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The PascGalois Summer
Undergraduate Research
Retreat
at New College of Florida
June 6- June 10, 2005
UPDATE: We have selected our
students for the 2005 PascGalois Retreat. We have a total of
13 students from around the country.Please click
here to meet our
participating
students as they introduce themselves in their own words.
TRAVEL INFORMATION: Press
here or the button on the
left side of the page.
We will support up to ten undergraduate students to join us for a
week in Sarasota at New College of Florida to investigate questions
related to the PascGalois project. Participants will receive:
Our target audience is students interested in mathematics,
mathematics education, computer science, and/or statistics. We hope
to have a mix of students, including some who are familiar with the
PascGalois project (perhaps have done some of our visualization
exercises in their course work) and others who are seeing it for the
first time at the retreat.
Participants will spend one week at New College working on
undergraduate-level research projects with Michael Bardzell
(Salisbury University-SU), Kathleen Shannon (SU), Eirini Poimenidou
(New College) and former undergraduate research student, Nicole
Miller (Virginia Tech), who has presented and published work related
to the project. Participants will attend short lectures on topics
related to the project and will be given ample time for exploration
and group work with other students. We will also arrange for outings
to explore the area and social events.
Travel dates: We will provide housing for arrival on
Sunday June 5th and departure on Saturday June 11th, 2005.
For more information contact the PI's:
Students interested in participating in this project must fill
out an application form and
ask two of their math professors to fill out
reference forms.
Screening will begin April 1, 2005 and continue until all the
slots are filled.
Support provided by the National Science Foundation award #'s
DUE-0087644 and DUE-0339477
What is PascGalois?
The PascGalois project has its origin in a simple exercise with
Pascal's
triangle. Take each entry in the triangle and replace it with its
congruent value mod n, where n is a positive integer larger than 1.
By assigning each of the values 0, 1, ..., n with a distinct color,
patterns reminiscent of fractals appear. These structures can be
treated as types of 1-dimensional cellular automata. Our interest in
this construction lies in the fact that arithmetic mod n is the
multiplication for the cyclic group Zn and the patterns seen in the
triangles are related to the structure of these groups. We
generalize
this construction using other groups. If G is a group with a; b 2 G,
then a PascGalois triangle is formed by placing a down the left side
of the triangle and b down the right. An entry in the interior of
the triangle is determined by multiplying the two entries above it
using the group multiplication.
Like Pascal's triangle mod n, PascGalois triangles can have
self-similar
properties. Furthermore, many of these properties can be described
using subgroups, quotients, and automorphisms of the group G. A
related structure is a 2-dimensional cellular automaton. 2-D
automata consist of rectangular grids of cells which take on various
state values that change discretely over time according to some
local rule. We use groups (and sometimes other algebraic structures)
as our alphabets and group multiplication for the various local
rules. The long term behavior of these systems can often be
understood in terms of the subgroup lattice of the underlying group.
The focus of this research will be on creating PascGalois triangles
and other 1-D and 2-D cellular automata generated using group and
ring multiplication rules, as well as rules over alphabets with
other algebraic structure. Undergraduate students from a variety of
backgrounds, including mathematics, statistics, computer science,
and secondary education have already completed successful PascGalois
research projects. Current areas of research interests include, but
are not limited to, the following:
Periodicity Properties of finite cellular
automata
Fractal Dimensions for Infnite cellular automata
Cellular Automata as algebraic Systems
Combinatorial and p-adic
approaches
Virtual Reality Rendering for Higher Dimensional
Systems
Information Entropy of Group Generated Cellular
Automata
Group Actions on Group Generated Systems
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