Contemporary Mathematics, Volume: 528
2010; 137 pp; softcover
ISBN-13: 978-0-8218-4804-3
This volume contains contributions from the INdAM School on Symmetry for Elliptic PDEs, which was held May 25-29, 2009, in Rome, Italy. The school marked "30 years after a conjecture of De Giorgi, and related problems" and provided an opportunity for experts to discuss the state of the art and open questions on the subject.
Motivated by the classical rigidity properties of the minimal surfaces, De Giorgi proposed the study of the one-dimensional symmetry of the monotone solutions of a semilinear, elliptic partial differential equation. Impressive advances have recently been made in this field, though many problems still remain open. Several generalizations to more complicated operators have attracted the attention of pure and applied mathematicians, both for their important theoretical problems and for their relation, among others, with the gradient theory of phase transitions and the dynamical systems.
Graduate students and research mathematicians interested in elliptic PDEs.
2010; 344 pp; hardcover
ISBN-13: 978-0-8218-4940-8
Expected publication date is January 5, 2011.
The 3x+1 problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer x is odd then "multiply by three and add one", while if it is even then "divide by two". The 3x+1 problem asks whether, starting from any positive integer, repeating this procedure over and over will eventually reach the number 1. Despite its simple appearance, this problem is unsolved. Generalizations of the problem are known to be undecidable, and the problem itself is believed to be extraordinarily difficult.
This book reports on what is known on this problem. It consists of a collection of papers, which can be read independently of each other. The book begins with two introductory papers, one giving an overview and current status, and the second giving history and basic results on the problem. These are followed by three survey papers on the problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computati
Clay Mathematics Proceedings, Volume: 11
2010; 675 pp; softcover
ISBN-13: 978-0-8218-5203-3
Expected publication date is January 14, 2011.
The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics.
Specific themes covered by the articles are as follows:
entropy in operator algebras, regular C^*-algebras of integral domains, properly infinite C^*-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces;
von Neumann algebras, fundamental Group of \mathrm{II}_1 factors, subfactors and planar algebras;
Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory;
cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem;
noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras;
Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities;
cyclotomy and analytic geometry over F_1, quantum modular forms;
differential K-theory, cyclic theory and S-cohomology.
Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Graduate students and research mathematicians interested in recent developments in various areas of mathematics.
P. Baum, N. Higson, and T. Schick -- A geometric description of equivariant K-homology for proper actions
D. Bisch, P. Das, and S. K. Ghosh -- The planar algebra of diagonal subfactors
E. Blanchard -- K_1-injectivity for properly infinite C^*-algebras
S. Brain and G. Landi -- Families of monads and instantons from a noncommutative ADHM construction
D. Burghelea -- Cyclic theory for commutative differential graded algebras and S-cohomology
P. Cartier -- Vinberg algebras, Lie groups and combinatorics
A. H. Chamseddine -- Noncommutative geometry as the key to unlock the secrets of space-time
J. Cuntz and X. Li -- The regular C^*-algebra of an integral domain
M. Dubois-Violette -- Noncommutative coordinate algebras
A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko -- Random matrices, free probability, planar algebras and subfactors
N. Higson -- The tangent groupoid and the index theorem
M. Karoubi -- Le theoreme de periodicite en K-theorie hermitienne
M. Khalkhali -- A short survey of cyclic cohomology
D. Kreimer -- The core Hopf algebra
V. Lafforgue -- Propriete (T) renforcee et conjecture de Baum-Connes
J.-L. Loday and M. Ronco -- Combinatorial Hopf algebras
Yu. I. Manin -- Cyclotomy and analytic geometry over \mathbb{F}_1
M. Marcolli -- Motivic renormalization and singularities
F. Martin and A. Valette -- Free groups and reduced 1-cohomology of unitary representations
H. Moscovici -- Local index formula and twisted spectral triples
A. Perez and C. Rovelli -- Observables in quantum gravity
S. Popa and S. Vaes -- On the fundamental group of II_1 factors and equivalence relations arising from group actions
M. A. Rieffel -- Leibniz seminorms for "matrix algebras converge to the sphere"
J. Simons and D. Sullivan -- Structured vector bundles define differential K-theory
E. Stormer -- Entropy in operator algebras
R. Wulkenhaar -- Non-compact spectral triples with finite volume
G. Yu -- A characterization of the image of the Baum-Connes map
D. Zagier -- Quantum modular forms
2010; 229 pp; hardcover
ISBN-13: 978-973-87899-6-8
The volume represents the proceedings of the 22nd International Conference on Operator Theory, held in Timi?oara, Romania, from July 3 to July 8, 2008. It includes a survey on Carleson measures and composition operators, as well as eighteen papers containing original research on a large variety of topics: single operator theory, Banach algebras, C^*-algebras, von Neumann algebras, moment problems, differential and integral operators, noncommutative probability, and spectral theory.
Graduate students and research mathematicians interested in analysis.
P. Albayrak and F. Celiker -- On invariant ideals on locally convex solid Riesz spaces
C. Benhida and D. Timotin -- Automorphism invariance properties for certain families of multioperators
H. Bercovici and L. Kerchy -- Spectral behaviour of C_{10}-contractions
M. R. Buneci -- Haar systems for double groupoids
G. Cassier and J. Verliat -- Stability for some operator classes by Aluthge transform
K. R. Davidson and E. G. Katsoulis -- Biholomorphisms of the unit ball of \mathbb{C}^n and semicrossed products
J. Janas -- Asymptotic of solutions of some linear difference equations and applications to unbounded Jacobi matrices
A. Juratoni -- On operator representations of weak*-Dirichlet algebras
L. D. Lemle -- L^\infty-uniqueness for one-dimensional diffusions
L. Lemnete-Ninulescu -- Positive-definite operator-valued functions and the moment problem
F. Pop and R. R. Smith -- On the cohomology groups of certain type I von Neumann algebras with coefficients in K(H)
M. Popa -- Non-crossing linked partitions and multiplication of free random variables
G. T. Pr?jitur? -- The geometry of an orbit
H. Queffelec -- Carleson measures and composition operators
M. ?abac -- Commuting systems of bounded operators, factorizations, and spectral correspondence
A. Skripka -- On the centralizer of a one-parameter representation
S. M. Stoian -- Spectrum of bounded operators on locally convex spaces
C. Stoica and M. Megan -- Nonuniform behaviors for skew-evolution semiflows in Banach spaces
L. Suciu and N. Suciu -- Selection of semispectral measures for bicontractions
Student Mathematical Library, Volume: 55
2010; 156 pp; softcover
ISBN-13: 978-0-8218-4829-6
Expected publication date is December 9, 2010.
The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation by considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equation and the closely related notion of harmonic functions from a probabilistic perspective.
The theme of the first two chapters of the book is the relationship between random walks and the heat equation. The first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set.
The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Undergraduate students interested in probability and connections between probability and classical analysis.