© Slezak: Courtesy of NYU Photo Bureau


The scientific applications of my research concentrate broadly in two areas:

MATERIALS SCIENCE studies how a great many small building blocks (like atoms, colloids, sand grains, blood cells, etc) come together to form something that we perceive on the macroscale as a continuous material. Understanding how the microscopic interactions between the building blocks give rise to the macroscopic properties of a material, or how a material evolves and responds to stress once formed, or how to design microscopic interactions to achieve a desired structure, are all areas where applied mathematics can play an important role. Mathematics can help to construct theories that simplify and make sense of patterns in complex systems, as well as to build computational tools to simulate systems that can't be fully understood with pencil and paper.

OCEANOGRAPHY studies patterns of fluid flow in the ocean basins, partly to understand our current climate and predict how it might respond to changes in forcing (like a warmer sea surface temperature.) But flow in the ocean is complicated, with dynamics at a multitude of scales that are intertwined with no sharp separations, so understanding the dynamics at one scale requires understanding it at many. It is difficult and expensive to measure what is happening in the ocean, and it is not possible to model nor to numerically simulate all of the scales simultaneously, so theoretical models of the processes acting at each scale are critical for interpreting measurements and parameterizing the processes in at other scales. One of my interests is in small-scale processes, like internal waves and small-scale bottom topography, which, because they are so complex and ubiquitous, are naturally modelled stochastically.

Below are some descriptions of specific projects in these areas. As I am interested in almost any problem where mathematics can help to elucidate a physical, biological, or social phenomenon there are also other projects that couldn't quite fit in these boxes.

Colloid self-assembly

Colloids are particles with diameters of nano- to micro-metres that typically live in a solution, where they move around at random like a Brownian motion, and interact with each other in various ways. Colloids are studied experimentally for a lot of reasons, but one reason is their potential to form materials with novel properties, like new kinds of crystals, containers for drug delivery, nanoscale robots, materials that heal or replicate themselves, etc. Experimentalists have gained a wealth of control over features such as shape, size, charge, specificity, and valency of the particles, so an enormous variety of structures could potentially be assembled; theories and computations are required to explore the vast parameter space of possibilities and make sense of these complex systems.

One major feature of many colloids (and other mesoscale particles) is that their interactions are very short-ranged compared to their diameters. This means that traditional models of energy landscapes, which characterize a high-dimensional energy surface by its local minima and transition states, no longer adequately describe the landscape nor the dynamics on top of it — a colloidal energy landscape has too many "flat" parts whose the geometry and topology must be taken into consideration.

My group is developing a new framework for analyzing colloidal energy landscapes and dynamics, which allows one to consider a variety of particle interaction structures, geometries, and other experimental constraints. This review gives an overview of the issues and framework:

The framework is based on considering the sticky limit of short-ranged, strong interactions. In this limit interactions become constraints, such as distance constraints between particles, and the energy landscape looks schematically like a high-dimensional polyhedron, with a high-dimensional interior, lower-dimensional faces, even lower-dimensional edges, etc. Mathematically it forms a stratification: a collection of manifolds of different dimensions, glued together at their boundaries. We are developing the theoretical tools to describe the energies and dynamics on this landscape (e.g. what are the right equations to write down?), as well as the computational tools to implement it (e.g. how do we actually compute the quantities we need to make predictions for a specific system?). We are also investigating some physical models of colloids along the way. Here is a selection of topics we are interested in:
  • How should colloids be programmed so a desired structure, e.g. a cluster of a crystal, is the thermodynamic ground state? How can we design colloids so this stucture is kinetically accessible? And how can all of this be framed as an inverse problem so it can be done automatically?
  • What happens when constraints become linearly dependent, so there are infinitesimal motions which may or may not extend to finite motions — what are the thermodynamics of such states, and what do they tell us about how the energy landscape changes as we vary parameters?
  • How can we sample points and calculate integrals over manifolds?
  • When particles are connected by sticky DNA strands, how can we describe their kinetics — do they move relative to each other by rolling or by sliding? How does this preference affect kinetic quantities, like transition paths between crystals? How can we account for particle deformability? etc.
These tools and questions apply to other systems that evolve stochastically subject to constraints, as could be the case for origami, flexible polyhedra, nanoscale robotics, viruses, etc. I am always interested to hear about new examples.

