6,846 research outputs found
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Sunflowers of Convex Open Sets
A sunflower is a collection of sets such that the
pairwise intersection is the same for all choices of distinct
and . We study sunflowers of convex open sets in , and provide
a Helly-type theorem describing a certain "rigidity" that they possess. In
particular we show that if is a sunflower in , then any hyperplane that intersects all must also intersect
. We use our results to describe a combinatorial code
for all which is on the one hand minimally non-convex,
and on the other hand has no local obstructions. Along the way we further
develop the theory of morphisms of codes, and establish results on the covering
relation in the poset
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