Maila Hallare | Searcharxiv

arXiv2026

A Discrete-Time Model of the Academic Pipeline in Mathematical Sciences with Constrained Hiring in the United States

The field of the mathematical sciences relies on a continuous academic pipeline in which individuals progress from undergraduate study through graduate training and postdoctoral program to long term faculty employment. National statistics report trends in bachelor's, master's, and doctoral degree awards, but these data alone do not explain how individuals move through the academic system or how structural constraints shape downstream career outcomes. Persistent growth in postdoctoral appointments alongside relatively stable faculty employment indicates that degree production alone is insufficient to characterize workforce dynamics. In this study, we develop a discrete time compartmental model of the academic pipeline in the field of the mathematical sciences that links observed degree flows to latent population stocks. Undergraduate and graduate populations are reconstructed directly from nationally reported degree data, allowing postdoctoral and faculty dynamics to be examined under completion, exit, and hiring processes. Advancement to faculty positions is modeled as vacancy limited, with competition for permanent positions depending on downstream population size. Numerical simulations show that increases in degree inflow do not translate into proportional faculty growth when hiring is constrained by limited turnover. Instead, excess supply accumulates primarily at the postdoctoral stage, leading to sustained congestion and elevated competition. Sensitivity analyses indicate that long run workforce outcomes are governed mainly by faculty exit rates and hiring capacity rather than by degree production alone. These results demonstrate the central role of vacancy limited hiring in shaping academic career trajectories within the field of the mathematical sciences.

math.DS↗

arXiv2026

A dynamical model of the U.S. mathematics graduate degree pipeline

We present a latent-stock compartmental framework for modeling degree production systems when only completion flows, rather than enrollments, are observed. Applied to U.S.\ mathematics degrees from 1969 to 2017, the model treats master's and PhD populations as latent compartments -- unobserved state variables that are inferred indirectly because they generate the observed completion flows -- with time-varying routing fractions and completion hazards. Using information-criterion model comparison across a grid of specifications, we find strong support for smooth nonlinear time variation in routing fractions and hazards, while models with explicit international forcing are disfavored. The preferred model achieves a log-scale root mean squared error of approximately 0.036, corresponding to a typical multiplicative error of about 4\% in fitted degree counts, and highlights key structural shifts in the graduate pipeline: the master's pathway became increasingly central to PhD production through the late twentieth century before weakening, while direct bachelor's-to-PhD entry remained small but persistent. Estimated completion hazards for both degrees rise over time, indicating faster effective turnover in the graduate compartments. Methodologically, our main contribution is a latent stock dynamical approach that recasts linked degreecompletion time series as a coherent stock-flow system when intermediate enrollments are unobserved, making explicit both what features of pipeline dynamics are identifiable from completion data alone and what limitations such data impose.

math.DS↗