Given some class of "geometric spaces", we can make a ring as follows. Additive structure: when U is an open subset a space X, [X] = [U] + [X - U]. Multiplicative structure: [X][Y] = [XxY]. In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. This is joint work with Melanie Matchett Wood.
- Algebraic and Symplectic Geometry Seminar