Jamil Mortada

                                                                                    Email: jmortada@ece.arizona.edu

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     This research project is continuation of the project I have worked on during the Fall

 

semester 2002 under Dr. David T. Gay’s supervision. So far, we have studied different

 

functions acting on topological surfaces, mainly, smooth, invertible, functions from a

 

surface to itself. The research has been focused on a specific class of functions on

 

surfaces, namely, compositions of twists along simple closed curves. The braid relations,

 

Lantern relation, and the Chain relation, which have been proven in the course of the

 

project, are good example of working with such functions.   

 

   

      The research on this project will be primarily focused on finding boundary-interior

 

relations for different surfaces as well as proving these relations by means of diagrams

 

similar to the ones already used. The main interest will be finding twelve curves Ci, i = 1,

 

…, 12 in a torus with nine boundary components such that the composition of twists

 

along C1, …, C12 equals one right twist along each boundary of the nine punctured torus.

 

    

     To insure progress, I will meet with Dr. Gay on a weekly basis. During each meeting,

 

Dr. Gay will check my work and suggest any alternative ways, if any, that could simplify

 

the work. At the conclusion of my research project, I intend to write a full report suitable

 

for publication.