Hyman Bass

The Mathematical Neighborhoods of School Mathematics visits regions beyond, but proximal to and accessible from school mathematics. Its aim is to give readers a glimpse of not just the rich diversity and adaptability of mathematics, but, most importantly, its interconnections and overall coherence, a perspective not easily available from the school curriculum. This aim entailed assembling a variety of substantial mathematical domains that do not typically cohabit the same volume. The book begins with an in-depth treatment of topics in the school curriculum, often with novel approaches and connections. A unifying thread is the group theoretic study of addition and multiplication in the various number systems of school mathematics. The exposition is mathematically rigorous, including proofs of many fundamental theorems not otherwise easily available in mathematically accessible form.The Mathematical Neighborhoods of School Mathematics is intended to be a conceptual contribution to mathematics education. It will be a valuable resource in professional development of mathematics teachers, and in mathematical enrichment programs, for both students and teachers. In this regard, many of the chapters are relatively self-contained. It could also serve as a text for undergraduate mathematics majors with an interest in teaching. The exceptional Chapter 11 presents some novel instructional designs for problem-solving activities meant to cultivate ""connection-oriented mathematical thinking.

[UPDATED 6/6/2000] Group actions on trees furnish a unified geometric way of recasting the chapter of combinatorial group theory dealing with free groups, amalgams, and HNN extensions. Some of the principal examples arise from rank one simple Lie groups over a non-archimedean local field acting on their Bruhat--Tits trees. In particular this leads to a powerful method for studying lattices in such Lie groups. This monograph extends this approach to the more general investigation of $X$-lattices $\Gamma$, where $X$ is a locally finite tree and $\Gamma$ is a discrete group of automorphisms of $X$ of finite covolume. These "tree lattices" are the main object of study. Special attention is given to both parallels and contrasts with the case of Lie groups. Beyond the Lie group connection, the theory has applications to combinatorics and number theory. The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Non-uniform tree lattices are much more complicated than unifrom ones; thus a good deal of attention is given to the construction and study of diverse examples. Some interesting new phenomena are observed here which cannot occur in the case of Lie groups. The fundamental technique is the encoding of tree actions in terms of the corresponding quotient "graph of groups." {\it Tree Lattices} should be a helpful resource to researchers in the field, and may also be used for a graduate course in geometric group theory.

The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study "cyclic renormalization", a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of such renormalizations is naturally represented by a rooted tree, together with a "spherically transitive" automorphism; the infinite case corresponds to maps with an invariant Cantor set, a class of particular interest for its relevance to the description of the transition to chaos and of the Mandelbrot set. The normal subgroup structure of the automorphism group of such "spherically homogeneous" rooted trees is investigated in some detail. This work will be of interest to researchers in both dynamical systems and group theory.