Number Theory - Summer SchoolGulbenkian Foundation, LISBON — 11 to 15 JULY 2011
sketch for the panel Começar by José de Almada Negreiros - image reproduced with kind permission of the Gulbenkian Foundation ProgramThis school comprises three 5-lecture courses (in English) aimed at 1st- or 2nd-year undergraduate students of mathematics, complemented by problem sessions.CoursesConics and Elliptic Curvesby Brian CONRAD (Stanford University)Abstract: Elliptic curves (not ellipses!) constitute a meeting ground for many parts of modern mathematics: complex analysis, number theory, algebraic geometry, and much more. The importance of elliptic curves stems from their mixture of accessibility (we can describe them with an explicit equation) and as a testing ground for very fundamental conjectures. They often pop up in classical problems which have no a-priori reason to be related to elliptic curves. In the first part of the course, we will discuss the classical story of Pythagorean triples and its generalization to the study of rational points on conics (culminating in the statement of Gauss' definitive result on such matters). Then we will move on to the study of elliptic curves. Beginning with some examples of elliptic curves, the theory will be illustrated via its connection to one of the oldest unsolved problems from the ancient Greeks, the so-called "congruent number problem", a problem about right triangles (not congruences!). In particular, we will see how one of the million-dollar Clay Math Problems is related to this ancient question. In the final part of the course we will show how elliptic curves allow for a conceptual understanding of Fermat's proof of his "last theorem" for exponent 4. (Elliptic curves are also central in Wiles' proof of Fermat's Last Theorem, but that is in an entirely different way which is rather beyond the level of the course.) The course will assume knowledge of elementary "modular arithmetic" (though we review what we need) and the concept of an abelian group. Number Theory in Quadratic Fieldsby Keith CONRAD (University of Connecticut)Abstract: One of the early important steps in the development of number theory was the discovery by Gauss that the set of all "complex integers" (that is, numbers of the form a+bi where a and b are both integers) has many properties in common with the integers: primes, unique factorization, and modular arithmetic can all be developed in this larger number system. The algebraic structure of the "complex integers" is not just interesting for its own sake, but can be the key to solving problems that at first involve only the integers. In this course we will study what happens when we adjoin quadratic irrational numbers (like i or √2) to the integers. Sometimes unique factorization remains valid and sometimes it does not. Repairing the failure of unique factorization will lead us to factorization of ideals and then to one of the central concepts of number theory: the ideal class group. We will see how to compute the ideal class group and some applications of it to Diophantine equations. The background for the course is modular arithmetic and the notions of a ring and field. P-adic Numbersby Christopher SKINNER (Princeton University)Abstract: P-adic numbers have become an indispensable tool for number theorists. They provide a convenient way of packaging congruences modulo all powers of a prime number p (by the Chinese remainder theorem, all congruences can be reduced to congruences modulo powers of primes). By enlarging the numbers considered from the rational numbers to the p-adic numbers, number theorists are able to combine both congruence information with techniques from analysis to study problems, especially Diophantine equations. This course will develop the basic properties of the p-adic numbers and the existence of p-adic solutions to various Diophantine equations. Background: The only necessary background for this course is some familiarity with modular arithmetic and the notions of a ring and field. Schedule
On Saturday, July 16, there will be student mini-conference at the Gulbenkian Foundation, including a roundtable with the school speakers. All school participants are invited to attend this mini-conference. Schedule for mini-conference (16/July/2011) AssistantsJoão Guerreiro () and Ana Rita Pires ()VenueThe school takes place in the headquarters of the Gulbenkian Foundation (Avenida de Berna, Lisbon), with lectures in Auditório 3 and problem sessions in Sala 4, except for the last problem session which will be in Sala 1.There is a map under Local Information. |
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