Discrepancy convergence for the drunkard's walk on the sphere

We analyze the drunkard's walk on the unit sphere with step size theta and show that the walk converges in order constant/sin^2(theta) steps in the discrepancy metric. This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at http://www.math.hmc.edu/~su/papers.htm

Consensus-halving via Theorems of Borsuk-Ulam and Tucker

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believe the portions are equally split. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

Guidelines for Good Mathematical Writing

Communicating mathematics well is an important part of doing mathematics. Many of us know from writing papers or giving talks that communicating effectively not only serves our audience but also clarifies and structures our own thinking. There is an art and elegance to good writing that every writer should strive for. And writing, as a work of art, can bring a person great personal satisfaction. Within the MAA, we value exposition and mathematical communication. In this column, I’m sharing the advice I give my students to help them write well. There are more extensive treatments (e.g., see Paul Halmos’s How to Write Mathematics), but I wanted a shorter introduction. So I developed the guidelines below

Mathematics for Human Flourishing

Why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish

Race, Space, and the Conflict Inside Us

Talking about race is hard. Our nation is wrestling with some open wounds about race. These sores have been around a while, but they have been brought to light recently by technology, politics, and an increasingly diverse population. And regardless of the outcome of the U.S. presidential election, we will all need to work at healing these sores, not just in our personal lives, but in our classrooms and in our profession

Two-player envy-free multi-cake division

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The resulting allocations of pieces to players are Pareto-optimal with respect to the division. We use a generalization of Sperner's lemma on the polytope of divisions to locate solutions to our generalized cake-cutting problem.Comment: 15 pages, 7 figures, see related work at http://www.math.hmc.edu/~su/papers.htm