# Title: Cube sections, Eulerian numbers and the Laplace-Pólya integral

## Speaker: Gergely Ambrus, Rényi Institute & University of Szeged

## Date and time: 2pm (New York time), Tuesday, March 12, 2024

## Place: On Zoom (details provided on the seminar mailing list)

Critical central sections of the cube.

Volumes of central hyperplane sections of the $d$-dimensional cube $Q_d$
have been studied for over a century: it is known that minimal
sections are parallel to a facet, while K. Ball proved in 1986 that
the sections of maximal volume are normal to the main diagonal of a
2-dimensional face. Many of the related results were achieved by using
analytic methods: the volumes in question can be expressed by the
so-called Laplace-Pólya integrals. These are also in connection with
Eulerian numbers of the first kind, and hence provide a connection
between geometric estimates and combinatorial inequalities. In our
joint work with Barnabás Gárgyán, we establish new bounds for the
Laplace-Pólya integrals by entirely combinatorial means, which then
imply new results in both fields. In particular, we prove the
existence of full-dimensional, non-diagonal critical central sections
of the cube in every dimension at least 4, and generalize asymptotic
bounds on Eulerian numbers obtained by Leusier and Nicolas in 1992.