demailly - Universite de Grenoble , 2007
Basic concepts of complex geometry, coherent sheaves and complex analytic spaces, positive currents and potential theory, sheaf cohomology and spectral sequences, Hermitian vector bundles, Hodge theory, positive vector bundles, etc.
Algebraic Curves: an Introduction to Algebraic Geometry
by William Fulton - Benjamin , 1969
These notes develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. It assumed that the reader is familiar with some basic properties of rings, ideals, and polynomials.
Homogeneous Spaces and Equivariant Embeddings
by Dmitri A. Timashev - arXiv , 2006
A monograph on homogeneous spaces of algebraic groups and their equivariant embeddings. Some results are supplied with proofs, the other are cited with references to the original papers. The style is intermediate between survey and detailed monograph.
by Kiran S. Kedlaya , 2006
This is not a typical math textbook, it does not present full developments of key theorems, but it leaves strategic gaps in the text for the reader to fill in. The original text underlying this book was a set of notes for the Math Olympiad Program.