Tayeb Aissiou: Shor's quantum factorisation algorithm

Anne-Sophie Charest: Elliptic curve factorisation algorithms

Mathieu Guay-Paquet: Schoof's algorithm for counting points on elliptic curves over finite fields

Ilya Hekimi: Elliptic curve cryptosystems

Louis-Francois Preville-Ratelle: Topics in combinatorial number theory.

Mireille Schnitzer: Primes in arithmetic progressions

Suggested topics for the seminar.

Congratulations on receiving the NSERC Undergraduate Summer Research Award!

I have decided to coordinate the activities of the USRA recipients working with me by running a weekly seminar, which will be a chance for people to interact, describe what they are working on, and discuss questions that arise.

The USRA program is very unstructured: you will be given an office and a monthly stipend, your sole duties being to learn some new mathematics, try to come up with a problem that you find interesting, and then make some progress on it. These conditions are (dauntingly, perhaps) similar to those of a mathematics PhD student, and you might not feel completely ready! Here are some words of advice, which also apply to students embarking on their PhD studies.

1. This website contains a list of suggestions for topics that are sufficiently self-contained that they can be absorbed and explored in a summer. Try fixing your choice, as early as possible, on a topic that catches your imagination. You are free to change your course along the way if, say, you decide to delve into a sub-topic that you find intriguing, or even to branch out into something completely different.

2. You will have to absorb alot of material, much of it on your own. Don't do this passively. For example, when confronted with a theorem, put off reading and understanding the proof until you have worked out a few concrete examples that illustrate the result, and can give you a feeling for how it works. This will often make the proof, when you get to it, seem more natural.

3. Do not isolate yourself. Being stuck is less discouraging when you can share your confusion with others! Although you are not required to show up at McGill beyond your obligatory attendance of the weekly seminar, you are encouraged to come to your office regularly, as this will allow frequent exchanges with the other USRA recipients.

4. Because of the unstructured nature of the program, you can only hope to get as much out of the USRA as you put in. A USRA can be a pleasant, stimulating, and rewarding experience, and will hopefully allow you to get a taste of more advanced mathematics, and even mathematical research.

I wish you the best of success in making the most of this exciting opportunity!