Kent E. Morrison

The Fisher-Hartwig conjecture and Toeplitz eigenvalues

Abstract The conjecture of Fisher and Hartwig, published in 1968, describes the asymptotic expansion of Toeplitz determinants with singular generating functions. For more than twenty years progress was made in extending the validity of the conjecture, but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture. This paper describes the history of the problem, several numerical examples, and the revised conjecture.

Cutting Cakes Carefully

Summary This paper surveys the fascinating mathematics of fair division, and provides a suite of examples using basic ideas from algebra, calculus, and probability which can be used to examine and test new and sometimes complex mathematical theories and claims involving fair division. Conversely, the classical cut-and-choose and moving-knife algorithms show it is often possible to express practical, yet clean, clear, and beautiful logical conclusions without using highly technical language.

The Extended Fisher-Hartwig Conjecture for Symbols with Multiple Jump Discontinuities

The asymptotic expansions of Toeplitz determinants of certain symbols with multiple jump discontinuities are shown to satisfy a revised version of the conjecture of Fisher and Hartwig.

Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup

Discrete Yang-Mills connections

{(S,\cdot)}

The polynomial analogue of a theorem of Rényi

as follows: Players 1 and 2 choose elements

Geometric equivalence relations on modules

{x \in S}