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.

With new renovations completed over the 2014-15 winter break, the UK Mathskeller unveiled its new look at an open house on Wednesday, March 4, 2015 - hosted by the Department of Mathematics and College of Arts and Sciences, in room 63 in the basement of the White Hall Classroom Building. 

Opened in 2001 with 20 computers and a large printing budget, the Mathskeller, a computing and mathematics learning center managed by the Department of Mathematics and the Mathematical Sciences Computing Facility, was established to implement a technology-assisted instructional model. Fourteen years later, the center is home to only four computers, printers aren't used nearly as much, and the facility looks nothing like a basement classroom.

Instead, the center resembles a modern, collective learning space. And while there may be fewer wires and less printing, technology still has a leading role at the center.

Today's students, at least UK students utilizing the revitalized Mathskeller, are also taking advantage of the multiple mobile workspaces, bright LED-lit atmosphere, comfortable seating, tutors and chalkboard-lined walls. The renovated Mathskeller still features a kitchenette and group study or meeting room, and has added more storage, new carpet, additional study tables by removing a closet, and even a new computerized sign-in method. 

>>View a photo album of the renovations