Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov's theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz's quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We con...

We show that a quantum deformation of quantum mechanics given in a previous work is equivalent to quantum mechanics on a nonlinear lattice with step size ∆x = (1− q)x. Then, based on this, we develop the basic formalism of quantum group Schrödinger field theory in one spatial quantum dimension, and explicitly exhibit the SUq(2) covariant algebras satisfied by the q-bosonic and q-fermionic Schrö...

algebra is an important course in the undergraduate mathematics curriculum. For some undergraduates, abstract algebra is the first mathematics course in which they must move beyond learning templates and procedures for solving common classes of problems (Dubinsky, Dautermann, Leron, and Zazkis, 1994). For most undergraduates, this course is one of their earliest experiences in coping with the d...

The renormalization group has played an important role in the physics of the second half of the twentieth century both as a conceptual and a calculational tool. In particular it provided the key ideas for the construction of a qualitative and quantitative theory of the critical point in phase transitions and started a new era in statistical mechanics. Probability theory lies at the foundation o...

One of the main themes of geometric group theory is to study a (finitely generated) group G in terms of the geometric properties of the Cayley graph of G. These “geometric properties” come in the form of quasi-isometry invariants. Our goal this semester is to look as some specific quasiisometry invariants such as Dehn functions, hyperbolicity, growth functions, and amenabiliity and try to under...

This paper presents a model for linguistic description based on group theory. A grammar in this model, or G-grammar is a collection of lexical expressions which are products of logical forms, phonolog-ical forms, and their inverses. Phrasal descriptions are obtained by forming products of lexical expressions and by cancelling contiguous elements which are inverses of each other. We show applica...