Lecture 1: Introduction
An overview will be given about where and why new kinds of mathematical issues arise in the social and behavioral sciences. (As it will be shown, many of these concerns are shared with the statistical, biological and engineering sciences.) This includes paired decision problems in psychology, sociology, and economics, it includes the consensus problem when constructing networks and trees, it includes decision problems in business and engineering. In a different direction, even though “change” (which means dynamics and dynamical systems) is of central importance to the mathematics of all of the social and behavioral sciences, many current models emphasize only “equilibria” (which are determined by unexpected types of fixed point theorems) rather than any description about how the equilibria are, or are not approached. A strong reason for this is the current lack of understanding of how to design appropriate mathematical models. The lecture will provide concrete examples along with a sample listing of open problems.
Lecture 2: Symmetries and paired comparison mysteries
"Paired comparisons" are among the more commonly used decision methods; this is true in the social and behavioral sciences as well as in statistics, engineering, and other areas. But in all of these areas, it is known that the outcomes can have serious problems including cycles.
The mathematical question is to understand why this is so. Here, just the phrase "paired comparisons" suggests that some sort of a product of Z2 symmetry groups plays a central role in analyzing these systems. As developed in this lecture, the use of appropriate symmetry groups and linear algebra provides new ways to completely analyze paired comparison issues. This includes explaining the source of the cycles and other difficulties.
Lecture 3: Impossibility theorems
A common, frustrating aspect shared by many of the social and behavioral sciences are the "impossibility theorems" stating that what we fully expect can be done is impossible to do. This ranges from the seminal “Arrow’s Theorem” in decision theory often described as asserting that “no voting rule is fair,” to Sen’s assertion (in economics and philosophy) about a fundamental conflict between individual decisions and societal concerns, to impossibility assertions about using particular and natural ways to construct consensus graphs (e.g., in biology, the "Tree of Life"), etc.
Adopting the attitude that an impossibility assertion means that the "obvious approach" is incorrect, the natural next step is to examine the underlying mathematical structures of these various systems to determine whether positive assertions are possible, and then develop them. By using this approach, it will be shown how to discover some positive conclusions; e.g., a “positive” version of Arrow’s Theorem will be derived to replace the standard negative statement. Open problems will be described.
Lectures 4 and 5: Symmetries and voting problems.
"Voting rules" are used everywhere, but they are accompanied with many puzzles. Indeed, for more than the last two centuries, it has been recognized that election rankings can more accurately reflect the choice of the election rule rather than the views of the voters. The mathematical question is to understand why this is the case, and to provide guidance on the choice of a rule.
As developed in these two lectures, answers follow by extracting properties of the orbits of appropriate symmetry groups. These orbits can be used to create a coordinate system; once this system is developed, it becomes possible to analyze a variety of problems by now using nothing more than linear algebra.
Lecture 6: Applications of symmetries to more general decision problems
Lessons learned from the previous two lectures will be applied to a wider selection of decision and aggregation settings. In this lecture, the above voting rules are treated as prototypes for more general aggregation methods; that is, individual preferences of candidates are aggregated into the election ranking for the group. In this manner, lessons learned from voting rules will be applied to more general systems, such as those that occur in non-parametric statistics, or price mechanisms in economics.
As another example, one of the early pioneers in mathematical psychology, R. D. Luce, developed an axiomatic approach to analyze decision behavior in cognitive behavior and psychology. But mysteries remain about his approach. It will be shown how to extract the hidden symmetries from these systems and then use this information to develop sharper conclusions.
Lectures 7 and 8: Dynamics of social and behavioral systems
Standard ways to understand dynamics include formulating appropriate x' = f(x) or xn+1 = f(xn) expressions where f(x) represents the behavioral changes. In the physical, engineering, and biological sciences, the choice of f(x), such as with Newton’s equations for the n-body problem, often is known and has been experimentally determined or verified in many ways. But in the social and behavioral sciences, a major difficulty is that the way in which the behavior changes is generally not known. Indeed, the intent of analyzing such an x' = f(x) expression is to understand what is the underlying behavior.
Herein lies a potential conflict; whether the x' = f(x) expression comes from evolutionary game theory, changes in economic processes, or modifications in social norms, the selection of a particular f(x) implicitly assumes--and imposes--specific characteristics about the unknown behavior. As such, using standard approaches carry the danger of assuming an answer to prove it; assumptions about the structure of behavior (the choice of f(x)) will affect statements about the unknown behavior. To avoid this difficulty, a new approach coming from algebraic topology will be described; this approach introduces a "qualitative" dynamic process. An interesting feature is how this method identifies what kinds of data must be gathered to verify and refine the dynamical model.
Lecture 9: Topological structures applied to psychology
A difficulty in mathematical psychology is that many results are experimentally determined; this usually means that the conclusions must be local in nature. A natural mathematical question, then, is to explore whether it is possible to create mathematical methods to connect these local conclusions into a global structure. Some elementary algebraic topology will be used to handle a selection of these kinds of questions that arise about the brain’s interpretations of color and of emotion.
Lecture 10: Overview and open problems
This final lecture will connect the main tools that have been developed over these lectures, and show where further mathematical development is necessary. Open problems and concerns will be discussed.