I am an applied mathematician working in complex and nonlinear systems. I study a variety of social and biological systems with mathematical models. My goal is to create and communicate mathematics in a way that is exciting, relevant, and inclusive.
I am currently a CAM Assistant Professor (a postdoctoral position) in the Department of Mathematics at UCLA, where my mentor is Mason Porter. I received my Ph.D. in Mathematics from the University of Utah in 2018, where I was advised by Paul Bressloff.
I use a combination of analytical and computational techniques to study phenomena in social and biological applications. I pair tools from dynamical systems and differential equations (including linear and weakly nonlinear analysis, perturbation theory, adiabatic reduction, and symmetric bifurcation theory), network theory, and stochastic processes with numerical simulation and data-driven computational techniques (like agent-based models and dynamic mode decomposition). Read more about my projects here.
I am passionate about the teaching and learning of mathematics. I am a current Project NExT Fellow (Silver '19 cohort) and I was a graduate fellow for the Center for Teaching and Learning Excellence at the University of Utah. I strive to create an inclusive environment where my students can engage meaningfully with challenging problems. My teaching portfolio is here.
As a first-generation college student and woman in mathematics, I strive to be an active advocate for historically underserved people in the math community. I am a proud member of AWM and the Spectra Ally List, and I am in the process of cofounding a "Womxn in Complex and Nonlinear Systems" research network.
I was born in Idaho and raised in Utah. I played classical flute for many years (I was originally a music major!). Outside of math and music, I love climbing, hiking, skiing, rafting, trivia, podcasts, and coffee.
I am interested in both theory and application of bounded-confidence models on graphs. I use these models to study the effects of ideology and quality in content spread online, and how media can influence these dynamics. I also use mean-field approximations to study the existence and dynamics of solutions, including bifurcations and basins of attraction of the steady states. I am developing generalizations of bounded-confidence models for multilayer networks and simplicial complexes.
Collaborators: Mason Porter (UCLA), Franca Hoffmann (Caltech), Alex Pan (Caltech)
I created a reaction hybrid-transport model to study development of synapses in the ventral cord of the worm C. elegans. We posit that the synapse structure is due to the transport of two interacting proteins: one undergoing rapid active transport on microtubules, and the other freely diffusing. I have also studied transport-driven instability of this model on growing domains and in two dimensions, when active transport is restricted to a lattice network of microtubules.
Collaborators: Paul Bressloff (U. Utah), Sam Carroll (U. Utah)
My collaborators and I use agent-based models to study parasite spread in dynamic social grooming networks. We examine how parasites alter the optimal social affiliation strategies for a population’s evolutionary fitness. We use network centrality measures to explore how parasite spread differs on networks for these different affiliation strategies. Using this model, we explore how system parameters such as parasite reproduction and grooming effectiveness alter the infection burden.
Collaborators: Nina Fefferman (U. Tennessee), Maryann Hohn (Pomona), Candice Price (Smith), Ami Radunskaya (Pomona), Suzanne Sindi (UC Merced), Shelby Wilson (U. Maryland), Nakeya Williams (US Merchant Marine Academy)
I am interested in how randomness can be exploited in biological systems for robustness and functionality. Using a stochastic model where continuous random variables for voltage are coupled with discrete random variables for ion channels, I studied how random fluctuations in ion channels can enhance subthreshold oscillations in neurons. I have also investigated randomly switching intracellular transport in the context of virus trafficking using SDEs.
Collaborators: Paul Bressloff (U. Utah), Sean Lawley (U. Utah), Marie Tuft (U. Utah, now U. Pittsburgh)
I have a number of current and forthcoming projects using a combination of modeling and data to study human social systems, including
Dynamic mode decomposition of survey data to assess the effectiveness of social programs for gang reduction
The effects of homophily in professional networks on gender parity
Modeling mass shooting events in the United States
Mathematical analysis of the impact of social structure on ectoparasite load in allogrooming populations: HZB, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Suzanne S. Sindi, Nakeya D. Williams, Shelby N. Wilson, and Nina H. Fefferman. In Understanding Complex Biological Systems in Mathematics, Springer (2018), pp. 47-62.
