Five recent book reviews
The main aim of this volume of the ‘Mémoires de la SMF’ is to study properties of cohomology groups for an arithmetic manifold and its totally geodesic submanifolds. The classical Lefschetz hyperplane theorem states that the restriction map induced on suitable cohomology groups by a (generic) hyperplane section of a projective manifold is injective. In the book, the author discusses a (broad) analogy of the classical Lefschetz theorem for arithmetic manifolds associated to the unitary and orthogonal groups U(p,q) and O(p,q). There are two types of results: injectivity of the map associated with a (virtual) restriction and the one associated with ‘a cup product with the class of the cycle’. The methods used are based on representation theory and a review of required results can be found in the early sections of the book.
This collection of papers is centred around topics of research of the late Andrei A. Bolibrukh. It starts with two short survey papers. The first one (written by Y. Ilyashenko) describes two Bolibrukh results (a sufficient condition for the solvability of the Riemann-Hilbert problem for Fuchsian systems and a discussion of the reduction to Birkhoff standard form). The second one (by C. Sabbah) contains a discussion of isomonodromic deformations and isomonodromic confluences.
Then there are ten other research contributions on various themes including a study of relations between two notions of integrability (M. Audin), formal power series solutions of the heat equation (W. Balser), master functions and the Schubert calculus on flag manifolds (P. Belkale, E. Mukhin and A. Varchenko), explicit solutions of the Riemann-Hilbert problem (P. Boalch), Galois theory for linear differential equations of a certain type (P. J. Cassidy, M. F. Singer), reductions of the Schlesinger equations (B. Dubrovin, M. Mazzocco), the Riemann-Hilbert correspondence for generalized KZ equations (V. A. Golubeva), monodromy groups of regular systems (V. P. Kostov), monodromy of Cherednik-Kohno-Veselov connections (V. P. Leksin) and finally a review of basic ideas of general differential Galois theory for ordinary differential equations (H. Umemura). The whole volume is dedicated to the memory of A. A. Bolibrukh and it gives an overview of recent results in the fields of his research interests.
This is a book describing the mathematics, and primarily the geometry, needed for the general theory of relativity. The first five chapters describe very basic notions. The exposition starts with groups, some linear algebra (including the Jordan canonical form of an endomorphism) and an introduction to the notion of Lie algebra, followed by metric and topological spaces, their various properties and finally the notion of a fundamental group and a covering space. The next part is devoted to differentiable manifolds and basic structures on them. Finally, the last section of this introductory part is devoted to Lie groups. This part, in fact, can be read by anybody who intends to learn the above mentioned topics. It should be mentioned that this introductory text is very nicely written.
The more specialized second part of the book starts with a chapter on the Lorentz group. It provides a lot of information about this group in compact form. It would hardly be possible to find all these facts together in one source elsewhere. Then the notion of a space-time (a 4-dimensional manifold endowed with a pseudo-Riemannian metric with signature + + + –) appears and the rest of the book deals with these space-time manifolds. Here, there are many results concerning this special branch of differential geometry: firstly the algebraic classification of space-time manifolds (above all the Petrov classification), than an investigation of the holonomy groups of these manifolds, an investigation of connections and the corresponding curvature tensors on these manifolds and finally symmetries. Isometries, homotheties, conformal symmetries, projective symmetries and curvature collineations are all studied. This second part, which represents the core of the book, will be attractive for physicists, who can revise their knowledge in a more mathematical form, but also for mathematicians working in differential geometry whose field is a little aside from space-time manifolds. Because of the first introductory part, the book is to a great extent self-contained and consequently can be recommended for students, possibly even undergraduate students with enough endurance. In the bibliography there are many references for further reading and for the study of relativity theory.
This book presents a number of important examples and constructions of pathological sets and functions as well as their properties and applications. Although the author is interested in all types of strange sets and functions, the central topic of the monograph is the question of the existence of continuous functions that are not differentiable with respect to various concepts of generalized derivatives. The book is organized as follows. After recalling basic facts from set theory, topology and measure theory, examples of Cantor and Peano type functions are constructed. The author continues with basic properties of the space of Baire-one functions and with applications to separately continuous functions. Particular subclasses of semicontinuous functions are analysed in the next chapter. The next part is devoted to a study of the differentiability properties of real functions. Monotone functions are investigated and some pathological examples of monotone functions are constructed. The next chapters deal with everywhere differentiable nowhere monotone functions and nowhere approximately differentiable functions.
A connection between category and measurability is another important part of the monograph. Blumberg's theorem and Sierpinski-Zygmund functions are followed by examples and properties of Lebesgue nonmeasurable functions and functions without the Baire property. After that, bad solutions of the Cauchy functional equation are discussed. Luzin and Sierpinski sets along with applications are presented in the next chapter. Interesting relations between absolutely nonmeasurable functions and the measure extension problem are shown, followed by examples witnessing the fail of Egorov type theorems for pathological functions. Several results connected with Sierpinski's partition of the Euclidean plane follow. Examples of bad functions defined on second category sets conclude this part of the book. The next chapter is devoted to sup-measurable and weakly sup-measurable functions and to applications in the theory of ordinary differential equations.
In the final part of the book, the family of continuous nondifferentiable functions is considered from the point of view of category and measure. A short scheme for constructing the classical Wiener measure is presented along with some simple but useful statements from the theory of stochastic processes. Since the book is self-contained and the proofs are explained in detail, it should be accessible for all kinds of interested readers. All chapters are endowed with a large number of exercises that vary from elementary to difficult and provide a deeper insight into topics presented in the book. Thus the monograph is a good companion in the realm of pathological objects and counterexamples in real analysis.
The aim of this memoir is to define the homotopy category of (Noetherian, separated and admitting an ample family) schemes and to show that this category plays the same role for smooth (Noetherian, separated and admitting an ample family) schemes as the classical homotopy category does for differentiable (topological) varieties. For schemes of finite Krull dimension, the combinatorial definition of Morel agrees with the topological approach (based on the concept of sheaves in the Nisnevich topology) of Voevodsky. The main results of this monograph are the homotopic purity theorems, which are the foundation of localization exact sequences for any oriented cohomology theory and Poincare duality; the Thom space of closed immersions between two smooth schemes of the above type depends only on the normal bundle of the immersion. The main technical obstacle in comparison to the case of differentiable varieties is that one does not have the notion of a tubular neighbourhood. A suitable replacement of this device relies on techniques of deformations of the normal bundle.