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.

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

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 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.

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.

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 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.

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 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.

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.

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 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.