Title: Some Special Extension Rings
Speaker: Alan Loper, OSU
Abstract: We focus on the field of commutative ring theory in the area of abstract algebra. The study of rings is interesting in its own right and is important in algebraic geometry and algebraic number theory. Specifically, we start with a localization of the ring of polynomials in two or more variables over the complex numbers. We build local extension rings (by adding some well-chosen fractions - so still inside the quotient field of rational functions) known as quadratic transforms. The process of building quadratic transforms can be iterated, so we build a quadratic transform tree of successively larger and larger rings - in lots of different directions. Abhyankar showed in 1956 that in the two-variable case the union of the rings comprising any one infinite branch is an especially nice type of ring known as a valuation domain (defined in the talk - it is simple). This is not true in dimensions three or higher. In this talk we look at what does happen in dimension three. The question of which branches do lead to valuation domains in three dimensions has surprising connections to matrix theory, number theory (continued fractions, farey numbers) and a fractal. This is a wide open problem. The presentation will be a combination of theorems and conjectures.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy PhD students can sign up for this lecture series by registering for one or two credit hours of Math 6193 with Professor Henri Moscovici.