Monte Carlo on manifolds

Relatedly, my group and collaborators are developing Monte Carlo methods to sample points and integrate functions on manifolds. The manifolds are defined by sets of constraints (levels sets of functions), so they are submanifolds of a higher-dimensional Euclidean space. Constraints arise when modelling the configuration spaces of a variety of physical systems, like colloids and other systems described above, so we are developing the methods with these systems in mind. However, the question of how to sample points in a high-dimensional, nonconvex, curved geometric object is of mathematical and numerical interest in itself.

Currently we have methods to sample and integrate over a single manifold, provided it is geometrically nice enough. We are developing methods to sample a stratification: several manifolds of different dimensions, sampled all at once. Eventually we hope to sample kinetic, path-dependent quantities, using Metropolis-style algorithms. Of course, there is much work still to be done in making the methods efficient for geometrically complex configuration spaces, and in adapting them to different situations.

Friction and noise

Friction (the kind you feel when you push a box across a floor) is at the heart of much of our lives: it is the reason why we can walk, drive, skate, build sand castles, etc. So is noise: random jiggling of particles at small scales provides the energy needed to overcome energy barriers, so our proteins can fold, our blood can mix, our lungs can expel waste, our salt can form nice crystals, etc. Both friction and noise have been studied for centuries separately, but what happens when a system (say a collection of particles) is both frictional, and noisy? Such systems are rarely been examined, though they are increasingly important, especially in materials science.

My group is interested in the mathematics of systems that are both frictional and noisy. What are the right equations to write down for a collection of particles that interact with friction? And how do you analyze them, and come up with specific solutions in specific settings, or understand the solutions' statistical properties? These turn out to be challenging, mathematically subtle questions that bring together some disparate ideas in stochastic analysis, differential geometry, and PDE theory, as well as challenge our physical understandings of statistical mechanics and the origin of friction; there is much work still to do.

© Slezak: Courtesy of NYU Photo Bureau

We are also interested in physical questions involving the collective effects of noisy frictional particles. For example, try this experiment: take a container (like a wide teacup, or a flat bowl), put in a few balls (like marbles or cedar moth balls), and swirl the container (translate it around in a circle.) The balls will move around the container in the same direction as your swirling. Now add more balls, so the density is quite high, and repeat. The balls will now move in the opposite direction as the swirling. Why?

The answer turns out to be richer than one might think, with connections to the wide field of active matter, and shows how you can begin to understand nonequilibrium systems on your own countertop.

Sphere packing

How many ways are there to arrange N hard spheres in space, so they form a rigid cluster? This is a mathematical question, but the solution could bring insight to a range of problems, including what structures colloids can assemble into and how they might do it. It is also a challenging computational problem that brings up interesting issues in rigidity theory and numerical algebraic geometry. My algorithm for generating data for this problem was described here:

The data itself can be found on the sphere packing data page. Currently we are extending the algorithm to handle spheres of different sizes and in different dimensions. We hope to use the data both to ask how energy landscapes vary under changes in parameters, as well as to give intuition into conjectures in geometry.

Applications of rigidity theory

I am interested in the mathematical theory of rigidity, and how this can be applied to physical systems like clusters of sticky spheres, origami, and jammed or glassy systems. Roughly, rigidity theory studies graphs made of points and edges, and determines for a given set of edge lengths whether the corresponding graph would be rigid or flexible. I am particularly interested in non-generic rigidity, and am developing a numerical tool to determine the flexibility properties of a non-generic framework. The tool will be based on convex programming and will account for numerical imprecision. Other topics are of interest too and I am glad to discuss them.

Random parking

What happens if you throw objects one-by-one at a surface and they stick to it irreversibly? This problem is called random sequential adsorption, or just random parking (this is different from random packing!) We looked at whether random parking can be used to build a given kind of colloidal cluster with high yield. Although the process is random, so you would expect to get a distribution of kinds of clusters, we showed that in a certain regime in parameter space it is theoretically possible to create clusters of 4 particles with 100% yield! We then demonstrated this high yield experimentally. Recently, Beth Chen and I considered random parking on other curved spaces: the sphere, plane, hyperbolic plane, and projective plane, and shown that random parkings can have some counterintuitive properties; for example, the average density in a compact space does not always increase monotonically as the discs shrink.

