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2000 participants and projects

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.