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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 |
Description
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