Factors of IID on Trees

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur

Analyzing the Gender Gap on an Entrance Exam for Mathematically Talented Students

We investigate the qualifying entrance exam for the University of Minnesota Talented Youth Mathematics Program (UMTYMP), a five-year accelerated program covering high school- and undergraduate-level mathematics. The exam is used to assess the computational, numerical reasoning, and geometric skills of hundreds of fifth-, sixth-, and seventh-grade students annually. It has accurately identified qualified students in past years, but female participants consistently have had lower overall scores. Based on our belief that they are equally well qualified, in 2011 we began an extensive investigation into the structure and content of the exam to determine the possible sources for these differences. After gathering and analyzing data, we made relatively modest changes in 2012 which essentially eliminated the gender bias on one version of the entrance exam, increasing the percentage of females who qualified. The other unmodified versions in 2012 exhibited the typical gender difference from previous years. We continue to analyze the possible reasons for the gender differences while monitoring the overall student performance upon entering the Program

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.