Series: UNITEXT, Vol. 66
Subseries: La Matematica per il 3+2
2013, XII, 101 p. 126 illus.
Softcover Information
Softcover
ISBN 978-3-642-36242-2
Due: April 30, 2013
Written by one of the most eminent mathematicians of the 20th and 21st centuries
Masterful exposition
Lecture courses do not get better than this
This book is concerned with one of the most fundamental questions of mathematics: the relationship between algebraic formulas and geometric images.
At one of the first international mathematical congresses (in Paris in 1900), Hilbert stated a special case of this question in the form of his 16th problem (from his list of 23 problems left over from the nineteenth century as a legacy for the twentieth century).
In spite of the simplicity and importance of this problem (including its numerous applications), it remains unsolved to this day (although, as you will now see, many remarkable results have been discovered).
Content Level ≫ Lower undergraduate
Keywords ≫ algebraic curves - conic sections - projective geometry
Related subjects ≫ Algebra - Geometry & Topology - Theoretical, Mathematical & Computational Physics
Publisher's Foreword.- Editors' Foreword.- Introduction.- 2 Geometry of Conic Sections.- 3 The Physics of Conic Sections and Ellipsoids.- 4 Projective Geometry.- 5 Complex Algebraic Curves.- 6 A Problem for School Pupils.- A Into How Many Parts do n Lines Divide the Plane?- Editors' Comments on Gudkov's Conjecture.- Notes
Series: Advanced Courses in Mathematics - CRM Barcelona
2013, Approx. 270 p.
Softcover
ISBN 978-3-0348-0617-6
Due: April 30, 2013
About this textbook.
The book contains the first published notes on the recent developments and major changes in Galois deformation theory during the last decade (deformations of pseudo-representations, framed deformations, groupoids, etc.)
A survey on the parity conjecture is presented
Computational aspects of Hilbert modular forms are presented by the people responsible for the most powerful and widely spread algorithms available
The notes in this volume correspond to advanced courses given at the Centre de Recerca Matematica (Bellaterra, Barcelona, Spain) during the Research Programme in Arithmetic Geometry that took place during the academic year 2009-2010. They appear in printed form due to the many requests received by the organizers to make the content of the courses publicly available. The material covers the theory of p-adic Galois representations and Fontaine rings, Galois deformation theory, arithmetic and computational aspects of Hilbert modular forms, and the parity conjecture for elliptic curves.?
Content Level ≫ Graduate
Keywords ≫ Galois representations - Hilbert modular forms - elliptic curves
Related subjects ≫ Birkhauser Mathematics
Part I: Galois Deformations.- On p-adic Galois Representations.- Deformations of Galois Representations.- Part II: Hilbert Modular Forms.- Arithmetic Aspects of Hilbert Modular Forms and Varieties.- Explicit Methods for Hilbert Modular Forms.- Part III: Elliptic Curves.- Notes on the Parity Conjecture.
Series: Springer Monographs in Mathematics
2013, X, 523 p. 78 illus.
Hardcover
ISBN 978-4-431-54396-1
Due: August 15, 2013
The perfect book for acquiring fundamental knowledge of simplicial complexes and the theories of dimension and retracts
Many proofs are illustrated by figures or diagrams for easier understanding
Fascinating problems in the final chapter enable readers to understand how deeply related the theories of dimension and retracts are
This book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars.
Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X × I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are.
Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.
Content Level ≫ Research
Related subjects ≫ Analysis - Geometry & Topology
Series: Lecture Notes in Mathematics, Vol. 2074
2013, X, 292 p. 11 illus., 2 in color.
Softcover
ISBN 978-3-642-36432-7
Due: April 30, 2013
It contains a quick and up to date introduction to viscosity solutions of Hamilton-Jacobi equations for graduate students or young researchers.
Two approaches to large time behavior of periodic solutions of Hamilton-Jacobi Equations are given: PDE and weak KAM theory.
Contributions on hot topics like for example numerical for mean field games should be of interest for many researchers
These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).
Content Level ≫ Research
Keywords ≫ 49-02, 35F21, 49L25, 35B40, 49N70, 91A13, 14T05,15A80 - Hamilton Jacobi Bellman equations - Large time behavior - Mean field games - Tropical and idempotent analysis
Related subjects ≫ Applications - Computational Science & Engineering - Dynamical Systems & Differential Equations - Mathematics
Finite Difference Methods For Mean Field Games.- An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications.- A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations.- Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations.
2013, XII, 320 p. 11 illus.
Hardcover
ISBN 978-1-4614-6481-5
Due: May 31, 2013
Leading experts report on recent advances in higher-dimensional birational geometry, with special regard to arithmetic applications
Highlights the tight connections between arithmetic and geometry
Documents the central role of the theory of rational curves
This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families.
This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.
Content Level ≫ Research
Keywords ≫ arithmetic of higher-dimensional algebraic varieties - birational geometry - cone of curves - rational curves - rationality - unirationality
Related subjects ≫ Algebra - Geometry & Topology - Number Theory and Discrete Mathematics
Foreword.- Introduction.- A. Bertram and I. Coskun, The birational geometry of the Hilbert scheme of points on surfaces.- F. Bogomolov and Ch. Bohning, Isoclinism and stable cohomology of wreath products.- F. Bogomolov, I. Karzhemanov, and K. Kuyumzhiyan, Unirationality and existence of infinitely transitive models.- I. Cheltsov, L. Katzarkov, and V. Przyjalkowski, Birational geometry via moduli spaces.- O. Debarre, Curves of low degrees on projective varieties.- S. Kebekus, Uniruledness criteria and applications.- S. Kovacs, The cone of curves of K3 surfaces revisited.- V. Lazi?, Around and beyond the canonical class.- C. Liedtke, Algebraic surfaces in positive characteristic.- A. Varilly-Alvarado, Arithmetic of Del Pezzo surfaces.