My work falls in three broad classes of mathematical approaches:

© Slezak: Courtesy of NYU Photo Bureau

Scientifically, much of my work is motivated by problems in materials science and soft-matter physics. These disciplines study 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.

One system of particular interest in my group are colloids. 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.

Colloids are the same scale as many processes in biology, a connection my group is beginning to explore.

This page is not an exhaustive description of my interests— I like learning new things, so if you have an interesting problem, please contact me about it!

Physical Modelling

© Slezak: Courtesy of NYU Photo Bureau

Many physical systems are incredibly complicated; if we wrote down equations describing all of the physics involved, we would never make sense of them, either computationally or analytically. Physical modeling involves figuring out what is the best set of equations that simply but accurately describes a particular system, and then, making predictions about physical phenomena from these equations. Often this involves writing down complicated equations, and then identifying small or large parameters so the equations can be simplified, for example using perturbation theory or homogenization techniques. Some questions that my group is interested in include:

  • How can we model the free energy landscape and dynamics of particles with very short-ranged attractive interactions? Such is the case for colloids, and traditional theories to address such questions don't work as effectively for such interactions.
  • How can we describe the dynamics of colloids coated with sticky DNA strands (a common technique to "design" interactions between colloids), without tracking the dynamics of each individual DNA strand?
  • How does hydrodynamics (interactions between colloids caused by the ambient fluid) affect the dynamics of colloids?
  • How can we incorporate friction between colloids, into a description of their stochastic dynamics?
  • Why does a small number of marbles swirled in a teacup, rotate in the same direction as the swirling, but a large number of marbles rotates in the opposite direction? Amazingly, this question also has something to do with why and how a hoola-hoop levitates.

Computational Statistical Mechanics

My group builds algorithms to compute the statistical mechanical properties of physical systems subject to constraints. One common example of a constraint is a distance constraint between particles. Mathematically, this means we build algorithms to perform calculations on manifolds, or their generalization, stratifications, which are collections of manifolds of different dimensions that are glued together at their boundaries. We develop algorithms to sample a probability distribution on a manifold or a stratification, and to simulate a stochastic differential equation on a manifold or a stratitication. We are also interested in solving inverse problems in statistical mechanics, for example to ask how can one design particles so a particular target state is low free energy, kinetically accessible, or both?

Computational Geometry and Rigidity Theory

Much of my group's work is informed by geometry, and particularly rigidity theory. We study the configuration spaces of frameworks, graphs embedded in a particular space with distance constraints on the edges, and their generalization tensegrities, frameworks where some edges can stretch and some can compress. We study how ideas developed to understand frameworks and tensegrities can bring new insight into the statistical mechanics of colloids, how they can lead to better simulation strategies for systems with constraints, and, we are interested in designing frameworks and tensegrities to perform various functions, or that can "assemble themselves."

One simple question where geometry gives insight into materials is: 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. The data generated by my algorithm to attack this problem is on the sphere packing data page.