MathAcrossCampus

The use of observations and models to predict the future state of a system is a hallmark of the scientific method that often has practical application. As a result, estimation and prediction are central pursuits across a vast range of disciplines, including the physical, biological, and social sciences; engineering; and finance. In many cases the system of interest is composed of a large number of interacting components that render estimation and prediction difficult. This challenge motivates this talk in which I will review essential aspects of, and the basic theory for, estimating and predicting complex systems. One such system, Earth's atmosphere, will be used to illustrate techniques that deal with complexity. The success of these methods for reducing uncertainty in weather forecasts will be contrasted against a failure to reduce uncertainty in climate-change forecasts. This contrast motivates a mathematically based reconsideration of model formulation and calibration for complex systems.

An evolutionary tree (a phylogeny) is a graph that shows the sequence of speciation events where one species splits into two. Two other types of tree have also become common in studies of molecular evolution. Coalescents are trees of copies of genes within a single species, and trees of gene duplication show the origin of new genes from old ones. All these trees are interrelated, and they "live" in unusual and difficult spaces. Although biologists now understand that we need mathematics and statistics to think clearly about inferring these trees, mathematics has as yet contributed few important insights about them.

Combinatorial optimization exploded on the mathematical and scientific scene in the 1950s. In this lecture I will briefly survey its development for a wide audience. Theoretical design and analysis of algorithms dominated the early development of the field, while computational progress has been particularly significant in the last twenty years. These theoretical and computational achievements, combined with successful modeling of applications, have made it possible today to solve real-world problems of breathtaking size and diversity. The majority of the lecture will report on success stories in areas such as telecommunication, transportation, traffic and logistics. These results are based on ongoing cooperation between industry, the DFG Research Center MATHEON and my research group at Konrad-Zuse-Zentrum (ZIB).

We shall consider two topics involving coalitions and voting. Each topic involves open questions both in mathematics (probability theory) and in political science.

1. Individuals in a committee or election can increase their voting power by forming coalitions. This behavior yields a prisoner's dilemma, in which a subset of voters can increase their power, while reducing average voting power for the electorate as a whole. This is an unusual form of the prisoner's dilemma in that cooperation is the selfish act that hurts the larger group. The result should be an ever-changing pattern of coalitions, thus implying a potential theoretical explanation for political instability.
2. In an electoral system with fixed coalition structure (such as the U.S. Electoral College, the United Nations, or the European Union), people in diferent states will have different voting power. We discuss some flawed models for voting power that have been used in the past, and consider the challenges of setting up more reasonable mathematical models involving stochastic processes on trees or networks.