About

Gregory Joseph Herschlag is an Associate Research Professor of Mathematics at Duke University, affiliated with the Trinity College of Arts & Sciences. His research interests include techniques to understand fairness in redistricting, computational fluid dynamics, and high-performance computing. He is involved in studying algorithms and methods for analyzing and auditing redistricting plans, including the use of ensembles of neutrally drawn plans to evaluate partisan fairness. Herschlag has published work on sampling measures on spanning forests for redistricting and multiscale parallel tempering for fast sampling on redistricting plans. In addition to his research, he serves as the Assistant Director of Curricular Engagement of the Information Initiative at Duke, contributing to academic support and university initiatives.

Research topics

• Computer Science
• Algorithm
• Archaeology
• Mathematics
• Operations research
• Library science
• Geography
• Parallel computing
• Computational science
• Database

Selected publications

• A Cycle Walk for Sampling Measures on Spanning Forests for Redistricting

  ArXiv.org · 2025-09-10

  preprintOpen access

  We introduce a new Markov Chain called the Cycle Walk for sampling measures of graph partitions where the partition elements have roughly equal size. Such Markov Chains are of current interest in the generation and evaluation of political districts. We present numerical evidence that this chain can efficiently sample target distributions that have been difficult for existing sampling Markov chains.

  PublisherOA PDFDOI

• Multiscale Parallel Tempering for Fast Sampling on Redistricting Plans

  Multiscale Modeling and Simulation · 2025-10-07

  article
  PublisherDOI

• Multiscale Parallel Tempering for Fast Sampling on Redistricting Plans

  arXiv (Cornell University) · 2024-01-30

  preprintOpen access

  When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given plan, one must ensure that the non-partisan criteria are matched so that we may conclude that partisan differences come from bias rather than, for example, levels of compactness or differences in community preservation. Certain sampling algorithms allow one to explicitly state the policy-based probability distribution on plans, however, these algorithms have shown poor mixing times for large graphs (i.e. redistricting spaces) for all but a few specialized measures. In this work, we generate a multiscale parallel tempering approach that makes local moves at each scale. The local moves allow us to adopt a wide variety of policy-based measures. We examine our method in the state of Connecticut and succeed at achieving fast mixing on a policy-based distribution that has never before been sampled at this scale. Our algorithm shows promise to expand to a significantly wider class of measures that will (i) allow for more principled and situation-based comparisons and (ii) probe for the typical partisan impact that policy can have on redistricting.

  PublisherOA PDFDOI

• Where Data Science and the Disciplines Meet: Innovations in Linking Doctoral Students With Masters-Level Data Science Education

  Harvard Data Science Review · 2024-08-21

  articleOpen access

  Although the need for data science methodological training is widely recognized across many disciplines, data science training is often absent from PhD programs. At the same time, Masters-level data science educational programs have seen incredible growth and investment. In 2018, Duke initiated a National Science Foundation (NSF)-funded program to determine whether Masters-level data science programs that universities have already invested in could be leveraged to reduce data science education barriers doctoral students face. Doctoral Fellows from diverse fields worked with teams of master’s students from Duke’s Master in Interdisciplinary Data Science program on applied Capstone projects focused on the doctoral Fellows’ own disciplines and dissertation research. Fellows also gained access to the Master program’s courses and professional development resources. We examined the implementation, experience, and effect of this integration into Master of Data Science program infrastructure using qualitative data collection with doctoral Fellows, master’s students, and Fellows’ doctoral advisors. Master’s students participating in doctoral-led Capstones benefited from their doctoral Fellows’ mentorship, project management, and content knowledge. Participating doctoral students showed increased learning of data science techniques and professional skills development. While some Fellows’ research was advanced through the Capstones, data also showed mismatches between selected master’s program goals and doctoral students’ needs. Overall, this pilot indicated potential promise in harnessing existing Master in Data Science programs to bolster doctoral students’ data science learning and professional readiness while also identifying areas for improving future such efforts.

  PublisherOA PDFDOI

• Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions

  SIAM Journal on Applied Mathematics · 2023-07-05 · 15 citations

  article

  .We develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis–Hastings method. Our resulting algorithm is able to sample from a specified measure on partitions or spanning forests. Being able to sample from a specified measure is a requirement of what we consider as the gold standard in quantifying the extent to which a particular map is a gerrymander. Our proposal chain modifies the recently developed method called recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the computational efficiency by augmenting the statespace from partitions to spanning forests. The extra information accelerates the computation of the forward and backward proposal probabilities which are required for the Metropolis–Hastings algorithm. We demonstrate this method by sampling redistricting plans on several measures of interest and find promising convergence results on several key observables of interest. We also explore some limitations in the measures that are efficient to sample from and investigate the feasibility of using parallel tempering to extend this space of measures.KeywordsMarkov chain Monte Carlobalanced graph partitioningMetropolis-Hastingsspanning treesMSC codes65C0591D1091D2091F10

  PublisherDOI

• Mathematically Quantifying Non-responsiveness of the 2021 Georgia Congressional Districting Plan

  2022-10-06 · 4 citations

  articleOpen accessSenior author

  To audit political district maps for partisan gerrymandering, one may determine a baseline for the expected distribution of partisan outcomes by sampling an ensemble of maps. One approach to sampling is to use redistricting policy as a guide to precisely codify preferences between maps. Such preferences give rise to a probability distribution on the space of redistricting plans, and Metropolis-Hastings methods allow one to sample ensembles of maps from the specified distribution. Although these approaches have nice theoretical properties and have successfully detected gerrymandering in legal settings, sampling from commonly-used policy-driven distributions is often computationally difficult. As of yet, there is no algorithm that can be used off-the-shelf for checking maps under generic redistricting criteria. In this work, we mitigate the computational challenges in a Metropolized-sampling technique through a parallel tempering method combined with ReCom[11] and, for the first time, validate that such techniques are effective on these problems at the scale of statewide precinct graphs for more policy informed measures. We develop these improvements through the first case study of district plans in Georgia. Our analysis projects that any election in Georgia will reliably elect 9 Republicans and 5 Democrats under the enacted plan. This result is largely fixed even as public opinion shifts toward either party and the partisan outcome of the enacted plan does not respond to the will of the people. Only 0.12% of the ∼ 160K plans in our ensemble were similarly non-responsive.

