**Eric A. Autry, Daniel Carter, Gregory Herschlag, Zach Hunter, Jonathan C. Mattingly "**

We develop a Multi-Scale Merge-Split Markov chain on redistricting plans. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number of elements which each correspond to a different district. The districts satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. The multi-scale algorithm is similar to our previously developed Merge-Split proposal, however, this algorithm provides improved scaling properties and may also be used to preserve nested communities of interest such as counties and precincts. Both works use a proposal which extends the ReCom algorithm which leveraged spanning trees merge and split districts. In this work we extend the state space so that each district is defined by a hierarchy of trees. In this sense, the proposal step in both algorithms can be seen as a "Forest ReCom." We also expand the state space to include edges that link specified districts, which further improves the computational efficiency of our algorithm. The collection of plans sampled by the MCMC algorithm can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection of plans, the given plan may have been gerrymandered and is labeled as an outlier.

*Multi-Scale Merge-Split Markov Chain Monte Carlo for Redistricting"*arXiv:2008.08054**Autry, E., Carter, D., Herschlag, G., Hunter, Z., & Mattingly, J.**SIAM Journal on Applied Mathematics (submitted October, 2020).

*A Merge-Split Proposal for Reversible Monte Carlo Markov Chain Sampling of Redistricting Plans*.We describe a Markov chain on redistricting plans that makes relatively global moves. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number elements which each correspond to a different district. The partitions satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. When these constraints and criteria are chosen to align well with classical legal redistricting criteria, the algorithm can be used to generate a collection of non-partisan, neutral plans. This collection of plans can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection plans, the given plan may have been gerrymandered and is labeled as an outlier.

**Sara M. Clifton, Kaitlin Hill, Avinash J. Karamchandani, Eric A. Autry, Patrick McMahon, and Grace Sun. “Mathematical model of gender bias and homophily in professional hierarchies.”**

*Chaos*(Woodbury, N.Y.), vol. 29, no. 2, Feb. 2019. Epmc, doi:10.1063/1.5066450. Women have become better represented in business, academia, and government over time, yet a dearth of women at the highest levels of leadership remains. Sociologists have attributed the leaky progression of women through professional hierarchies to various cultural and psychological factors, such as self-segregation and bias. Here, we present a minimal mathematical model that reveals the relative role that bias and homophily (self-seeking) may play in the ascension of women through professional hierarchies. Unlike previous models, our novel model predicts that gender parity is not inevitable, and deliberate intervention may be required to achieve gender balance in several fields. To validate the model, we analyze a new database of gender fractionation over time for 16 professional hierarchies. We quantify the degree of homophily and bias in each professional hierarchy, and we propose specific interventions to achieve gender parity more quickly.

**Autry, EA; Bayliss, A; Volpert, VA. “Biological control with nonlocal interactions.”**

*Mathematical Biosciences,*vol. 301, July 2018, pp. 129–46. Epmc, doi:10.1016/j.mbs.2018.05.008.In this paper, we consider a three-species food chain model with ratio-dependent predation, where species u is preyed upon by species v, which in turn is preyed upon by species w. Our primary focus is on biological control, where the bottom species u is an important crop, and v is a pest that has infested the crop. The superpredator w is introduced into this pest-infested environment in an attempt to restore the system to a pest-free state. We assume that the species can behave nonlocally, where individuals will interact over a distance, and incorporate this nonlocality into the model. For this model, we consider two types of nonlocality: one where the crop species u competes nonlocally with itself, and the other where the superpredator w is assumed to be highly mobile and therefore preys upon the pest v in a nonlocal fashion. We examine how biological control can prove to be highly susceptible to noise, and can fail outright if the pest species is highly diffusive. We show, however, that control can be restored if the superpredator is sufficiently diffusive, and that robust partial control can occur if the superpredator behaves nonlocally. Since the superpredator is generally introduced artificially, our results point to properties of the superpredator which can lead to successful control.

**E A Autry, A Bayliss and V A Volpert “Traveling waves in a nonlocal, piecewise linear reaction-diffusion population model.”**

*Nonlinearity*, vol. 30, no. 8, July 2017, pp. 3304–31. Scopus, doi:10.1088/1361-6544/aa7b95.We consider an analytically tractable switching model that is a simplification of a nonlocal, nonlinear reaction–diffusion model of population growth where we take the source term to be piecewise linear. The form of this source term allows us to consider both the monostable and bistable versions of the problem. By transforming to a traveling frame and choosing specific kernel functions, we are able to reduce the problem to a system of algebraic equations. We construct solutions and examine the propagation speed and monotonicity of the resulting waves.