This project concerns probability models evolving and coalescing in one dimension. Such models arise in diverse areas of application; the models in this project are taken from materials science, social science, genetics, and distributed computing. Specific examples are: an interface between regions of liquid in a pipe; candidates sequentially dropping out of a primary election; tree-search for approximate evaluation of a Boolean function. Common to all these models is an underlying mathematical structure in which initial conditions are random, after which the evolution is governed by a non-random mechanism. Analyses of evolutions without randomness is often considerably more difficult than analysis of those with randomness. The focus in this project is on creating tools to circumvent this problem. In most cases the PI seeks qualitative answers to questions such as: what does the system look like after a long time, how long does it take to reach this state, and how robust is the description to changes in the initial conditions? The project's broader impacts beyond the potential applications to the aforementioned areas, include the training of graduate students in the mathematical sciences at the University of Pennsylvania who will have opportunities to engage in research on highly non-trivial topics in probability theory. The PI will also devote some effort to improving STEM education at a broader level through the development of innovative techniques for calculus instruction and curriculum development. The PI will investigate the systematic mathematical study of certain common practices which have suffered from a lack of methodological validation, as a further broader impact.

The mathematical techniques to be used in this project involve the introduction of time reversals. In a time reversal, randomness of initial conditions becomes randomness of the time-reversed path. Once there is randomness in the evolution, it is possible to use techniques from Markov chains, statistical mechanics, and other applications of probability theory. There are a number of Markov chains that describe time reversals of a given system. Suppose one's concern is to describe the time-zero distribution of a system evolving deterministically from random conditions at time minus infinity. Among possible Markovian time-reversals, the one that describes this is the one with the maximum entropy. The PI then plans to compute this using a variational principle.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
1612674
Program Officer
Tomek Bartoszynski
Project Start
Project End
Budget Start
2016-09-01
Budget End
2019-08-31
Support Year
Fiscal Year
2016
Total Cost
$179,999
Indirect Cost
Name
University of Pennsylvania
Department
Type
DUNS #
City
Philadelphia
State
PA
Country
United States
Zip Code
19104