Math.
Dept.

Kazem Mahdavi, Ph.D.Professor of Mathematics Work in ProgressI am involved in both doing research and writing a book.1. SUMMARY OF RESEARCH ON AUTOMATIC GROUP IN PROGRESS: A group can be studied from a few different
perspectives, including: Using finite state automata to study a group is a new idea that was introduced by W. Thurston, based on the work of J. W. Cannon [2]. The concept of finite state automaton [4] has emerged as a significant tool in many branches of human knowledge and understanding including: linguistics, computer science, philosophy, biology, logic, etc, and in particular, recently, group theory. The language on a finite set of symbols, A, accepted by a finite state automaton is called a regular language. We say a group is automatic if for a set of semigroup
generators A of a group G there is a regular subset R of
A*(words on A) which can be mapped onto G under an
appropriate evaluation map. My REU students and I have been able to study a class of groups that wecall virtually biautomatic, i.e., there is a finite index biautomatic subgroup inside the group. Our results are generalizing both Gersten Short's paper [3], and Mosher's paper [5]. Gersten and Short show that centralizers of finite subsets of biautomatic groups are biautomatic. Mosher shows that quotients of a biautomatic group by central subgroups are biautomatic. We prove for virtually biautomatic groups, by showing that the quotient of any group G by any normal, finitely generated, virtually abelian, subgroup H is virtually biautomatic if and only if G/H is virtually biautomatic. We also prove that normalizers of finitely generated subgroups of virtually biautomatic groups with finite index are virtually biautomatic, see[1]. This paper is under review for possible publication in the International Journal of Algebra and Computation. There are many other open problems in this field that I
am currently working on. Some of these include: References: 2. SUMMARY OF BOOK IN PROGRESS: I am writing a mathematical physics text book with I. Schensted and E. Ryan. The book would be suitable for advanced undergraduates, graduate students, and scholars in mathematics and mathematical physics. Here I give the chapters of this book. Many of these chapters have been written, some are under revision, and a few remain to be written. Chapter I Chapter IX 3. SUMMARY OF RESEARCH IN BIOLOGICAL SCIENCES: I am collaborating with my colleagues in the Biology Department, SUNY Potsdam, and from Clarkson University to study and model the population dynamics of biological systems in particular schools of fish. This project is at the beginning stage. We plan to send a proposal to NSF for funding soon. I. 