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Discrete CATS Seminar--Dissertation Defense--Clifford Taylor

Title:  Deletion-Induced Triangulations

Abstract:   Let $d > 0$ be  a fixed integer and let $\A \subseteq \mathbb{R}^d$ be a collection of $n \geq d+2$ points which we lift into $\mathbb{R}^{d+1}$. Further let $k$ be an integer satisfying $0 \leq k \leq n-(d+2)$ and assign to each $k$-subset of the points of $\A$ a (regular) triangulation obtained by deleting the specified $k$-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector outlined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope $\Sigma_k(\A) \subseteq \mathbb{R}^{| \A |}$ by taking the convex hull of all obtainable compound GKZ-vectors by various liftings of $\A$, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to $\A$. We will see that by varying $k$, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal $k$ corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case $k=d=1$, in which we can outline a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.

POT 745
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In the summer of 2014, several undergraduate and graduate students from the College of Arts and Sciences received a grant from the National Science Foundation. This NSF grant gave them the means to pursue research in various fields as they explored their interests and prepared for their potential futures.In this podcast, Josiah Hanna, a recent graduate in Mathematics and Computer Science, tells us about his research interests and the impact that the NSF grant will have upon his future.

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