Date/Time
Date(s) - 07/02/2025
3:30 pm - 4:30 pm

Speaker: Will Traves (US Naval Academy)
Research Area: Algebra

Location: Hamilton Hall, Room 305

Title: Interpolation and Incidence

Abstract: I will give an overview of several geometric problems and how their solutions depend on incidence results involving linear spaces. Extending results of Pappus and Pascal on conic curves, we’ll look at an incidence construction (developed jointly with David Wehlau) that determines when 10 points lie on a cubic curve. Our construction uses an important tool from algebraic geometry: the Cayley-Bacharach Theorem. In the 18th century  Hermann Grassmann came up with his own solution to this problem that used only elementary methods. I’ll explain Grassmann’s approach and show how to extend his results. In particular, we will see that a simple extension of his ideas allows us to locate the base-point of a 1-parameter family of cubic curves with just a straightedge: if two cubic curves intersect in 8 known points and we know one additional point on each curve, we can locate their ninth point of intersection with an unmarked ruler. We will also consider a  three-dimensional geometry problem originally posed at l’Academie de Bruxelles in 1825. This problem asks for an incidence construction that determines whether 10 points lie on a quadric surface. I’ll give an overview of my recent solution to the Bruxelles problem and point to many other open problems in this area. 

 

Coffee will be served in the same room, HH 305 at 3:00pm. All are welcome.