Ion-bombarded surfaces

One promising method of creating structures on surfaces is to bombard them with ions. The ions transmit energy to the surface, causing the atoms to locally redistribute, or if they have enough energy, to sputter away, creating a small crater near the impact location of each ion. The average effect over many ion impacts is to cause the surface height to erode. Because it erodes with different rates depending on the slope of the surface, it can develop intricate patterns depending on the initial shape of the surface and the parameters involved in ion bombardment. These patterns can have features at a very small scale, so if we understand how they form we can learn how to exploit them to create desired target patterns. Uniform ion bombardment is cheap compared to other methods of nanofabrication, so this is a promising avenue to pursue.

My aim is to further a theoretical understanding of the structures that can be formed, in particular by the nonlinear dynamics. Along with experimental and numerical collaborators, we have shown have the striking property that steep, sharp features can arise spontaneously on a surface provided it is initially gently patterned on the macroscale. These features have length scales on the order of nanometers, much smaller than anything realizable by linear instabilities. They can be fully understood by a discrete set of traveling-wave solutions to the PDE governing the macroscale dynamics, that are special in that they are undercompressive (they violate the Lax entropy condition). Because of this they act as attractors for the dynamics, and allow for low-dimensional parameterizations of the nonlinear dynamics. This parameterization allows one to characterize the nonlinear dynamics using only 3 material-dependent parameters, that are much easier to measure experimentally than the full erosion function and smoothing physics. Additionally, we have used it to develop a computationally efficient method for solving the inverse problem of finding the initial condition whose evolution leads to a desired target pattern.

Tidal scattering

As the tides slosh back and forth in the ocean's irregular basins, they generate energetic internal motions that cascade to smaller scales and eventually dissipate. The mixing that results is critical to ocean circulation patterns, providing roughly half of the amount required to maintain the global abyssal density distribution, so understanding the mechanisms and geography of the different steps in the tidal energy cascade is fundamental to our overall understanding of ocean dynamics.

One question is in the role that topography can play in transferring energy to small scales, and in particular in the small-scale topography the baroclinic waves encounter away from the large features where they were generated. We have looked at tidal scattering by small-scale random topography, and studied energy dissipation over subcritical (low-slope) topography, as well as the presence of wave "attractors" over supercritical (high-slope) topography. Our studies however were limited to one-dimensional subcritical topography, however, and it remains an interesting question what energy dissipation looks like for the (clearly more realistic) case of two-dimensional topography.

Mixing by internal waves

The ocean is filled with an energetic background of small-scale motions called internal waves that are too small to be resolved in basin-scale ocean models, so their effects on large-scale circulation patterns must be parameterized. It is natural to take a stochastic approach, by modelling the internal waves as a random wave field, and to consider how nonlinear wave-wave or wave-vortical interactions can contribute to mean dynamics at larger scales. One possibility is in mixing at horizontal scales of 1--10km, over time scales of days to weeks — measurements show that the mixing rates are much higher than any explainable by known mechanisms.

We considered the role that internal waves might play in mixing at these scales, by calculating the contribution from the nonlinear wave-wave interactions in both the shallow-water and Boussinesq equations. This yielded an expression that we can apply to any Gaussian random wave field in these systems. However, when we applied this formula to oceanic conditions described by a typical model wave spectrum, it gave values well below those observed in measurements. Subsequent numerical experiments and theoretical analyses suggested that the nonlinear effects are actually negligible as soon as a small amount of viscosity is added to the system; this singular perturbation causes the linear wave field to do most of the mixing. More work is needed to determine the appropriate dynamical and statistical model to properly evaluate the mixing due to internal waves in the ocean.

Length of a lava tube

Lava tubes are tunnels that form during a volcanic eruption to carry hot liquid lava away from the source. They can be extremely long -- the Mauna Loa flow tube in Hawaii is over 50 km long, while the Toomba and Undara flows in Queensland, Australia are 123km and 160km long respectively. The longest known tubes are found on Mars and are over 200km. What sets the length of a lava tube? Could they be arbitrarily long, if there were no oceans or other geometrical obstructions? Or are there physical constraints that govern their lengths? We analyzed an idealized model of a lava tube, and found that the length is very sensitive to conditions at the entrance: if the source has constant flux, the tube can be arbitrarily long, but if the source is at constant pressure, then there is a maximum length that we can calculate exactly. Experiments with wax have verified our model.