Mathematics Colloquia and Seminars

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The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a $d$-dimensional polytope with $n$ facets cannot have diameter greater than $n - d$. Despite being one of the most fundamental, basic and old problems in combinatorial geometry and optimization, what we know is quite scarce. Most notably, not even a polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. We survey results on both the positive and on the negative side of the conjecture. No prior knowledge will be assumed, and this talk will be of interest to those who like topological triangulations!