American Mathematical Society

Probability.

Designed for a cohesive undergraduate one-semester course in measure-theoretic probability for students whose schedules of study may not accommodate a one-year course. The brevity and conciseness with which this is written nevertheless covers the material at a moderately deep level. Khoshnevisan (mathematics, U. of Utah)also accommodates single-semester graduate courses as he covers the basics in terms of classical probability, including discrete and conditional probability, independence, discrete distributions, absolutely continuous distributions, expectations and variance, as well as Bernoulli trials, measure theory, integration, product spaces, the central limit theorem, martingales, Brownian motion, and stochastic integration. He includes background materials on Hilbert spaces and Fourier series in appendices along with exercises for each chapter. (Annotation ©2007 Book News Inc. Portland, OR)

Pseudo-differential operators and the Nash-Moser theorem.

The two topics are generally treated separately, each being of interest in its own right, but Alinhac and Gérard (not further identified) combine them partly in order to show the deep connection between them, especially in terms of the Littlewood-Paley theory, dyadic analysis, and the para-differential calculus and its application to interpolated inequalities. The treatment can be used as a textbook for a graduate course on nonlinear partial differential equations for geometry. Opérateurs pseudo-différentiels et thèoréme de Nash-Moser was published in 1991 by EDP Science, Paris. (Annotation ©2007 Book News Inc. Portland, OR)

The Ricci flow; techniques and applications, pt.1: Geometric aspects.

This is the first volume of a two-part sequel of The Ricci Flow: An Introduction by the same authors, which laid out the foundations for the study of Richard Hamilton's Ricci flow, an evolution equation which deforms Riemannian metrics by evolving them in the direction of minus the Ricci tensor. In this volume, they continue to explore the fundamental properties of the Ricci flow, with particular emphasis on their application to the study of singularities. The volume's eight chapters discuss Ricci solitons; Kähler-Ricci flow and Kähler- Ricci solitons; the compactness theorem for Ricci flow and its proof; energy, monotonicity, and breathers; entropy and non local collapsing; the reduced distance and its applications; and the basic topology of 3-manifolds. (Annotation ©2007 Book News Inc. Portland, OR)

Singularities in geometry and topology, 2004.

As evidence of the continued partnership of French and Japanese mathematicians, this collection of 14 articles includes significant advances taken in collaboration or in support of each others' work in leading-edge studies of geometry and topology. These papers, drawn from the Singularities in Geometry and Topology Third Franco-Japanese Symposium on Singularities held December 2004 in Sapporo, Japan, cover a range of topics in English, including papers on the cobordism of fibered knots and related topics, the proportionality of indices of I-forms on singular varieties, motivic sheaves and intersection cohomology, hyperbolic perturbations of algebraic links and small Mahler measure, stably hyperbolic polynomials, weighted-degrees for algebraic local cohomologies associated with semi-quasi-homogeneous singularities, perverse shaves and Milno fibers over singular varieties, Horn-Kapranov uniformity of the criminantal loci, duality of Euler data for affine varieties, plane curve singularities whose Milnor and Tjurina numbers differ by three and characteristic classes of (pro)algebraic varieties. Distributed in the US by the American Mathematical Society. (Annotation ©2007 Book News Inc. Portland, OR)

Singularity theory and its applications; proceedings.

Drawn from those submitted to the September 2003 conference, these papers represent a broad range of topics in singularity theory and its applications, including motivic integrations, Thom polynomials, complex analytic singularity theory, Lagrangian singularity theory and generic differential geometry. The papers, in English and French, include such subjects as invariants of combinatorial line arrangements and Rybnikov's example, time averaged optimization of dynamic inequalities on a circle, Thom polynomial computing strategies, quasi-convex decomposition in o-minimal structures, homotopy groups of complements to ample divisions, Massey products of complex hypersurface complements, weighted homogeneous polynomials and blow-analytic equivalence, an infinitesimal criterion for topological triviality of families of sections of analytical variants, valuations and local uniformization, and finite Dehn surgery along A'Campo's divide knots. Distributed in the US by the American Mathematical Society. (Annotation ©2007 Book News Inc. Portland, OR)

Spectral theory and mathematical physics; proceedings; 2v.

This combination proceedings from the symposia in March 2006 and festschrift in honor of Dr. Barry Simon reflects Simon's myriad interests and areas of expertise. This set includes recent research in quantum field theory, statistical mechanics and nonrelativistic quantum systems in the first volume and ergodic Schrödinger operators, singular spectrum, orthogonal polynomials and inverse spectral theory in the second. General topics include the aforesaid as well as nonrelativistic two-body and N-body quantum systems, resonances, electrical and magnetic fields, semiclassical limit, random ergodic Schrödinger operators and singular continuous spectrum. The editors have provided a selection of "Barry stories," all of which are mathematical and pure, a genealogy of Simon's mathematical heritage and, a poster of the event, and, all kidding aside, an impressive list of his publications. (Annotation ©2007 Book News Inc. Portland, OR)

Systolic geometry and topology.

