## Grants and Special Projects

## Table of Contents

Madelaine Bates and Roman Kossak:

Graphing Workshops/ Presidential Faculty Staff Development Grant (2000)

Mathematics inAdvanced Technological Education (ATE)Programs/NSF-sponsored project (1999)CUNY Logic Workshop/ Faculty Development Program Grant from the Office for Instructional Technology and External Programs of CUNY Graduate Center. (1998-2000)

Structure of Nonstandard Models of Arithmetic/ PSC-CUNY Grant

Riemannian Manifolds associated with two step nilpotent Lie groups(With A. Koranyi, Distinguished Professor, Z. Szabo, Professor, Lehman College, M.Moskowitz, Graduate Center)/ CUNY Collaborative Research Grant 2000-2002

Asymptotic Behavior of Maxwell-Klein-Gordon Equations in 4-d Minkowski Space/ PSC-CUNY Research Grant, 2000-2002

Automorphisms of Riemann Surfaces/ PSC-CUNY Grant (1999)

A Relationship between Vertices and Quasi-isomorphisms for a Class of Bracket Groups/ PSC-CUNY Grant (1999)

On Quasi-representing Graphs for a Class of Butler Groups/ PSC-CUNY Grant (2000)

## Abstracts

Graphing Workshop (M. Bates & R. Kossak): During the workshops, students explore the relation between a function and its graph using graphing calculator. The understanding of this concept is essential for precalculus and calculus. These workshops are open to any student in MTH 05 or MTH 06. The workshop is held for two hours a week for six weeks.

ATE Programs (S. Forman):A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community.

CUNY Logic Workshop (R. Kossak):The workshop has been financed in 1998-2000 by the Faculty Development Program Grant from the Office for Instructional Technology and External Programs of CUNY Graduate Center. Roman Kossak of Bronx Community College and Joel Hamkins of the College of Staten Island are the co-directors of the workshop.

Structure of Nonstandard Models of Arithmetic(R. Kossak):This is a two year grant to support preparation of the book with the same title. The book will cover the last 20years of research in model theory of arithmetic, in particular the study of the lattices of elementary substructures and the automorphism groups of nonstandard models of Peano Arithmetic. This is a joint project with Professor James Schmerl of the University of Connecticut.

Riemannian Manifolds associated with two step nilpotent Lie groups (M. Psarelli):This is a CUNY Collaborative Research Grant with investigators A. Koranyi, Distinguished Professor, Z. Szabo, Professor, Lehman College. The investigators plan a collaboration between Bronx Community College, Lehman College and the Graduate Center in the study of differential geometry and topology of Riemannian Manifolds associated with certain 2-step nilpotent Lie groups, which include as a special case groups of Heisenberg type.

Asymptotic Behavior of Maxwell-Klein-Gordon Equations in 4-d Minkowski Space (M. Psarelli):This research project is concerned with issues that arise in the electrodynamics of continuous media and are described within the framework of gauge field theory. The focus is on the study of the solutions of coupled Maxwell-Klein-Gordon fields equations in the 4-dimensional Minkowski space-time in the presence of mass and arbitrary size initial data that have charge. The problem belongs in the area of non-linear systems of hyperbolic partial differential equations and abelian gauge field theory.

Automorphisms of Riemann Surfaces (A. Weaver): I will pursue several different approaches to the problem of matching a finite group with the the set of genera of surfaces on which it acts as a group of automorphisms. During the current grant cycle, I plan to (1) determine the genus spectrum of the cyclic group of order $pq$, where $p$ and $q$ are primes; (2) determine the genus spectrum of the dihedral $2$-groups; (3) determine the complete list of odd-order groups acting in general less than $50$; (4) classify the actions of elementary abelian $p$ groups up to topological equivalence.

A Relationship between Vertices and Quasi-isomorphisms for a Class of Bracket Groups (P. Yom):I will characterize the class of Bracket groups up to quasi-isomorphisms by showing there is a sequence of vertex switches, where Bracket group is the cokernel of the diagonal embedding of intersection of $n$ subgroups $Ai$of $Q$ to the direct sum of $Ai$. That is, two Bracket groups $[A1,A2,...,An]$ and $[B1,B2,...,Bn]$ are quasi-isomorphic if and only if there is a sequence of vertex switches which successfully replaces $Ai$ by $Bi$.

On Quasi-representing Graphs for a Class of Butler Groups (P. Yom):Represent a group of the form $G(A1,...,An)$ in terms of type-labelled graph and analyze the quasi-isomorphisms of two groups using graph theoretic properties.