Research Synopses

My research interests lie in nonlinear and complex systems and in the applications of the techniques from these fields to the physical, engineering, biological, social, and information sciences (i.e., everything). I am especially interested in networks, dynamics on networks, and dynamics of networks; and I have also done a lot of work in nonlinear waves and on other phenomena in nonlinear and complex systems. My main research thrust right now is on theory and applications of multilayer networks.

My idea of interesting and viable research is to first model a system and then analyze that model both analytically and computationally (though some of my papers are purely computational; it depends on the topic). This approach is interdisciplinary in nature, as many of the same methods and structures arise in superficially distinct scientific disciplines, which allows one to better understand the structure and dynamics of the systems under study. My research stretches from methodological development and model formulation to data analysis. I enjoy all of these approaches, and all of them are important.

This web page briefly discusses some of the topics that I have investigated.


  • Wikipedia entry
  • In the study of networks, analytical and computational techniques from subjects such as statistical mechanics and dynamical systems are employed to study graphs that embody natural and man-made networks. Examples include the World Wide Web, ecosystems, protein interaction networks, granular force networks, online social networks, citation networks, and myriad more. My collaborators and I have spent a lot of time studying mesoscopic network structures such as community structure and "core-periphery structure" to try to understand structural properties better. My group has conducted both data-driven investigations and studies in which we have developed new methodological tools and models. Some of the many applications that we have considered are baseball networks, legislative networks, protein interaction networks, functional brain networks, granular force networks, and Facebook networks. (There are many more, but hopefully this short list provides a decent indication of the breadth of my interests in networks.)
  • The networks wiki of the collaboration between Peter Mucha and me

  • Multilayer Networks

  • Review article on multilayer networks

  • Many networks are time-dependent (i.e., temporal) or "multiplex" (i.e., possess multiple types of edges), and it is important to develop and employ methods and mathematical structures beyond the usual graphs that are used for ordinary static networks. Temporal and multiplex networks can be represented as "multilayer networks", which can in turn be represented as tensors. Together with students and collaborators, I have been developing a framework and accompanying tools and methods to study multilayer networks (and dynamical systems on multilayer networks). For example, my collaborators and I have developed a method to study community structure in multilayer networks, and we have applied this method to areas such as neuroscience and political science. My group is also currently applying multilayer community detection to data from disease propagation, consumer shopping, and finance.

  • Dynamics on Networks

  • The study of "dynamics on networks" concerns how nontrivial connectivity influences dynamical systems running on top of a network. Example dynamical systems that are interesting to consider include toy models of biological or social epidemics, nonlinear oscillators, percolation, and more. My work in this area has included the examination of the efficacy of methodology like mean-field and locally tree-like approximation on networks as well as the development of new models of social influence. I have focused thus far on the examination of tractable models, but I also want to incorporate real data more thoroughly into these studies. I also seek to examine the interaction between dynamics on networks and dynamics of networks.

  • Bose-Einstein Condensation

  • Wikipedia entry
  • Review article on nonlinear waves in Bose-Einstein condensates
  • Bose-Einstein condensates (BECs) are described at the mean-field level by the cubic nonlinear Schrodinger equation with a potential. I am interested in the dynamics and manipulation of solutions to this equation (both solitary waves and spatially extended solutions). I have spent a lot of time with potentials that arise from optical lattices and superlattices, so a key theme in this research has entailed competition between nonlinearity and periodicity. I have also studied "collisionally inhomogeneous condensates" (in which the coefficient of the nonlinearity is spatially-dependent) and vortices in two-component condensates.

  • Granular Crystals

  • Wikipedia entry
  • A "phononic crystal" refers to a chain of beads (in one dimension, so this can be called a "granular lattice") and is technically only an appropriate term if one applies some precompression (think of putting beads in a clamp, as always used to happen to Curly of The Three Stooges) so that band gaps open up. By hitting the chain, one excites a nonlinear wave that propagates through the chain. I have studied such waves in both ordered chains (e.g. diatomic chains) and disordered chains, and I am keenly interested in building on the prior research on disordered chains. I have also studied intrinsic localized modes and defect modes in one-dimensional granular crystals. I hope to work on two-dimensional granular crystals and various other projects in this general area. My projects on granular crystals have all been joint with experimentalists.

