Five recent book reviews
The main topic treated in this book is a study of the properties of finite reductive groups with disconnected centre. Let G be a connected reductive group over a finite field Fq of characteristic p with Frobenius map F. The Lustig conjecture relates almost characters of the finite reductive group GF to characteristic functions of character sheaves of G (up to a scalar). If the corresponding scalars can be computed, the conjecture makes it possible to compute irreducible characters of GF. The Lustig conjecture was proved by T. Shoji for the case when the group G has a connected centre and by J.-L. Waldspurger for Sp2n and On (with p and q big enough). In the book, a circle of related problems (a parametrization of irreducible characters, a parametrization of character sheaves and computations of the character table) are treated for the case when the centre of G is not connected. It contains a review of results obtained by several authors as well as new results. In particular, the (generalized) Lusztig conjecture is proved here for the special linear group and for the special unitary group in the case that p is arbitrary and q is large enough.
Random Matrix Theory (RMT) was created by physicists in a study of statistical properties of energy levels of atomic nuclei. The book contains a series of 22 lectures on relations of RMT to problems in number theory and it clearly shows the amazing richness of the subject. Many papers are related to lectures given at a workshop on the topic organised at the Isaac Newton Institute in 2004. An important feature of the book is that it contains a number of excellent survey papers, including the paper by C. Delaunay (probabilistic group theory), D. W. Farmer (families of L-functions), A. Gamburd (symmetric functions and RMT), E. Kowalski (families of elliptic curves and random matrices), F. Rodrigues-Villegas (central values of L-functions), A. Silverberg (ranks of elliptic curves), P. Swinnerton-Dyer (2-descent on elliptic curves), D. Ulmer (functions fields and random matrices) and M. P. Young (analytic number theory and ranks of elliptic curves). Due to these expository lectures, the book may well be of help to newcomers to the field.
This interesting book is based on the author’s course given at the University of Zürich. It is addressed to people who are interested in geometric measure theory. The main aim of the book is to provide a self-contained proof of the celebrated Preiss theorem, which in particular characterizes k-dimensional rectifiable sets in an n-dimensional Euclidean space using existence of k-dimensional densities. The Preiss theorem (proved in 1987) is the culmination of the long deep research started by Besicovich in 1938. In its full generality, it gives a rectifiability criterion for measures in terms of upper and lower densities. The author of the book presents a simpler proof of its special case dealing with the density of measures. In this way, he succeeded in presenting a proof accessible for people who are not experts in the field. However, most of deep ideas that are used in the Preiss proof (in particular the method of tangent measures) are needed in this simpler proof and are carefully explained. The first four chapters contain an introduction to rectifiable sets and measures in Euclidean spaces, including basic properties of tangent measures and the most elementary rectifiability criterions. The fifth chapter contains the subtler Marstrand-Mattila rectifiability criterion. In the sixth chapter, the Preiss strategy is explained and its ingredients are proved in the following three chapters. The last chapter presents several open problems related to the topic.
The book contains a few introductory texts of the authors on an irregular version of Hodge theory, the formal reduction of differential equations with irregular singularities and Gevrey filtration on the Picard-Vessiot group of irregular differential equations. The general theme behind these expositions comprises topics like the theory of linear differential equations with irregular singularities, asymptotics of their solutions, Stokes sheaves and Galois theory of differential fields. However, the core of the book is based on reprints of correspondence among the authors (with a few exceptions) on the topics mentioned above in the period 1976-1991 (with a few exceptions). In this way, an interested reader can gain a direct insight into the evolution of the subject, culminating in applications like the geometric Langlands program.
Mathematical gauge theories investigate solution spaces of partial differential equations defined with the help of a principal bundle connection. These partial differential equations are generalizing (nowadays classical) equations introduced by physicists Yang and Mills in the realm of the strong interaction or, mathematically speaking, in the realm of principal SU(3)-bundles over four dimensional manifolds. After fundamental works by Simon Donaldson, Nathan Seiberg and Edward Witten, the gauge theory approach is extensively used in investigations of the (low dimensional) topology of manifolds. There is a strong relationship of these theories to symplectic geometry. The Floer construction is throwing light on this relationship; it can be applied either to define symplectic invariants of Lagrangian submanifolds or to define certain invariants of 3-manifolds. The Heegaard Floer homology is derived from an application of the Lagrangian Floer homology and it is conjecturally equivalent to the Seiberg-Witten theory. However, it is much more combinatorial and often more suitable for calculations.
These striking interplays between the low dimensional topology and symplectic geometry are the main subject of this book, which is freely based on lecture courses given at the Clay Mathematics Institute Summer School in Budapest, Hungary, in 2004. Preparatory and introductory parts contained in these proceedings are written in an intelligible way, often with proofs. Also, many examples and figures are included in the book. In this way it can serve for researchers active in the subject and related fields as well as for graduate students. Specific chapters of the book are devoted to the Heegaard Floer homology and knot theory, Floer homologies, contact structures, symplectic 4-manifold and Seiberg-Witten invariants. The authors are the following experts in the mentioned research field: P. Oszváth, Z. Szabó H. Goda, J. Etnyre, A. Stipsicz, P. Lisca, T. Ekholm, R. Fintushel, R. Stern, J. Park, Tian-Jun Li, D. Auroux and I. Smith.