Ordered Bell numbers, Hermite polynomials, Skew Young Tableaux, and Borel orbits

We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety \ms{X} of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on \ms{X} previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.Comment: We revised the manuscrip

Weak Order on Complete Quadrics

Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.Comment: Some typos of the earlier version are fixed, a new section adde

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library