  PublisherOA PDFDOI

• Mathematically Quantifying Non-responsiveness of the 2021 Georgia Congressional Districting Plan

  arXiv (Cornell University) · 2022-03-13

  preprintOpen accessSenior author

  To audit political district maps for partisan gerrymandering, one may determine a baseline for the expected distribution of partisan outcomes by sampling an ensemble of maps. One approach to sampling is to use redistricting policy as a guide to precisely codify preferences between maps. Such preferences give rise to a probability distribution on the space of redistricting plans, and Metropolis-Hastings methods allow one to sample ensembles of maps from the specified distribution. Although these approaches have nice theoretical properties and have successfully detected gerrymandering in legal settings, sampling from commonly-used policy-driven distributions is often computationally difficult. As of yet, there is no algorithm that can be used off-the-shelf for checking maps under generic redistricting criteria. In this work, we mitigate the computational challenges in a Metropolized-sampling technique through a parallel tempering method combined with ReCom[12] and, for the first time, validate that such techniques are effective on these problems at the scale of statewide precinct graphs for more policy informed measures. We develop these improvements through the first case study of district plans in Georgia. Our analysis projects that any election in Georgia will reliably elect 9 Republicans and 5 Democrats under the enacted plan. This result is largely fixed even as public opinion shifts toward either party and the partisan outcome of the enacted plan does not respond to the will of the people. Only 0.12% of the $\sim$160K plans in our ensemble were similarly non-responsive.

  PublisherOA PDFDOI

• A consistent hierarchy of generalized kinetic equation approximations to the master equation applied to surface catalysis

  UNC Libraries · 2021-07-03

  articleOpen access

  We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.

  PublisherDOI

• Optimal Legislative County Clustering in North Carolina

  Figshare · 2021-01-01 · 4 citations

  datasetOpen access

  North Carolina’s constitution requires that state legislative districts should not split counties. However, counties must be split to comply with the “one person, one vote” mandate of the U.S. Supreme Court. Given that counties must be split, the North Carolina legislature and the courts have provided guidelines that seek to reduce counties split across districts while also complying with the “one person, one vote” criterion. Under these guidelines, the counties are separated into clusters; each cluster contains a specified number of districts and that are drawn independent from other clusters. The primary goal of this work is to develop, present, and publicly release an algorithm to optimally cluster counties according to the guidelines set by the court in 2015. We use this tool to investigate the optimality and uniqueness of the enacted clusters under the 2017 redistricting process. We verify that the enacted clusters are optimal, but find other optimal choices. We emphasize that the tool we provide lists <i>all</i> possible optimal county clusterings. We also explore the stability of clustering under changing statewide populations and project what the county clusters may look like in the next redistricting cycle beginning in 2020/2021. Supplementary materials for this article are available online.

  PublisherDOI

• Reynolds number limits for jet propulsion: A numerical study of simplified jellyfish

  UNC Libraries · 2021-08-14

  articleOpen accessSenior author

  The Scallop Theorem states that reciprocal methods of locomotion, such as jet propulsion or paddling, will not work in Stokes flow (Reynolds number = 0). In nature the effective limit of jet propulsion is still in the range where inertial forces are signi cant. It appears that almost all animals that use jet propulsion swim at Reynolds numbers (Re) of about 5 or more. Juvenile squid and octopods hatch from the egg already swimming in this inertial regime. Juvenile jellyfi sh, or ephyrae, break off from polyps swimming at Re greater than 5. Many other organisms, such as scallops, rarely swim at Re less than 100. The limitations of jet propulsion at intermediate Re is explored here using the immersed boundary method to solve the two-dimensional Navier Stokes equations coupled to the motion of a simpli ed jelly fish. The contraction and expansion kinematics are prescribed, but the forward and backward swimming motions of the idealized jellyfi sh are emergent properties determined by the resulting fluid dynamics. Simulations are performed for both an oblate bell shape using a paddling mode of swimming and a prolate bell shape using jet propulsion. Average forward velocities and work put into the system are calculated for Re between 1 and 320. The results show that forward velocities rapidly decay with decreasing Re for all bell shapes when Re < 10. Similarly, the work required to generate the pulsing motion increases significantly for Re < 10. When compared actual organisms, the swimming velocities and vortex separation patterns for the model prolate agree with those observed in Nemopsis bachei. The forward swimming velocities of the model oblate jelly sh after two pulse cycles are comparable to those reported for Aurelia aurita, but discrepancies are observed in the vortex dynamics between when the 2D model oblate jelly sh and the organism. This discrepancy is likely due to a combination of the differences between the 3D reality of the jellyfi sh verses the 2D simpli cation, as well as the rigidity of the time varying geometry imposed by the idealized model.

  PublisherDOI

Frequent coauthors

• Jonathan C. Mattingly

  20 shared

• Zach Hunter

  9 shared

• Daniel Carter

  Wichita State University

  8 shared

• Robert Ravier

  Sarcos (United States)

  6 shared

• Ryan M. Huang

  Duke University

  6 shared

• Christy V. Graves

  Princeton University

  5 shared

• Sachet Bangia

  Duke University

  5 shared

• Jessica Sperling

  Social Science Research Council

  4 shared

Labs

• Gregory Joseph HerschlagPI