Our next Monday Lectures will take place on May 30, in attendance, at 14:15 at FU Berlin. There will be TWO lectures. _ATTENTION! Changed Location_ Hörsaal A Institut für Mathematik Arnimallee 22 14195 Berlin (across the street from the usual location) _Time_: Monday, May 30, 2022 - 14:15 _First Lecture_: János Pach (Rényi Institute, Budapest) _Title_: *Facets of Simplicity* _Abstract_:We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures are of bounded complexity: they can be embedded in a bounded-dimensional space, or have small VC-dimension, or a short algebraic description. What are the advantages of low complexity? I will suggest a few possible answers to this question, and illustrate them with classical examples.
*_Coffee & Tea Break_: Inner yard of Arnimallee 22 _Time_: Monday, May 30 - 16:00 _Second Lecture_: Imre Bárány (Rényi Institute, Budapest) _Title_: *Cells in the box and a hyperplane* _Abstract_:It is well known that a line can intersect at most 2n−1 cells of the n×n chessboard. What happens in higher dimensions: how many cells of the d-dimensional box [0,n]^d can a hyperplane intersect? We answer this question asymptotically. We also prove the integer analogue of the following fact. If K,L are convex bodies in R^d and K ⊂ L, then the surface area K is smaller than that of L. This is joint work with Péter Frankl.