Research Interests

In general, I am interested in creating models from basic principles, using asymptotic analysis to study their behavior, and simulating complex systems numerically.

Dynamical Systems

Working with my PhD advisor Danny Abrams, I developed a model to describe religious shift in a society where each person belongs to one of two groups. To verify our model, we compared the model’s predictions to census data from a variety of countries around the world. Using perturbation methods and numerical simulation, we showed that even when society is structured into two nearly independent cliques, the society will still favor one group in the end. For more details, see our publication:

D.M. Abrams, H.A. Yaple, R.J. Wiener. Dynamics of social group competition: modeling the decline of religious affiliation. Phys. Rev. Lett., 107 (2011), p. 088701. [http://prl.aps.org/abstract/PRL/v107/i8/e088701]

I worked with a senior mathematics major at Carthage to use the two-group competition model to study shifts in political party affiliation in the United States. She found interesting data that shows the rise and fall of partisanship across the U.S., found the best-fit model parameters using Mathematica, and analyzed the theory behind the two-group model. Her work could be extended to further study the oscillation of partisanship (which aligns with the presidential election cycle) and compare between elections.

We have further extended this group competition model to describe the dynamics of ferromagnetism, based on the Ising model. Instead of modeling people in a society who may belong to one group or another, we look at particles in a material which may switch magnetic polarities. We are investigating how different networks coupling particles to one another change the behavior of the system as it evolves in time. I've worked with students on this project during summers 2016 and 2017, and currently have a student devoting his senior thesis in math to this topic.

A related project tracks internet usage in countries around the world. We hypothesize that countries influence one another, possibly due to business and trade factors or social and travel factors. We are working with an undergraduate student to study our group competition model as set on an empirical network coupling countries, and compare results to a data set tracking internet use.

Network Science and Epidemiology

One project investigated how an epidemic responds to changes in network structure; specifically, when infected hubs are quarantined and their previous connections rewire. This is in analogy to real-life networks such as transportation: if one route is cut off, travelers will find another. We have found that there is a threshold in the quarantine rate: before the threshold the overall effect of quarantine is minimal, while after the threshold the epidemic is eliminated, with a discontinuous transition between the two cases.

In 2014 I participated in a Math Research Community in Network Science, where I worked with collaborators to study temporal networks, or networks with structure that changes over time. This project has grown to include undergraduate students, who published the results of their summer research.

Math in Industry

I have expanded my research interests by attending mathematical modeling workshops, where I've learned about topics ranging from slime mold, to fluid dynamics, to machine learning.

PIC Math Program

• “Preparation for Industrial Careers in Mathematical Sciences,” facilitating student research with industrial partners, including preparatory workshop, spring research class, and conference presentations.

Mathematical Problems in Industry Workshop

• Medical data machine learning, sponsored by Revon Systems, Inc. in 2018 (hosted by Claremont Graduate College)

• Mathematical model for air filtering by "scrubbers" in a smoke stack, presented by Gore in 2016 (hosted by Duke University)

• Flow of glass as it undergoes tempering, presented by an industry representative from Corning Glass in 2012 (hosted by University of Delaware)

• Working with a representative from Corning Glass, this year the topic was the flow of glass as it is pulled to form a fiber optic cable. 2011 (hosted by NJIT)

• My first year at MPI, I worked in the Corning Glass group to study the manufacture of sheets of glass, such as those used for flat-screen televisions. 2010 (hosted by RPI)

Mathematical Modeling in Industry Workshop

• At this workshop I studied machine learning, under the guidance of an industry representative from Google. With no prior knowledge of the field, I worked in a team of about a dozen graduate students to understand current machine learning techniques and propose improvements to existing algorithms. 2011 (hosted by IMA, Minneapolis, MN)