  • Nonlinear Optics

  • Wikipedia entry
  • Jointly with experimental colleagues, I have studied "nonlinearity management" in the propagation of pulses. In an appropriate regime, these pulses satisfy the cubic nonlinear Schrodinger equation (so I am considering what is known as the Kerr effect), so this work is in a sense coupled to my research on Bose-Einstein condensates. Nonlinearity management, whose analog in BECs is obtained via Feshbach resonance management, can be achieved by propagating the pulse through layered media (say, a multi-layer sandwich of glass and air). This allows pulses to last longer and creates extra modulational instability bands.
  • A good basic introduction to some of my work in this field can be seen in this Physical Review Focus article.

  • Quantum Chaos

  • Wikipedia entry
  • Scholarpedia entry
  • An excellent expository introduction by Martin Gutzwiller, who is one of the subject's pioneers.
  • An excellent expository article by Ze'ev Rudnick that gives a mathematical introduction to quantum chaos.
  • In 2001, I wrote an expository article on quantum chaos that gives a good idea of some of the aspects of this subject.
  • Quantum chaos refers to the study of the quantization of classically chaotic systems, which exhibit fundamentally different behavior than the quantizations of integrable (regular) systems. This can be seen in, for example, their spectral statistics, scarring/antiscarring in their wavefunction amplitudes, etc. Much of the research in quantum chaos is concered with the behavior of quantum chaotic systems in semiclassical regimes in order to consider correspondence with corresponding classical dynamics. For my doctoral thesis, I studied "semiquantum" models of small molecules (via vibrating quantum billiards), in which the slow ("nuclear") degrees-of-freedom (the billiard boundaries) are modeled classically and the fast ("electronic") ones (the confined particle) are modeled quantum-mechanically. More recently, my group studied the quantization of systems with mixed regular and chaotic classical dynamics.

  • Billiard Systems

  • Wikipedia entry
  • Scholarpedia entry
  • In a classical billiard, one has a particle (usually given by a point) confined by a boundary of some shape and colliding perfectly elastically against it. The trajectories describing the particle dynamics are thus given by unions of specular reflection and free (straight-line) motion. In quantum billiards, one studies the Schrodinger equation with homogeneous Dirichlet boundary conditions (i.e., the wavefunction vanishes on the boundary). For classical billiards that behave chaotically (or exhibit mixed regular-chaotic dynamics), the study of their quantizations is very important in the field of quantum chaos. In recent years, I have studied multi-particle billiard systems, and I would like to delve deeper in my investigations of them.

  • Synchronization

  • Wikipedia entry
  • Scholarpedia entry
  • Synchronization in coupled oscillators occurs when they start to move together in some way. This can occur, for example, via phase locking in interacting phase-only oscillators. (See, for example, Wikipedia's discussion of the Kuramoto model.) My group has studied synchronization in cows, which we represent using coupled piecewise-smooth dynamical systems.

  • Mathematical Biology

  • Scholarpedia entry
  • Biology has been called the science of the 21st century. Mathematical biology has become an increasingly big field in recent years (and, to a lesser extent, decades), and I am interested in becoming more involved in it. My group's work on mathematical biology includes a paper on mathematical modelling of bipolar disorder, studies of plankton dynamics (e.g., using piecewise smooth dynamic systems), and investigation of biological networks in areas such as protein biology and neuroscience. I have many current projects in neuroscience, and I plan for that to continue being the case for quite a while. Thus far, my group has looked predominantly at problems related to functional connectivity, but I of course would like to examine other aspects of neuroscience as well.

  • Mathematical Finance

  • Wikipedia entry
  • Large financial systems are ubiquitous, and mathematical tools are crucial to attempting to understand how they work. My group has studied several financial systems using ideas from networks, and I have also used tools from data analytics and random matrix theory. I have also examined limit order books (see this paper for my group's survey article on limit order books) and hope to eventually do that using a network approach.

  • Piecewise-Smooth Dynamical Systems

  • Scholarpedia entry
  • Piecewise-smooth dynamical systems are dynamical systems in which the right-hand-side takes a different form in different situations. For example, when modelling animal locomotion, one can use equations of motion that are like an inverted pendulum for a foot that is off of the ground but like a spring for a foot that is pressing against the ground. My group has used piecewise-smooth systems to study synchronization in cattle, for which one can use a different right-hand-side depending on whether a cow is eating, lying down, or just standing. Additionally, one of my students is using such a framework to study plankton dynamics.