Georgi S. Medvedev
Georgi Medvedev
Professor



Department of Mathematics
Drexel University
3141 Chestnut Street
Philadelphia, PA 19104
phone: (215) 895-6612
fax: (215) 895-1582
email: medvedev@drexel.edu

Office: Korman 222


Georgi Medvedev received Ph.D. in Mathematics from Boston University in 1999. Before coming to Drexel University in 2002, he was a Veblen Research Instructor at Princeton University and at the Institute for Advanced Study.

Dr. Medvedev teaches courses at all levels. He developed an interdisciplinary graduate course MATH 723 Mathematical Neuroscience.

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


Research Experience for Undergraduates


Research

Dr. Medvedev's research areas include dynamical systems, stochastic analysis, numerical methods, and mathematical neuroscience. Dr. Medvedev pioneered the use of graphons to study interacting dynamical systems. His recent work focuses on continuum description of large dynamical networks with applications to synchronization and pattern formation in systems of coupled oscillators.

NSF Dr. Medvedev's research has been supported by National Science Foundation through grants

  • #2009233    Large Deviations and Metastability in Dynamical Networks,   2020-23,
  • #1715161    Mean Field Analysis of Dynamical Networks,   2017-20,
  • #1412066    Dynamics of Large Networks,                             2014-17,
  • #1109367    Mathematical Analysis of Synchronization,        2011-14,
  • #0417624    Irregular Firing in Dopaminergic Neurons,       2004-08.


Selected Publications ( full list)

Graphon Dynamical Systems

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

* 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

* 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, The semilinear heat equation on sparse random graphs, SIAM J. Math. Anal., 49(2), 1333-1355, 2017.   abstract   pdf

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

Synchronization and Pattern Formation

* H. Chiba, G.S. Medvedev, M.S. Mizuhara, Bifurcations and patterns in the Kuramoto model with inertia, submitted, ( 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 ).

* G.S. Medvedev and M. Mizuhara, Chimeras unfolded, J. Stat. Phys., 86, 2022.

* G.S. Medvedev and M. Mizuhara, Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs , J. Stat. Physics, 2021, ( 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

* H. Chiba, G.S. Medvedev, and M. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos 28, 073109 (2018),   abstract,   arxiv

* G.S. Medvedev and X. Tang, The Kuramoto model on power law graphs: Synchronization and Contrast States, Journal of Nonlinear Science, 2018,   abstract   arxiv

G.S. Medvedev and X. Tang, Stability of twisted states in the Kuramoto model on Cayley and random graphs, Journal of Nonlinear Science, 2015.   abstract   arxiv

G.S. Medvedev, Small-world networks of Kuramoto oscillators, Physica D 266 (2014), 13-22. abstract   arXiv preprint

Numerical Methods

* G.S. Medvedev and G. Simpson, A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise, Stochastic Partial Differential Equations: Analysis and Computation, accepted, ( arxiv ).

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, Sparse Monte Carlo method for nonlocal diffusion equations, submitted arxiv

Random Perturbations

* P. Hitczenko and G.S. Medvedev, The Poincare map of randomly perturbed periodic motion, J. Nonlin. Sci., Vol. 23(5), pp. 835-861, 2013. (abstract) (arXiv preprint)

* P. Hitczenko and G.S. Medvedev, Stability of equilibria of randomly perturbed maps, Discrete and Continuous Dynamical Systems - B, 22(2), 2017.   abstract   pdf

Mathematical Neuroscience

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

* G.S. Medvedev, Synchronization of coupled limit cycles, Journal of Nonlinear Science, 2011. PDF

* 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, J.E. Cisternas, Multimodal regimes in a compartmental model of the dopamine neuron, Physica D, 194(3-4), 333-356, 2004. ( PDF )


Upcoming conferences

International Conference on Difference Equations and Applications, Paris-Saclay, France, July 18-22, 2022.

SIAM Conference on Nonlinear Waves and Coherent Structures, University of Bremen, Germany, August 30 - September 2, 2022.


Presentations

Graphon Dynamical Systems: a Law of Large Numbers, Large Deviations, and Metastability, Computational and Applied Mathematics Colloquium, Penn State, April 2022.


Last modified June 11, 2022.