Title: \(A^1\)-Milnor number: counting zeros arithmetically
Speaker: Kirsten Wickelgren
Speaker Info: Georgia Tech
Brief Description:
Special Note:
Abstract:
The Milnor fibration is a fibration associated to a point p of a hypersurface f = 0 over the complex numbers. Its fiber is homotopy equivalent to a wedge of spheres and the number of these spheres has an interpretation in singularity theory: if p is a singularity, it is possible to deform p into nodes, which are the simplest type of singularities, and the number of these nodes is the same as the number of spheres, or the Milnor number. Nodes over fields other than C have arithmetic information associated with them. We use \(A^1\)-homotopy theory to give an enrichment of Milnor's theorem equating the number of nodes with the number of spheres to an equality in the Grothendieck-Witt group of a field of characteristic not 2. To do this, we prove that the local \(A^1\)-Brouwer degree equals the quadratic form of Eisenbud--Khimshiashvili--Levine, answering a question posed by David Eisenbud in 1978. This is joint work with Jesse Kass.Date: Saturday, February 06, 2016