Teacher: Wushi Goldring
Office: Room 110 in House 6
Course dates: Thursdays 31/8, 7/9, 14/9
Meeting time and place: 15:30-17:30 Room 16 (Kraftriket House 5)
Office Hours for the course: 14:15-15:15 on Thursdays before class
Grading: Based on a homework assignment which will be due on 28/9. You can either email your solutions or send them to Samuel under the "Combinatorics" module of the course. I will use the following grading scheme: 90% -100% is A, 80% -90% is B 70% -80% is C, 60% -70% is D, 50% -60% is E, <50% Is F
Overview: I would like to discuss the theme of reciprocity in number theory. The origins of reciprocity lie with Quadratic Reciprocity, Gauss 'golden theorem' ('teorema aureum'). This a statement that says that whether an odd prime is a square modulo a second odd prime q is related in a very simple way to whether q is a square modulo p. Thus the roles of p and q are inverted, or reciprocated, hence Reciprocity . This phenomenon was experimentally observed by Euler; Gauss gave several proofs throughout his career in order to understand why this phenomenon holds. In so doing, Gauss exposes two fundamental themes in numerical theory. These have never stopped growing in the prominent side of Gauss, and they are a critical source of motivation and inspiration in research today.
The first theme uncovered by Gauss is the "underlying structure" or "roots of the tree" theme: Quadratic reciprocity concerns ordinary natural numbers - primes - but to get to the bottom of it, we get into the ground and find a series of levels Or new structures of increasing complexity and wealth. One of the first structures that Gauss discovered were these roots of unity: solutions for x^n=1.
2. Methods of analysis and the theory of special functions, because the roots of unity are special values of the exponential function
x --> e^x,
3. Methods of geometry, because roots of unity lie symmetrically on the unit circle in the complex plane.
The second theme discovered by Gauss is that "reciprocity is everywhere", the question about squares is just the beginning. Just as we study squares modulo p, we can study cubes, fourth powers, etc. Each of these questions leads to a new reciprocity law: Cubic Reciprocity, Quartic Reciprocity etc.
So what is the real source of reciprocity? Deep reflection on this problem has shaped research in number theory today and leaves us with some of the most mysterious wide-open problems in the subject.
I will try to motivate the beginning of this progression in elementary terms, starting with Euler's examples and then following Gauss's path to roots of unity and beyond.