Fundamental Groups of Random Clique Complexes

Clique complexes of Erd\H{o}s-R\'{e}nyi random graphs with edge probability between $n^{-{1\over 3}}$ and $n^{-{1\over 2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex for which every subcomplex has fewer than three times as many edges as vertices must have the homotopy type of a wedge of circles, two spheres and real projective planes. Note that $n^{-{1\over 3}}$ is a threshold for simple connectivity and $n^{-{1\over 2}}$ is one for vanishing first \F_2 homology.Comment: 7 pages statement of Theorem 1.2 correcte

Meunier Conjecture

Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is addressed, leaving the question of connectivity for the color hypergraphs of such a multicolored simplex. Sperner's lemma asserts the existence of a simplex using all the colors for any vertex coloring of a subdivision of a large simplex with appropriate boundary conditions. Meunier's questions generalizes this to the situation of having several such colorings and asserts the existence of a simplex using enough different colors from each coloring

Lefschetz Properties and Basic Constructions on Simplicial Spheres

The well known $g$-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the $g$-conjecture for piecewise-linear spheres.Comment: 18 pages, no figure

Random Walks and Mixed Volumes of Hypersimplices

Below is a method for relating a mixed volume computation for polytopes sharing many facet directions to a symmetric random walk. The example of permutahedra and particularly hypersimplices is expanded upon.Comment: 6 page

Come Join Us! Volunteer Organizing From a Local-Union Base

[Excerpt] Four months later, May 3, 1991, Delta workers elected to join the UAW by a vote of 68 to 58. The small numbers belie the real significance of this achievement, for Delta is the largest Japanese-owned supplier yet organized by the UAW. Equally important, this victory highlights the potency of two overlapping strategies: the use of volunteer organizers, and the reliance on a local-union base for launching and sustaining a drive