An explicit construction for neighborly centrally symmetric polytopes

We give an explicit construction, based on Hadamard matrices, for an infinite series of floor{sqrt{d}/2}-neighborly centrally symmetric d-dimensional polytopes with 4d vertices. This appears to be the best explicit version yet of a recent probabilistic result due to Linial and Novik, who proved the existence of such polytopes with a neighborliness of d/400.Comment: 9 pages, no figure

Dissections, Hom-complexes and the Cayley trick

We show that certain canonical realizations of the complexes Hom(G,H) and Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of such projected Hom-complexes: the dissections of a convex polygon into k-gons, Postnikov's generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, and the graph of weak compositions of an integer into a fixed number of summands.Comment: 23 pages, 5 figures; improved exposition; accepted for publication in JCT

Kalai's squeezed 3-spheres are polytopal

In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.Comment: 11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometr

On the Monotone Upper Bound Problem

The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank (n<=d+2: G\"artner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have M_{ubt}(6,9)=30 vertices, but not more than 26 <= M(6,9) <= 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai's (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl's (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma.Comment: 15 pages; 6 figure

Extinguishing The Fuse of Teacher Burnout

In the 1970’s, Herbert Freudenberger, defined teacher burnout as being a state of exhaustion or displeasure brought on by professional relationships that failed to generate predictable rewards. Teacher burnout is defined by three themes, emotional exhaustion, depersonalization, and lack of personal accomplishment. Some indications of teacher burnout consist of sleeping and fatigue issues, constantly neglecting or having difficulties with focusing, weight and appetite concerns, and depression and anxiety (Tapp, 2021). In 2021, an evaluation was given to teachers highlighting how they are feeling and if they feel like leaving or are planning to leave the education world. They found that teacher\u27s rate of leaving within the first five years, increased to forty-one and three hundredth percent (Campbell, 2020). On top of teacher burnout in the past, Covid-19 has played an even bigger role by making a larger impact on teacher’s mental state as well as kids (Lizana et al., 2021). Research reveals there is a dire need to be mentally and physically stable in order to set our students up for success. When a teacher is mentally and physically exhausted and are not working to better themselves, it will rub off on students and create a negative environment for all

Showing non-realizability of spheres by distilling a tree

In [Zhe20a], Hailun Zheng constructs a combinatorial 3-sphere on 16 vertices whose graph is the complete 4-partite graph K4;4;4;4. Such a sphere seems unlikely to be realizable as the boundary complex of a 4-dimensional polytope, but all known techniques for proving this fail because there are just too many possibilities for the 16 4 = 64 coordinates of its vertices. Known results [PPS12] on polytopal realizability of graphs also do not cover multipartite graphs. In this paper, we level up the old idea of Grassmann{Pl ucker relations, and assemble them using integer programming into a new and more powerful structure, called positive Grassmann{Pl ucker trees, that proves the non-realizability of this example and many other previously inaccessible families of simplicial spheres. See [Pfe20] for the full versionPeer ReviewedPostprint (published version