How disease risks can impact the evolution of social behaviors and emergent population organization: Nakeya D. Williams, HZB, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Suzanne S. Sindi, Shelby N. Wilson, and Nina H. Fefferman. Preprint available at arXiv:1904.09238. In Understanding Complex Biological Systems in Mathematics, Springer (2018), pp. 31-46.
Turing mechanism for homeostatic control of synaptic density during C. elegans growth.: HZB and Paul C. Bressloff. Physical Review E, 96(1):012413, 2017.
A mechanism for Turing pattern formation with active and passive transport: HZB and Paul C. Bressloff. SIAM Journal on Applied Dynamical Systems, 15(4):1823-1843, 2016.
Coarse-graining intermittent intracellular transport: two and three-dimensional models: Sean D. Lawley, Marie Tuft, and HZB. Physical Review E, 92(4):042709, 2015.
Quasicycles in the stochastic hybrid Morris-Lecar neural model: HZB and Paul C. Bressloff. Physical Review E, 92(1):012704, 2015.
Influence of media on opinion dynamics in social networks: HZB and Mason A. Porter. Preprint available at arXiv:1904.09238. Submitted, 2019.
Bifurcation analysis of pattern formation in a two-dimensional hybrid reaction-transport model: Sam R. Carroll, HZB, and Paul C. Bressloff. Preprint available upon request. Submitted, 2019.
How emergent social patterns in allogrooming combat parasitic infections: Shelby N. Wilson, Suzanne S. Sindi, HZB, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Nakeya D. Williams, and Nina H. Fefferman. Preprint available upon request. Submitted, 2019.
In Fall 2019, I'm teaching Math 142: Mathematical Modeling. Our course materials can be found on Piazza.
I will also be supervising teams for the 2020 COMAP Math Contest in Modeling.
In Spring 2020, I will be teaching Math 168: Introduction to Networks.
Reading Course in Nonlinear Dynamics [Spring 2019, UCLA]
Linear and Nonlinear Systems of Differential Equations [Fall 2018 and Winter 2019, UCLA]
Vector Calculus and Partial Differential Equations [Spring 2018, U. Utah]
Calculus I, online course [Fall 2017, U. Utah]
Linear Algebra [Spring 2017, U. Utah]
Partial Differential Equations for Engineers [Summer 2016 and Fall 2016, U. Utah]
Calculus II for Biologists [Spring 2016, U. Utah]
Calculus I for Biologists [Fall 2015, U. Utah]
Differential Equations and Linear Algebra for Engineers [Fall 2013, U. Utah]
College Algebra [Summer 2013, U. Utah]
Intermediate Algebra [Spring 2013 and Spring 2014, U. Utah]
Intro to Quantitative Reasoning [Fall 2012, U. Utah]
Gabriella Lorenzi (UCLA): Modeling gender bias and homophily in professional networks (co-mentor Mason Porter)
Alex Pan (Caltech): Opinion formation on networks (co-mentors Mason Porter and Franca Hoffmann)
Zehan Chao (UCLA), Zheyuan Cui (UCLA), Avery Edson (UCLA), Yihuan Huang (UCLA), Cesar Guajardo (SMC), Xingjia Wang (UCLA), Zhanyuan Yin (UCLA): Evaluating effectiveness of the Gang Reduction and Youth Development program with dynamic mode decomposition
Marie Tuft (University of Utah): Quantitative analysis of virus trafficking in a biological cell (co-mentor Sean Lawley)
Oliver Richardson (University of Utah): Modeling learning on neural networks (co-mentor Sean Lawley)
Braden Schaer and Anand Singh (University of Utah): Modeling diffusion of neurotransmitters. (co-mentor Sean Lawley)
Video: Rehumanizing Mathematics (Rochelle Gutiérrez)
Something missing? Contact me to let me know if there are other links I should add.