Colloquium Publications, Volume: 55
2008; 785 pp; hardcover
ISBN-10: 0-8218-4210-2
ISBN-13: 978-0-8218-4210-2
Expected publication date is January 17, 2008.
The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces.
The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory.
The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions.
The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.
Readership
Graduate and research mathematicians interested in noncommutative geometry, quantum field theory and particle physics, number theory, and arithmetic algebraic geometry.
Table of Contents
Quantum fields, noncommutative spaces, and motives
The Riemann zeta function and noncommutative geometry
Quantum statistical mechanics and Galois symmetries
Endomotives, thermodynamics, and the Weil explicit formula
Appendix
Bibliography
Index
CBMS Regional Conference Series in Mathematics, Number: 108
2008; 107 pp; softcover
ISBN-10: 0-8218-4456-3
ISBN-13: 978-0-8218-4456-4
Expected publication date is January 18, 2008.
The study of convex bodies is a central part of geometry, and is particularly useful in applications to other areas of mathematics and the sciences. Recently, methods from Fourier analysis have been developed that greatly improve our understanding of the geometry of sections and projections of convex bodies. The idea of this approach is to express certain properties of bodies in terms of the Fourier transform and then to use methods of Fourier analysis to solve geometric problems. The results covered in the book include an analytic solution to the Busemann-Petty problem, which asks whether bodies with smaller areas of central hyperplane sections necessarily have smaller volume, characterizations of intersection bodies, extremal sections of certain classes of bodies, and a Fourier analytic solution to Shephard's problem on projections of convex bodies.
The book is written in the form of lectures accessible to graduate students. This approach allows the reader to clearly see the main ideas behind the method, rather than to dwell on technical difficulties. The book also contains discussions of the most recent advances in the subject. The first section of each lecture is a snapshot of that lecture. By reading each of these sections first, novices can gain an overview of the subject, then return to the full text for more details.
Readership
Graduate students and research mathematicians interested in convex geometry, emphasizing methods from harmonic analysis.
Table of Contents
Hyperplane sections of ell_p-balls
Volume and the Fourier transform
Intersection bodies
The Busemann-Petty problem
Projections and the Fourier transform
Intersection bodies and L_p-spaces
On the road between polar projection bodies and intersection bodies
Open problems
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 144
2008; 458 pp; hardcover
ISBN-10: 0-8218-4429-6
ISBN-13: 978-0-8218-4429-8
Expected publication date is January 10, 2008.
Geometric analysis has become one of the most important tools in geometry and topology. In their books on the Ricci flow, the authors reveal the depth and breadth of this flow method for understanding the structure of manifolds. With the present book, the authors focus on the analytic aspects of Ricci flow.
Some highlights of the presentation are weak and strong maximum principles for scalar heat-type equations and systems on manifolds, the classification by Bohm and Wilking of closed manifolds with 2-positive curvature operator, Bando's result that solutions to the Ricci flow are real analytic in the space variables, Shi's local derivative of curvature estimates and some variants, and differential Harnack estimates of Li-Yau-type including Hamilton's matrix estimate for the Ricci flow and Perelman's estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow.
The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. They have also attempted to give the appropriate references so that the reader may further pursue the statements and proofs of the various results.
Readership
Graduate students and research mathematicians interested in geometic analysis; geometry and topology.
Table of Contents
Weak maximum principles for scalars, tensors, and systems
Closed manifolds with positive curvature
Weak and strong maximum principles on noncompact manifolds
Qualitative behavior of classes of solutions
Local derivative of curvature estimates
Differential Harnack estimates of LYH-type
Perelman's differential Harnack estimate
An overview of aspects of Ricci flow
Aspects of geometric analysis related to Ricci flow
Tensor calculus on the frame bundle
Bibliography
Index
University Lecture Series, Volume: 42
2008; approx. 176 pp; softcover
ISBN-10: 0-8218-4434-2
ISBN-13: 978-0-8218-4434-2
Expected publication date is April 12, 2008.
This book provides careful and detailed introductions to some of the latest advances in three significant areas of rapid development in commutative algebra and its applications. The book is based on courses at the Winter School on Commutative Algebra and Applications held in Barcelona: Tight closure and vector bundles, by H. Brenner; Combinatorics and commutative algebra, by J. Herzog; and Constructive desingularization, by O. Villamayor.
The exposition is aimed at graduate students who have some experience with basic commutative algebra or algebraic geometry but may also serve as an introduction to these modern approaches for mathematicians already familiar with commutative algebra.
This book is copublished by the Real Sociedad Matematica Espanola and the American Mathematical Society.
Readership
Graduate students and research mathematicians interested in commutative algebra and algebraic geometry.
Table of Contents
H. Brenner -- Tight closure and vector bundles
J. Herzog -- Combinatorics and commutative algebra
O. Villamayor -- Notes on constructive desingularization
Publisher: Addison-Wesley
Copyright: 2008
Format: Cloth; 384 pp
ISBN-10: 0321390539
ISBN-13: 9780321390530
Published: 10/03/2007
Description
Mathematical Proofs: A Transition to Advanced Mathematics, Second Edition, prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.
Table of Contents
0. Communicating Mathematics
1. Sets
2. Logic
3. Direct Proof and Proof by Contrapositive
4. More on Direct Proof and Proof by Contrapositive
5. Existence and Proof by Contradiction
6. Mathematical Induction
7. Prove or Disprove
8. Equivalence Relations
9. Functions
10. Cardinalities of Sets
11. Proofs in Number Theory
12. Proofs in Calculus
13. Proofs in Group Theory
Answers and Hints to Selected Odd-Numbered Exercises
References
Index of Symbols
Index of Mathematical Terms