# Mathematics Colloquia and Seminars

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### Path-following via PL maps, Sperner's lemma, and hyperplane mass partitions

**Algebra & Discrete Mathematics**

Speaker: | Florian Frick, Cornell |

Related Webpage: | http://www.math.cornell.edu/~frick/ |

Location: | 1147 MSB |

Start time: | Mon, Jan 11 2016, 1:10PM |

Sperner's lemma guarantees the existence of a facet with no repeated colors in certain colorings of triangulations of the d-simplex. Quantitative versions of Sperner's lemma for triangulations of d-polytopes giving a lower bound for the number of colorful facets that is linear in the number of vertices of the polytope were obtained by De Loera, Peterson, and Su. I will somewhat strengthen their result. The proof method is inspired by the dissertation of Edgar Ramos, in which he used the PL version of the preimage theorem (that is, preimages of regular values are submanifolds) to prove results about equipartitions of masses by affine hyperplanes. I will also show how Ramos' point of view can shed light on number-theoretic conditions that show up in this mass partition problem