To many mathematicians, the phrase “dynamical systems” refers mainly to transformations and flows on smooth manifolds (read “smooth dynamics”). However, over time, the meaning has grown to include transformations and flows on topological spaces (read “topological dynamics”) and on measure spaces (read “ergodic theory”). Of course, in many situations, one might have more than one transformation ...

One route to investigating the differential topology of real manifolds is through Morse theory. A smooth manifold M is decomposed into the level-sets f−1(x) of a smooth real-valued function f : M → R, and the global topology of M emerges from the descriptions of how these level-sets change. One can understand the whole homotopy type of M from this point of view [Mil69], or one can pass more qui...

The book under review is an excellent introduction to the group theoretical and analytic aspects of the field by one of its pioneers. Before reviewing the book, we will provide an overview of the field. Integral geometry draws together analysis, geometry, and numerical mathematics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The f...

The romance of analysis and arithmetic is among the deepest and most enticing themes in all of mathematics. Though arithmetic, with all of its delicate subtleties, may at first seem an unlikely match for analysis with its powerful techniques, the relationship between the two has always been a vital one. In recent decades padic analysis—a hybrid of arithmetic and analysis—has emerged as a fascin...

In preparation for this review, I decided to remind myself how I became interested in automorphic forms as an undergraduate. I seem to remember that Eric Temple Bell [1] had something to do with it. Imagine my surprise when I found the following statement on page 333 of [1]: “The subject, elliptic functions, in which Jacobi did his first great work, has already been given what may seem like its...

Whitfield Diffie and Martin Hellman startled the computer security world in 1975 with their paper “New Directions in Cryptography”, which introduced public-key cryptography [7]. There is now evidence that three cryptographers at the British Government Communications Headquarters (GCHQ), the British equivalent of the National Security Agency, may have predated this work as well as the discovery ...

The two hundred twenty-first regular meeting of the Society was held at Columbia University on Saturday, February 25, extending through the usual morning and afternoon sessions. The attendance included the following seventy-five members: Alexander, Beal, Birkhoff, Borden, Bowden, E. W. Brown, T. H. Brown, E. T. Browne, B. H. Camp, G. A. Campbell, Cole, Coolidge, Cowley, Cronin, Dantzig, Douglas...

Elasticity theory is the central model of solid mechanics. Properly formulated, it gives rise to formidable nonlinear problems whose understanding is in many cases beyond the reach of present-day mathematics. Nevertheless, the last quartercentury has seen substantial advances in this understanding, due largely to the development and application of new methods of nonlinear analysis. An important...

The book under review is one of the first books on Asymptotic Group Theory—a new, quickly developing direction in modern mathematics which has links to many topics in Algebra, Analysis, Probability and Discrete Mathematics. Typically the subject of Asymptotic Group Theory is the study of the type of growth of various functions involving a natural parameter related to the group. Among the most i...

Complex analysis texts written in the early 1900’s, mostly by British authors such as E. T. Copson [2] and E. T. Whittaker and G. N. Watson [4], devote considerable space in later chapters to what we call special functions. These include the gamma, the hypergeometric, the elliptic, the modular, and the zeta functions. Paul Halmos [3] wrote an interesting discussion illustrating the neglect of t...