# Search Results for "introduction-to-toric-varieties-am-131-annals-of-mathematics-studies"

## Introduction to Toric Varieties. (AM-131)

**Author**: William Fulton**Publisher:**Princeton University Press**ISBN:**1400882524**Category:**Mathematics**Page:**180**View:**5315

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

## Das Kontinuum diskret berechnen

**Author**: Matthias Beck,Sinai Robins**Publisher:**Springer-Verlag**ISBN:**3540795960**Category:**Mathematics**Page:**242**View:**5413

Das Gebiet des „Zählens von Gitterpunkten in Polytopen", auch Ehrhart-Theorie genannt, bietet verschiedene Verbindungen: zu elementarer endlicher Fourier-Analysis, zum Münzenproblem von Frobenius, zu Raumwinkeln, magischen Quadraten, Dedekind-Summen und vielem mehr. Dies nutzen die Autoren und knüpfen einen roten Faden, der so die grundlegenden Ideen aus diskreter Geometrie, Kombinatorik und Zahlentheorie verbindet. 250 Aufgaben und offene Probleme sowie der ansprechende Stil der Autoren laden zum Mitdenken ein.

## Strings, Gauge Fields, and the Geometry Behind

*The Legacy of Maximilian Kreuzer*

**Author**: Anton Rebhan,Ludmil Katzarkov,Johanna Knapp,Radoslav Rashkov,Emanuel Scheidegger**Publisher:**World Scientific**ISBN:**9814412562**Category:**Mathematics**Page:**568**View:**5363

This book contains exclusively invited contributions from collaborators of Maximilian Kreuzer, giving accounts of his scientific legacy and original articles from renowned theoretical physicists and mathematicians, including Victor Batyrev, Philip Candelas, Michael Douglas, Alexei Morozov, Joseph Polchinski, Peter van Nieuwenhuizen, and Peter West. Besides a collection of review and research articles from high-profile researchers in string theory and related fields of mathematics (in particular, algebraic geometry) which discuss recent progress in the exploration of string theory vacua and corresponding mathematical developments, this book contains a pedagogical account of the important work of Brandt, Dragon, and Kreuzer on classification of anomalies in gauge theories. This highly cited work, which is also quoted in the textbook of Steven Weinberg on quantum field theory, has not yet been presented in full detail except in private lecture notes by Norbert Dragon. Similarly, the software package PALP (Package for Analyzing Lattice Polytopes with applications to toric geometry), which has been incorporated in the SAGE (Software for Algebra and Geometry Experimentation) project, has not yet been documented in full detail. This book contains a user manual for a new thoroughly revised version of PALP. By including these two very useful original contributions, researchers in quantum field theory, string theory, and mathematics will find added value in a pedagogical presentation of the classification of quantum gauge field anomalies, and the accompanying comprehensive manual and tutorial for the powerful software package PALP. Contents:Gauge Field Theory, Anomalies, and Supersymmetry:BRST Symmetry and Cohomology (N Dragon and F Brandt)Aspects of Supersymmetric BRST Cohomology (F Brandt)Character Expansion for HOMFLY Polynomials: Integrability and Difference Equations (A Mironov, A Morozov and A Morozov)Bicategories in Field Theories — An Invitation (T Nikolaus and C Schweigert)The Compactification of IIB Supergravity on S5 Revisited (P van Nieuwenhuizen)String Theory and Algebraic Geometry:Max Kreuzer's Contributions to the Study of Calabi–Yau Manifolds (P Candelas)Calabi–Yau Three-Folds: Poincaré Polynomials and Fractals (A Ashmore and Y-H He)Conifold Degenerations of Fano 3-Folds as Hypersurfaces in Toric Varieties (V Batyrev and M Kreuzer)Nonassociativity in String Theory (R Blumenhagen)Counting Points and Hilbert Series in String Theory (V Braun)Standard Models and Calabi–Yaus (R Donagi)The String Landscape and Low Energy Supersymmetry (M R Douglas)The Cardy–Cartan Modular Invariant (J Fuchs, C Schweigert and C Stigner)A Projection to the Pure Spinor Space (S Guttenberg)Mathieu Moonshine and Symmetries of K3 Sigma Models (S Hohenegger)Toric Deligne–Mumford Stacks and the Better Behaved Version of the GKZ Hypergeometric System (R P Horja)Fano Polytopes (A M Kasprzyk and B Nill)Dual Purpose Landscaping Tools: Small Extra Dimensions in AdS/CFT (J Polchinski and E Silverstein)Notes on the Relation Between Strings, Integrable Models and Gauge Theories (R C Rashkov)E11, Generalised Space-Time and IIA String Theory: The R ⊗ R Sector (A Rocén and P West)The Kreuzer Bi-Homomorphism (A N Schellekens)Emergent Spacetime and Black Hole Probes from Automorphic Forms (R Schimmrigk)How to Classify Reflexive Gorenstein Cones (H Skarke)PALP — A Package for Analyzing Lattice Polytopes:PALP — A User Manual (A P Braun, J Knapp, E Scheidegger, H Skarke and N-O Walliser) Readership: Graduate students and researchers in theoretical physics and mathematics. Keywords:String Theory;Gauge Theory;Algebraic Geometry;CalabiâYau Manifolds;Toric Geometry;Lattice Polytopes;BRST Symmetry;Cohomology;Anomalies;SupersymmetryKey Features:Original research articles contributed by prominent theoretical physicists and mathematicians (Victor Batyrev, Ralph Blumenhagen, Ron Donagi, Michael Douglas, Jürgen Fuchs, Alexei Morozov, Joseph Polchinski, Bert Schellekens, Christoph Schweigert, Eva Silverstein, Peter van Nieuwenhuizen, and Peter West, among others)Previously unpublished lecture notes on the classification of quantum gauge field anomalies by Friedemann Brandt and Norbert DragonA comprehensive manual and tutorial for the powerful software package PALP that was developed originally by Kreuzer and Skarke in connection with the classification of reflexive polytopes. Together with the publication of this memorial volume an overhauled version 2.1 of PALP will be released in the public domain

