Academics

Main Math Research Topics

The Significant-digit Phenomenon, or Benford's Law

A century-old empirical observation now called Benford's Law says that the significant digits of many real datasets are logarithmically distributed, rather than uniformly distributed, as might be expected. This research includes discoveries that help explain the ubiquity of Benford data sets. For example, it has now been shown that iterations of many common functions (including all polynomials, power, exponential, and trigonometric functions, as well as compositions thereof), dynamical systems and differential equations, geometric Brownian motion (hence most stock market models), mixtures of data from different sources, and Newton's method, all produce Benford sequences. These publications also include applications to fraud detection and to diagnostic tests for mathematical models, many examples, and open Benford-related problems in dynamical systems, probability, number theory, and differential equations.

Optimal Stopping Theory and Secretary Problems

In many basic processes in science (and in life), there is an element of chance involved, and a crucial problem is deciding when to stop. The process could be debugging large software programs, proofreading a paper, waiting to buy Google stocks, performing medical experiments, looking at houses to buy, gathering laboratory data, or interviewing for a new secretary (or spouse). The typical framework is that a sequence of random variables is being observed, and the objective is to decide when to stop in order to maximize the expected reward. These publications include basic "prophet inequalities" ( comparisons of the expected return of a gambler who has foresight, or inside information, with a gambler who does not' game-theoretic extensions to the classical secretary problem, and determiation of optimal rules and optimal bounds for generalized stopping.

Fair Division Problems

The general subject of this research is the question of whether an object (such as a cake or piece of land) can be divided among a number of people so that each receives a portion he considers a fair share, according to his own values. (Formally, there are n measures on the same object - a measurable space - and a typical goal is to find a partition of the object into n pieces so that the minimum value of the i-th measure on the i-th piece is as large as possible). These publications include generalizations of Steinhaus'classical "Ham Sandwich Problem", Neyman and Pearson's "Bisection Problem", and Fisher's "Problem of the Nile", determination of optimal bounds and extreme-case measures, generalizations of Lyapounov's Convexity Theorem, and applications to disarmament, dividing inheritances, and lotterized allocations of indivisible goods.

From: “Mathematical Devices for Getting a Fair Share”, American Scientist,
Volume: 88 Number: 4, July-August 2000, Page: 325

Current Collaborations

• Arno Berger, University of Canterbury, New Zealand
• Marco Dall'Aglio, U. of Pescara, Italy
• Ron Fox, Georgia Tech, Atlanta, GA
• Kent Morrison, Cal Poly, San Luis Obispo, CA
• Klaus Schuerger, U. Bonn, Germany

PhD Students

To link to the Mathetmatics Genealogy Project, click here.  

• Pieter Allaart, Vrije Universiteit Amsterdam (with Petrus Holewijn)
• Lisa Bloomer, Georgia Institute of Technology
• Frans Boshuizen, Vrije Universiteit Amsterdam (with Petrus Holewijn)
• Josephine Gouweleeuw, Vrije Universiteit Amsterdam (with Petrus Holewijn)   
• Martin Jones, Georgia Institute of Technology
• Zuzana Kuhn, Georgia Institute of Technology
• David Sitton, Georgia Institute of Technology