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We develop a combinatorial rule to compute the real geometry of type B Schubert curves S (λ ∙) in the orthogonal Grassmannian O G (n , C 2 n + 1) , which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A [J. Algebraic Combin. 45(1), 191–243 (2017)]. First, using the type B Wroński map studied in [Adv. Math. 224(3), 827–862 (2010)], we show that the real locus of the Schubert curve has a natural covering map to R P 1 , with monodromy operator ω defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute ω without rectifying the skew tableau. The first uses the crystal operators introduced in [Algebr. Comb. 3(3), 693–725 (2020)], while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves. [ABSTRACT FROM AUTHOR]

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math, 10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by "convex hypersurfaces" we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces X 1 , ... , X d k , n ⊂ R P n , where d k , n = (k + 1) (n - k) , are in random position if each one of them is randomly translated by elements g 1 , ... , g d k , n sampled independently from the orthogonal group with the uniform distribution. Denoting by τ k (X 1 , ... , X d k , n ) the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that τ k (X 1 , ... , X d k , n ) = δ k , n · ∏ i = 1 d k , n | Ω k (X i) | | Sch (k , n) | , where δ k , n is the expected degree from [6] (the average number of k-flats incident to d k , n many random (n - k - 1) -flats), | Sch (k , n) | is the volume of the Special Schubert variety of k-flats meeting a fixed (n - k - 1) -flat (computed in [6]) and | Ω k (X) | is the volume of the manifold Ω k (X) ⊂ G (k , n) of all k-flats tangent to X. We give a formula for the evaluation of | Ω k (X) | in terms of some curvature integral of the embedding X ↪ R P n and we relate it with the classical notion of intrinsic volumes of a convex set: | Ω k (∂ C) | | Sch (k , n) | = 4 V n - k - 1 (C) , k = 0 , ... , n - 1. As a consequence we prove the universal upper bound: τ k (X 1 , ... , X d k , n ) ≤ δ k , n · 4 d k , n . Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case k = 1 , n = 3 for every m > 0 we can provide examples of smooth convex hypersurfaces X 1 , ... , X 4 such that the intersection Ω 1 (X 1) ∩ ⋯ ∩ Ω 1 (X 4) ⊂ G (1 , 3) is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions. [ABSTRACT FROM AUTHOR]

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile's monotonicity conditions are not satisfied. [ABSTRACT FROM AUTHOR]