Cilanne Boulet
Home university:
University of Waterloo
Supervisor:
Ian Goulden
Project title:
Transitive Products of Transpositions
Research results:
Cilanne Boulet presented a talk about some of the background to her project at the 2000 Canadian Undergraduate Mathematics Conference, in June at McMaster University, entitled "Factorizations of (12...n) into Transpositions".
Student comment:
After learning the relevant background, we were able to give an algebraic proof of a recurrence for Hurwitz numbers that generalizes a recurrence that has previously been given by algebraic geometers. This recurrence has a very simple form when interpreted combinatorially, and we made some progress in finding a combinatorial construction to explain this recurrence directly, but no definitive results were established.
Update 2008:
Visiting Assistant Professor, Cornell University, Department of Mathematics
Sabin Cautis
Home university:
University of Waterloo
Supervisor:
D.M. Jackson and C. Stewart
Project title:
Walks on the integer sublattice of the real plane.
Research Results:
A research paper will be written.
Student comments:
A meander is a simple, closed loop which crosses a given line $2n$ times. Counting the number of distinct meanders is an open problem which interests both mathematicians and physicists. This problem is directly related to the Temperley-Lieb algebra which is a rich and beautiful area. We weren't able to enumerate meanders but our efforts yielded some neat observations regarding meanders, partitions and the Temperley-Lieb algebra.
Update 2008:
G.C. Evans Instructor, Rice University
Donny Cheung and Michael Brown
Home university:
University of Waterloo
Supervisor:
Alfred Menezes
Project title:
Secure
Electronic
Commerce
on
Constrained
Devices
Research
Results: A
Joint
Research
Paper
was
presented
at
the
USENIX
Security
Conference
in
August.
Student comments:
We studied various methods of efficiently implementing elliptic curve cryptography and finite field arithmetic. The algorithms were implemented on a PC as well as the Blackberry email pager from Research in Motion. Some of the code was also integrated with the OpenPGP and OpenSSL libraries.
Cedric Davies
Home university:
St. Mary's University
Supervisor:
J. Geelen
Project title:
4-connected graphs
Research results:
Joint research paper prepared.
Student comments:
My relationship with my supervisor made all the difference. We met frequently which was good. He kept his distance from the project and allowed me to discover things on my own, but at the same time pushed me in the right direction and aided me when I was stuck. There were several meetings near the end of the summer where we spent several hours together drawing graphs on the board attempting to make it over a hurdle in the proof. It was fun.
Graeme Kemkes
Home university:
University of Waterloo
Supervisor:
Bruce Richmond
Project title:
Stirling Numbers of the Second Kind
Research results:
Joint research note prepared.
Student comments:
I strongly recommend the Combinatorics and Optimization (C and O) Undergraduate Research Assistant (URA) experience. I worked closely with an experienced researcher on a problem that interested me. I enjoyed the variety of seminars presented by faculty, visitors, graduate students and the other URA students. I met brilliant undergraduates who share my interests. I now have a much better understanding of the academic environment. Very few undergraduates are given such a rare opportunity. I am quite grateful to have been selected. Thank you again.
Update 2008:
Postdoctoral Fellow, San Diego
Michael Ludkovski
Home university:
Simon Fraser University
Supervisor:
D.G. Wagner
Project title:
Ternary pseudo-random sequences
Research results:
One paper "New Families of Ideal 2-level Autocorrelation Ternary Sequences From Second Order DHT" has been submitted to the WCC (International Workshop on Coding and Cryptography) 2001 meeting in Paris, France. If the paper is accepted, the student will travel to Paris to present the paper at the conference. Another paper "Ternary Ideal 2-level Autocorrelation Sequences" is to be submitted to a combinatorics journal before the end of the year.
Student comments:
I have immensely enjoyed my summer at Waterloo. It was a great opportunity to see what's it like doing original math research. My own research topic was interesting, and more importantly, at the right level of difficulty. By the end of the summer we made fair progress and I even got to write two papers for publication. I have had good interaction with my supervisor, who was friendly and encouraging. The entire atmosphere was very relaxed and supportive, and I have made many friends with other URA students. I would strongly recommend Waterloo URA program for anyone interested in discrete math.
Update 2008:
Assistant Professor, Dept of Mathematics, University of Michigan
Anjayan Puvananathan
Home university:
University of Waterloo
Supervisor:
David Wagner
Project title:
Problems in Enumerativeraph Theory
Van Anh Truong
Home university:
University of Waterloo
Supervisor:
C. Godsil
Project title:
Problems in Algebraic Graph Theory
Research results:
The results are summarised in a technical report.
Student comments:
The
goal
of
my
project
was
to
gain
a
better
underdstanding
of
polynomials
coming
from
the
weight
systems
of
the
Kauffman,
Homfly,
and
Jones
Vassiliev
invariants.
We
wrote
a
computer
program
to
generate
a
large
numbers
of
these
polynomials
from
which
to
make
observations
and
draw
generalizations.
We
also
applied
results
from
algebraic
graph
theory
to
simplify
the
derivation
of
the
polynomials
and
delineate
their
close
connection
with
certain
underlying
graphs.
The
project
allowed
me
to
learn
much
about
how
research
is
conducted,
as
well
as
clarified
my
goals
and
interests
with
respect
to
future
study
in
mathematics.
I
especially
appreciated
the
opportunity
to
observe
and
receive
advice
from
my
supervisor
about
many
important
aspects
of
academic
life.
Update 2008:
PhD student, School of Operations Research and Industrial Engineering, Cornell University
Rob Waiser
Home university:
University of Waterloo
Supervisor:
D. Stinson
Project title:
Combinatorial Cryptography
Yuetai Wang
Home university:
Ottawa
Supervisor:
B. Richter
Project title:
Computing Crossing Numbers of Graphs
Research results:
The research entailed a detailed study of Colin de Verdiere's Graph Parameter and founding of proofs for two major theorems, one regarding outer planar graphs, and the other planar graphs.