Katz (mathematics, Bar Ilan U.) presents the systolic geometry of manifolds and polyhedra, starting with two classical inequalities and then proceeding to recent results, including a proof of Gromov's filling area conjecture in a hyperelliptic setting. Along the way he describes the literature and historical progression of the study, the work of Gauss, the global geometry of surfaces, inequalities of Loewner and Pu, systolic applications of integral geometry, surfaces at a basic level, Loewner's hyperelliptic surfaces, an optimal inequality for CAT(0) metrics, volume entropy and asymptotic upper bounds. He then moves to systolic geometry and topology in n dimensions, covering systoles and their category, Gromov's optimal stable systolic inequality and those dependent on Massey products, cup products and stable systoles, dual-critical lattices and systoles, generalized degree and Loewner-type inequalities, higher inequalities of the Loewner-Gromov type, Systolic inequalities for certain norms and Four-manifold systolic asymtotics. (Annotation ©2007 Book News Inc. Portland, OR)

Topological algebras and applications; proceedings.

The main topic of this summer 2005 conference was the general theory of topological algebras and its various applications, with emphasis on the non-normed case. In addition to studying the internal structure of non-normed and non-locally convex topological algebras, papers from the conference offer insight on applications to other branches of mathematics, such as differential geometry of smooth manifolds, and to mathematical physics, in areas such as quantum relativity and quantum cosmology. Operator theory of unbounded operators and related non-normed topological algebras are intensively examined. Other topics include topological homological algebra, topological algebraic geometry, sheaf theory, and L-theory. There is no subject index. (Annotation ©2007 Book News Inc. Portland, OR)

Universality and renormalization; from stochastic evolution to renormalization of quantum fields.

This collection of articles covers a wide range of phenomena in the natural sciences dominated by notions of universality and renormalization. They are equally broad in their approach to these phenomena, offering the mathematical view as well as the perspective of the applied sciences, exploring renormalization in quantum field theory and statistical physics and its connection to modern mathematics as well as physics on scales from the microscopic to the macroscopic as they cover manifestly gauge invariant exact renormalization groups, SLE(8/3) and Brownian excursions in annuli, Cardy's formula on the triangular lattice, Rota-Baxter algebras in renormalization of perturbative quantum field theory, practicalities of renormalizing quantum field theories, quantum field theory in curved spacetime, entropy of XY spin chain and block Toeplitz determinants, the quantitative boundary behavior of SLE, the configurational measure on mutually avoiding SLE paths, Dyson-Schwinger equations from Hopf algebras to number theory, two-sided SLE8/3 and the infinite self-avoiding polygon, using the renormalization group, short distance behavior of scaling functions for the planar Ising model, constructing conformal field theory models, the art of computing loop integrals and the transition temperature of the weakly interacting Bose gas. (Annotation ©2007 Book News Inc. Portland, OR)

The universal Kobayashi-Hitchin correspondence on Hermitian manifolds.

Lubke and Teleman prove a general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds and discuss differential geometric properties of the corresponding moduli spaces. Their Kobayashi-Hitchin correspondence relates the complex geometric concept of polystable oriented holomorphic pairs to the existence of a reduction solving a generalized Hermitian-Einstein equation, with proof based on the Uhlenbeck-Yau continuity method. Using ideas from Donaldson theory and other sources they cover the finite dimensional Kobayashi-Hitchin correspondence, a universal complex geometric classification problem, Hermitian-Einstein pairs, and polystable pairs that allow Hermitian-Einstein reductions and give examples and applications. (Annotation ©2007 Book News Inc. Portland, OR)

The Volterra chronicles; the life and times of an extraordinary mathematician, 1860-1940.

This is a biography of Vito Volterra (1860-1940), the Italian mathematician and physicist who rose from the Jewish ghettos to become the founder and president of Italy's National Research Council. While his academic career is naturally a strong focus of the biography, politics plays a major part as well, in that Volterra served time as an Italian Senator and was compelled to resign his university post for being one of only 12 Italian professors to refuse to sign an oath of loyalty to Benito Mussolini's fascist regime. (Annotation ©2007 Book News Inc. Portland, OR)

Weil-Petersson metric on the Universal Teichmüller space.

Takhtajan and Teo prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold and show that the connected component of the connected component of the identity of T(1), a Hilbert submanifold, is a topological group. They define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, execute its Riemann curvature tensor, and show that T(1) is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. They present their proof by describing curvature properties and Chern forms by way of the universal Teichmuller space, T(1) as a Hilbert manifold and To(1) as a topological group, Velling-Kirillov metrics, characteristic forms of the universal Teichmuller curve. First and second variations of the hyperbolic metric, and Riemann curvature tensor, followed by a comprehensive treatment of Kahler potential and period mapping. (Annotation ©2007 Book News Inc. Portland, OR)