SUNY Potsdam

Math. Dept.
Chair.htm

SUNY Potsdam

home page

Resume

More About Mahdavi

Teaching Philosophy

Grants Recieved

Kazem Mahdavi, Ph.D.

Professor of Mathematics
Math. Dept. SUNY Potsdam, Potsdam, NY 13676
Tel: 315 267 2095
Fax: 315 267 3176
email: mahdavk@potsdam.edu

 

Work in Progress

I 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:
(i)its action on a set, in particular its action on Euclidean and Hyperbolic spaces or more generally its action on a metric space,
ii) the maximum area of the loops of certain length in a Cayley Graph of the group(isoperimetric inequality), 
or
(iii) using finite state automata to study a given group.
Items (i), (ii), and (iii) are to some extent related. For more information see[2]. 

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. 
Any subset of R+R which consists of (v, u) where v= ua, for 'a` an element of A is regular.
Furthermore a group is called biautomatic if any subset of R+R which consists of (v, u) where v=ua, or v= au for a an element of A is regular.

 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:
Are there infinitely generated abelian subgroups of biautomatic groups?
Is the centralizer of any finite subset of any automatic group automatic?
Is the center of an automatic group finitely generated?
The answer to the first question could help us with our studies of Richard Thompson's group and other related problems on automatic groups. The answer to the second and third questions could help us solve some well-known open problems in automatic group theory. 

References:
[1] Cramer, W. Mahdavi, K. Gabe, M. Nguyen, L. Politi, H. Schedler, T., "Virtually abelian subgroups of biautomatic groups," under review for International Journal of Algebra and Computations..
[2] E D.B.A. Epstein, J.W. Cannon, D.F. Holt, S. Levy, M. Paterson, and W.P. Thurston, Word Processing in Groups Jones and Bartlett, Boston, 1992.
[3] Gersten, S., and Short, H. "Rational subgroups of biautomatic groups," Annals of Mathematics 134 (1991), 125-128.
[4] Hopcroft, J. and Ullman, J. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Phillipines, 1979.
[5] Mosher, L. it "Central quotients of biautomatic groups," Comment. Math. Helv. 72 (1997), no. 1, 16-29.

 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
A Review of Newtonian Mechanics

Chapter II
The Lagrangian Equations of Motions derived from the principle of least action 

Chapter III
The Lagrangian Equations Derived from Geometry

Chapter IV
The Application of the Lagrangian Equations to one, two, and More- body Problems

Chapter V
The Theory of Small Viberation

Chapter VI

Rotating Coordinate Systems and Rigid body Dynamics

Chapter VII

The Hamiltonian Formulation of Mechanics

Chapter VIII
The Hamiltonian Jacobi Theory

Chapter IX
Special Relativity Theory

Chapter X
General Relativity Theory

Chapter XI
Basic Manifold Theory

Chapter XII
Singularity Theory

Chapter XII
I
Noether's Theorem and Conservation Laws.

Chapter XIV

Symmetry and Super Symmetry.

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.

Email: Mahdavk@potsdam.edu