B. Shapiro and M. Shapiro's conjecture on real osculating Schubert calculus

Let $s_1,\ldots,s_n$ be distinct real numbers, and let $\gamma(s)$ be a parameterization of a rational normal curve in $\mathbf{R}^{m+p}$ of degree $m+p-1$. For integers $k_i$, define

$$K_i(s):=[\gamma(s),\gamma'(s),\gamma”(s),\ldots,\gamma^{(m+1-k_i)}(s)].$$

Thus, $K_i(s)$ is the $(m+1-k_i)$-plane osculating the curve at $s$. B. Shapiro and M. Shapiro's conjecture. Let $s_1,\ldots,s_n$ be distinct real numbers and suppose $K_i(s_i)$ osculates $\gamma$ at $s_i$ and $k_1+\cdots+k_n=mp$. Then each of the finitely many $p$-planes $X\subset \mathbf{C}^{m+p}$ satisfying $X\cap K_i(s_i)\neq \\{0\\}$ for $i=1,\ldots,n$ is defined over the reals. This conjecture links real output-feedback laws in control theory with real solutions in enumerative geometry; the claim concerns the reality of all solutions for Schubert conditions imposed by osculating planes at distinct real points.