## Introduction to Tropical Geometry

**Author**: Diane Maclagan,Bernd Sturmfels**Publisher:**American Mathematical Soc.**ISBN:**0821851985**Category:**Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra**Page:**363**View:**3270

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

## Recent Trends in Algebraic Combinatorics

**Author**: Hélène Barcelo,Gizem Karaali,Rosa Orellana**Publisher:**Springer**ISBN:**3030051412**Category:**Mathematics**Page:**362**View:**731

This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.

## Arrangements, Local Systems and Singularities

*CIMPA Summer School, Galatasaray University, Istanbul, 2007*

**Author**: Fouad El Zein,Alexander I. Suciu,Meral Tosun,Muhammed Uludag,Sergey Yuzvinsky**Publisher:**Springer Science & Business Media**ISBN:**9783034602099**Category:**Mathematics**Page:**305**View:**8810

This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements, Local systems and Singularities held at Galatasaray University, Istanbul during June 2007. The volume is intended for a large audience in pure mathematics, including researchers and graduate students working in algebraic geometry, singularity theory, topology and related fields. The reader will find a variety of open problems involving arrangements, local systems and singularities proposed by the lecturers at the end of the school.

## Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer 2003/2004

**Author**: Yuri Tschinkel**Publisher:**Universitätsverlag Göttingen**ISBN:**3930457512**Category:**Mathematics**Page:**229**View:**2722

## Singularités franco-japonaises

**Author**: Centre national de rencontres mathématiques (France)**Publisher:**N.A**ISBN:**N.A**Category:**Mathematics**Page:**460**View:**1547

## Memorandum

**Author**: N.A**Publisher:**N.A**ISBN:**N.A**Category:**Electrical engineering**Page:**N.A**View:**304

## Singularities and Jet Schemes

**Author**: Mircea Immanuel Mustaţǎ**Publisher:**N.A**ISBN:**N.A**Category:****Page:**104**View:**8094

## Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry

**Author**: Michel Coste**Publisher:**Societe Mathematique De France**ISBN:**9782856292365**Category:**Mathematics**Page:**125**View:**9069