Geometry Seminar

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Fan distributions via Tverberg’s theorem

Speaker: Steven Simon, Bard College

Location: Warren Weaver Hall 1314 in person and on Zoom

Videoconference link: https://youtu.be/eeYf4ZUuPcU

Date: Tuesday, March 18, 2025, 6 p.m.

Synopsis:

Equipartition theory, beginning with the classical ham sandwich theorem, seeks the fair division of finite point sets in R^d by full-dimensional regions determined by a prescribed geometric dissection of R^d. Here we examine equidistributions of finite point sets in R^d by prescribed low dimensional subsets of R^d. Our main result states that if r≥3 is a prime power, then for any m-coloring of a sufficiently small point set X in R^d, there exists a certain r-fan in R^d -- that is, the union of r ``half-flats'' of codimension r-2 centered about a common (r-1)-codimensional affine subspace -- which captures all the points of X in such a way that each half-flat contains at most an r-th of the points from each color class. The number of points in R^d we require for this is essentially sharp when m≥2. Additionally, we extend our equidistribution results to ``piercing''  distributions in a similar fashion to Dolnikov's hyperplane transversal generalization of the ham sandwich theorem. By analogy with recent work of Frick et al., our results are obtained by applying Gale duality to linear cases of topological Tverberg-type theorems. Finally, we extend our distribution results to multiple r-fans after establishing a multiple intersection version of a topological Tverberg-type theorem due to Sarkaria.

Notes:

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