I am a Professor of Mathematics at Drexel University. I received Ph.D. in Mathematics from Boston University in 1999. Before joining Drexel University in 2002, I was a Veblen Research Instructor at Princeton University and at the Institute for Advanced Study.

I use techniques from Dynamical Systems, Partial Differential Equations, Numerical Analysis, and Probability to study mathematical models in Physics and Biology.

My recent work focuses on interacting particle systems on networks and fractals (continuum limit and mean-field formalism), synchronization and pattern formation, randomly perturbed dynamical systems and metastability.

I teach courses at all levels. I developed an interdisciplinary graduate course MATH 723 Mathematical Neuroscience.

I serve on the editorial boards of Discrete and Continous Dynamical Systems (Series B) and Networks and Heterogeneous Media.

Research highlights

(B) As part of our research on the Kuramoto model of coupled phase oscillators on self-similar networks, Matthew Mizuhara (TCNJ) and I investigated harmonic maps from post-critically finite fractals to the circle. The figure below (courtesy of M. Mizuhara) illustrates a representative solution arising from this analysis. Further details and results are presented in our paper published in Studies in Applied Mathematics.

(C) In our recent paper Graphon Signal Processing for Spiking and Biological Neural Networks with Takuma Sumi, we develop a graphon-based approach to identify patterns of stimulation from the output of a neural network. We test our method on data generated by a computational model of synaptically coupled integrate-and-fire neurons and on calcium-imaging datasets of modular cultured neuronal networks. This work represents the first application of Graphon Signal Processing to experimental neuroscience.

Projects, Publications, and Funding

Recent papers and preprints

* G.S. Medvedev and E. Trelat, From PDEs on standard domains to self-similar particle systems on fractals, arxiv, 2026.

* G.S. Medvedev, Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates, arxiv , 2026.

* T. Sumi and G.S. Medvedev, Graphon Signal Processing for Spiking and Biological Neural Networks, Neural Computation, 2026, (arxiv).

* G.S. Medvedev and M. Mizuhara, The Kuramoto Model on the Sierpinski Gasket I: Harmonic Maps from Post-Critically Finite Fractals to the Circle , Studies in Applied Mathematics, 2026.

* G.S. Medvedev and M. Mizuhara, The Kuramoto Model on the Sierpinski Gasket II: Twisted States, Journal of Nonlinear Science, 2026.

* M. Ghandehari and G.S. Medvedev, The Large Deviation Principle for W-random spectral measures, Appl. Comp. Harmonic Anal., vol. 77, 2025; arxiv .

* N. Berglund, G.S. Medvedev, and G. Simpson, Metastability in the stochastic nearest-neighbour Kuramoto model of coupled phase oscillators, Nonlinearity, 38 095031, 2025.

* G.S. Medvedev and D.E. Pelinovsky, Turing bifurcation in the Swift-Hohenberg equation on deterministic and random graphs, J. Nonlin. Sci., 2024.

Selected publications

* G.S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Analysis, 46(4), 2743-2766, 2014. abstract   pdf

* G.S. Medvedev, The nonlinear heat equation on W-random graphs, Archive for Rational Mechanics and Analysis June 2014, Volume 212, Issue 3, pp 781-803, 2014.   abstract   pdf

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit, SIAM J. Math. Anal., 50 (2018), no. 3, 2441-2465, abstract   pdf

* P. Dupuis and G.S. Medvedev, The large deviation principle for interacting dynamical systems on random graphs, Comm. in Math. Phys., 390, (2022) ( arxiv).

* G.S. Medvedev, The continuum limit of the Kuramoto model on sparse random graphs, Communications in Mathematical Sciences, vol. 17 (2019), no. 4,pp. 883- 898.

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, Sparse Monte Carlo method for nonlocal diffusion equations, SIAM J. Numer. Anal., 2022, arxiv

* H. Chiba and G.S. Medvedev, Stability and bifurcation of mixing in the Kuramoto model with inertia, SIAM J. Math. Anal., 54(2), 2022 ( arxiv ).

* H. Chiba and G.S. Medvedev, The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discrete and Continuous Dynamical Systems - A, 39(1), 2019.   ( abstract )

* H. Chiba and G.S. Medvedev, The mean field analysis for the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations, Discrete and Continuous Dynamical Systems - A, 39(7), 2019.   ( abstract )   arxiv

* G.S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202(1-2), 37-59, 2005. ( PDF )

* G.S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonl. Sci., 2012. ( arXiv:1105.2801).

* G.S. Medvedev and N. Kopell, Synchronization and transient dynamics in the chains of electrically coupled FitzHugh-Nagumo oscillators, SIAM J. Appl. Math., vol. 61, No. 5, pp. 1762-1801. ( PDF )

Movies of the patterns generated by the Kuramoto model are available here.

Last modified March 2026.