Nickolas Hein

THE SECANT CONJECTURE IN THE REAL SCHUBERT CALCULUS

We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

The Monotone Secant Conjecture in the Real Schubert Calculus

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 terahertz-years of computing, and we discuss some of the phenomena we observed in our data.

Modular Catalan numbers

The Catalan number

A congruence modulo four for real Schubert calculus with isotropic flags

C_n

A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

enumerates parenthesizations of

Lower bounds in real Schubert calculus

x_0*\dotsb*x_n

A congruence modulo four in real Schubert calculus

where

CERTIFIABLE NUMERICAL COMPUTATIONS IN SCHUBERT CALCULUS

*

A lifted square formulation for certifiable Schubert calculus

is a binary operation. We introduce the modular Catalan number

Reality and Computation in Schubert Calculus

C_{k,n}