Mathematics Colloquium

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Cutting and Pasting in Algebraic Geometry

Speaker: Ravi Vakil, Stanford University

Location: Warren Weaver Hall 1302

Date: Monday, November 11, 2013, 3:45 p.m.

Synopsis:

Given some class of “geometric spaces,” we can make a ring as follows.

(i) (additive structure) When \(U\) is an open subset of such a space \(X\), \(\left [ X \right ] = \left [ U \right ] + \left [ \left ( X \setminus U \right ) \right ]\);

(ii) (multiplicative structure) \(\left [ X \times Y \right ] = \left [ X \right ]\left [ Y \right ]\).

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 talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.