Date/Time
Date(s) - 09/03/2023
10:30 am - 11:30 am

Speaker: Kevin Purbhoo

Title: The secant conjecture and other stories

Abstract: If we have a system of polynomial equations with real coefficients, we typically expect some of the solutions to be real, and some to be complex (non-real).  However, in some rare, special situations, we discover that all solutions are in fact real.  The eigenvalues of a real symmetric matrix are perhaps the best known example of this phenomenon. In the mid-1990’s, B. and M. Shapiro discovered another remarkable example of this reality phenomenon, in Schubert calculus.  This became known as the Shapiro-Shapiro conjecture.  Since then, many generalizations and related conjectures have been formulated.  One of the first of these was the secant conjecture, which relaxes the hypotheses of the Shapiro-Shapiro conjecture in a geometrically natural way. While the Shapiro-Shapiro conjecture has been proved (Mukhin-Tarasov-Varchenko 2009), many of the related conjectures have remained open.  I will talk about a new theorem, which we use to resolve several of these open conjectures, including the secant conjecture (in its original form).  This is joint work with Steven Karp.

Location:  Hamilton Hall, Room 207