Department of Mathematical Sciences
George Mason University
Get In Touch:
My office is 4458 Exploratory Hall.
My email is mholzer at gmu.edu.
My research interests are in differential equations, existence and stability of traveling waves, dynamics on complex networks and pattern formation. Before coming to Mason, I completed my Ph.D. at Boston University and was a postdoc at the University of Minnesota.
Outside of mathematics, I play basketball, hike, and spend time with my lovely wife and sons.
Spring 2023: Ordinary Differential Equations (Math 677)
Spring 2022: Differential Equations (Math 214) and Numerical Analysis (Math 446)
Fall 2021: Differential Equations (Math 214) and Probability (Math 351)
Spring 2021: Ordinary Differential Equations (Math 677) and Complex Variables (Math 411)
- Pushed and pulled fronts in a logistic Keller-Segel model with chemorepulsion, preprint, 2023 (with Montie Avery and Arnd Scheel). pdf
- Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularization, to appear in NoDEA, 2023 (with Matthew Kearney, Samuel Molseed, Katie Tuttle and David Wigginton). pdf
- Pushed-to-pulled front transitions: continuation, speed scalings, and hidden monotonicty. Journal of Nonlinear Science, 33(6), 2023 (with Montie Avery and Arnd Scheel). pdf    Matlab code
- Pattern formation in random networks using graphons. SIAM J. Math. Anal., 55(3), 2150-2185, 2022 (with Jason Bramburger). pdf
- Epidemic spreading on complex networks as front propagation into an unstable state. Bulletin of Mathematical Biology, 85(1), 4, 2023 (with Ashley Armbruster, Noah Roselli and Lena Underwood). pdf
- Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. DCDS-S, 15(9), 2467-2496, 2021 (with Grégory Faye and Thomas Giletti). pdf
- Locked fronts in a discrete time discrete space population model. Journal of Mathematical Biology, 85 (4), 1-31, 2022 (with Zachary Richey, Wyatt Rush and Samuel Schmidgall). pdf
- Invasion into remnant instability: a case study in front dynamics. Indiana University Mathematics Journal, 71(5), 1-78, 2022 (with Grégory Faye, Arnd Scheel and Lars Siemer). pdf
- Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model. Journal of Differential Equations, 269 (9), 6559-6601, 2020 (with Grégory Faye). pdf
- Asymptotic stability of the critical Fisher-KPP front using pointwise estimates. ZAMP, 70:13, 2019. (with Grégory Faye). pdf
- Estimating epidemic arrival times using linear spreading theory. Chaos, 28(1), 2018 (with Lawrence Chen and Anne Shapiro). pdf
- Pattern formation, traveling fronts and consensus versus fragmentation in a model of opinion dynamics. Physics Letters A, 381(37):3197-3202, 2017 (with Ratna Khatri). pdf
- Bifurcation to locked fronts in two component reaction-diffusion equations. Annales de l'Institut Henri Poincaré C, 36(2):545-584, 2019 (with Grégory Faye). pdf .
- Invasion fronts on graphs: the Fisher-KPP equation on homogeneous trees and Erdos-Reyni graphs. DCDS-B, 24(2):671-694, 2019. (with Aaron Hoffman). pdf
- Linear spreading speeds from nonlinear resonant interaction. Nonlinearity, 30(6):2403-2442, 2017. (with Grégory Faye and Arnd Scheel). pdf
- Wavetrain solutions of a reaction-diffusion-advection model of mussel-algae interaction. SIAM J. on Applied Dynamical Systems, 16(1):431-478, 2017 (with Nikola Popovic). pdf
- A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. DCDS-A, 36(4):2069-2084, 2016. pdf .
- Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach. Journal of Differential Equations, 258(7):2257-2289, 2015. (with Grégory Faye). pdf .
- Criteria for pointwise growth and their role in invasion processes. Journal of Nonlinear Science, 24(4):661-709, 2014. (with Arnd Scheel) pdf .
- Anomalous spreading in a system of coupled Fisher-KPP equations. Physica D, 270(1):1-10, 2014. pdf .
- Accelerated fronts in a two stage invasion process. SIAM J. Math. Anal. 46(1):397-427, 2014. (with Arnd Scheel) pdf
- An analysis of the renormalization group method for asymptotic
expansions with logarithmic switchback terms. Advances in Differential Equations, 19(3-4):254-282, 2014. (with Tasso Kaper) pdf
- A slow pushed front in a Lotka-Volterra competition model. Nonlinearity, 25:2151, 2012 (with Arnd Scheel) pdf
- Existence and Stability of Traveling Pulses in a Reaction-Diffusion-Mechanics System. Journal of Nonlinear Science, 23:129-177, 2013. (with Arjen Doelman and Tasso Kaper) pdf
Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations. Physica D, 237, 2008. (with Lee Deville, Tony Harkin, Kreso Josic and Tasso Kaper)
Singular perturbations of $z^n$. Transcendental dynamics and complex analysis, 111--137, London Math. Soc. Lecture Note Ser., 348, Cambridge Univ. Press, Cambridge, 2008. (with Bob Devaney, Dan Look, Monica Moreno Rocha and David Uminsky)
Blowup points and baby Mandelbrot sets for singularly perturbed rational maps. Complex dynamics, 51--62, Contemp. Math., 396, Amer. Math. Soc., Providence, RI, 2006. (with Bob Devaney and David Uminsky)
Phase locking in integrate-and-fire models with refractory periods and modulation. Journal of Mathematical Biology 49 (2004), no. 6, 577--603. (with Tomas Gedeon)
Attractor reconstruction from interspike intervals is incomplete. Physica D 178 (2003), no. 3-4, 149--172. (with Tomas Gedeon and Mark Pernarowski)
Undergraduate Research Mentored:
- Invasion fronts and pattern formation in a model of chemotaxis in one and two dimensions. SIAM Undergraduate Research Online. 6 (2013), 228-245. (Koushiki Bose, Tyler Cox, Stefano Silvestri and Patrick Varin) pdf