Spring 2019 Published by the Courant Institute of Mathematical Sciences at New York University Courant Newsletter Volume 14, Issue 2 The Physics of Blowing Bubbles See back cover p04 YEVGENIY DODIS ON RANDOMNESS AND CRYPTOGRAPHY p02 p09 MIRANDA HOLMES- IN MEMORIAM: CERFON ON SELF- DAVID CAI ASSEMBLY ALSO IN THIS ISSUE: n COMPUTER SCIENCE ANNOUNCES NEW n AN EMERGENCY MATHEMATICS MASTERS PROGRAM EDUCATION p06 p08 n GSTEM STUDENTS RANK IN TOP 25 n FACULTY HONORS PERCENT IN NATION-WIDE COMPETITION p10 p06 n STUDENT PRIZE AND FELLOWSHIP n SAVE THE DATE: MATHEMATICS IN RECIPIENTS p07 FINANCE CELEBRATES 20 YEARS p11 p06 HENRY MCKEAN RETIRES Geometric foundations of self-assembly with Miranda Holmes-Cerfon FACULTY RESEARCH by April Bacon Those applications include self- In order for the team to calculate the healing, self-replicating, and self-assembling rate at which these systems of interacting synthetics. Miranda is most compelled by self- particles change states, Miranda first needed assembly—i.e. the ability to spontaneously to develop a way to mathematically represent organize into a desired form, a property with their energy landscapes—a plot of the energy game-changing possibilities in medicine of the system—because traditional methods and nanotechnology. A piece of the puzzle is for doing so don’t work effectively for colloids. designing a system architecture that would Whereas the energy landscape of other enable the property to emerge, and so the systems can be graphed as a surface of © NYU Photo Bureau (Slezak) question is in part geometric. gentle hills, for colloids the landscape is a A number of phenomena can be in part flat line with narrow spikes and valleys. This explained by the geometric arrangement means that trying to describe the landscape of particles, including water freezing, gels by certain “critical points” doesn’t provide forming, and glass melting. There are also enough information. “You somehow need to Miranda Holmes-Cerfon has built geometric underpinnings to the ability of incorporate more global information about computational and theoretical tools that certain biological systems (such as proteins the whole geometry (shape, size, etc) of each hold promise for synthetic self-assembly and viruses) to fold, replicate, and assemble flat piece of the landscape,” says Miranda. and broaden our knowledge and means of themselves. The synthetics have to be To get around this, Miranda used investigating complex colloidal systems. designed so that the system starts at a higher the sticky sphere limit, which simplifies energetic state than the desired result, and so the landscape by shrinking the range of Colloids are a mixture of insoluble particles that there aren’t any traps that would make interaction between particles down to exactly in a medium—in milk, butterfat is dispersed the system assemble in a different way while zero. She offers the analogy of two tennis in an aqueous solution of carbohydrates; in en route. balls stuck together with velcro. Unlike a pair blood, cells are suspended in plasma; and in Her first ventures into the area began as of magnets, which will interact over a wider paint, pigment is held within turpentine. For a Postdoc at Harvard in fall 2010. In the labs distance, two tennis balls Velcro-ed together a few decades, theorists and experimentalists of chemical engineer and physicist Vinothan only interact so long as they are exactly stuck have been studying colloids, substances Manoharan and applied mathematician together. with particles small enough to exhibit erratic Michael Brenner, Miranda and collaborators “What happens in the sticky limit is that Brownian motion but large enough to be calculated the rate at which colloid particles the basic shape of the energy landscape is observed by light and escape the realm of transition from one cluster arrangement to determined by the shape of the particles, quantum mechanics. another, as seen modeled with sphere clusters and all the other parameters (interaction Miranda Holmes-Cerfon, an Assistant in Figure A and imaged through a microscope potential, temperature, etc, which are Professor of Mathematics at Courant who in Figure B. usually not known very accurately for also attained her Ph.D. through the Institute’s Colloids exhibit Brownian motion—that colloidal systems) collapse into just one or Center for Atmosphere Ocean Science (2010), is, their particles move around erratically and two parameters. Therefore, you can change was named a 2018 Sloan Fellow for her work incessantly due to constant bombardment interaction potentials but the basic shape on the subject. She explains that colloids by molecules coming from their surrounding of the landscape doesn’t change, so once are about the same scale as the wavelength medium. Much the way shaking a bucket you calculate the complicated geometrical of visible light, making them potentially a of sand will flatten its surface, this constant properties of it, you can use them to unique tool for designing new synthetic jiggling forces the colloids from one geometric understand the system over a wide range materials. configuration to the next. of conditions.” “One of the most popular applications Figure A to cite is an invisibility cloak. Obviously we haven’t gotten there yet,” she says, gesturing to her own visibility—“If I had an invisibility cloak I would be wearing it! Right now, a lot of the interest is in just understanding general Figure B principles of physics from colloids that could be used in material science, but there is a lot of potential for compelling applications.” 0 2 4 (sec) 2µm 2 — Courant Newsletter, Spring 2019 “I’m hoping that looking at the energy through the whole space, but on a computer says Miranda. “They can make a very landscape in this different way might lead we can guide the search.” complicated interaction matrix with different to more efficient ways to solve certain kinds “One thing that is nice about this kinds of interaction for different kinds of of problems. There are highly developed question [of enumerating rigid sphere particles. And they can do this purely by optimization algorithms that I could throw at packings] is that it’s such a simple question coating particles with strands of DNA.” the problem, but I’m also hoping to discover to formulate,” she says. She has done With Postdoc Emilio Zappa, Miranda is some kind of fundamental structure, a demonstrations with both high school looking at the information-theoretical part more general property [within statistical teachers and students, including at cSplash, of DNA-mediated systems—i.e. questions mechanics].” Courant’s day-long program for high such as how DNA can be used to program With this tricky variable pinned schoolers that Miranda co-founded in 2006 the particles to assemble in different ways, down, Miranda and the team at Harvard as a doctoral student. “I give [participants] what structures can be formed by mixing developed a model that determines the these toys and ask them to come up with an particles of different types, and if a system’s rate at which a system will transition from algorithm, and they usually do. They usually energy landscape places constraints on one state to another, the different paths a figure out what some of the rigid clusters what structures it can form. She is also system may take, and the likelihood of any are. It’s a great question for teaching people building computational models with several given arrangement appearing. While former about mathematics research and getting collaborators. For example, With Jonathan methods approaching this latter problem them engaged in solving problems.” Goodman and Nawaf Bou-Rabee, “I’m produced results that couldn’t resolve the fine In new work, Miranda has been looking developing algorithms to efficiently sample differences between states, Miranda’s results at problems related to physical properties and simulate dynamics on sticky energy are far more precise and, further, have been of colloidal systems, such as the effects of landscapes. With Steven Gortler [Harvard] experimentally verified within acceptable friction and the effects of coating colloidal and Louis Theran [St. Andrews], I’m limits of error. particles with DNA strands. developing algorithms to better understand Miranda realized her model offered Adding friction is challenging because the geometrical properties of the landscapes.” solutions for another problem as well: friction itself is not well understood. “My hope is that all these tools will one enumerating rigid sphere packings. Some Miranda gives an example of its complexity day be useful for other kinds of systems of the theoretical work surrounding colloids introduced to her by Tadashi Tokieda as well,” she says. The ideas she has been requires knowing a total set of the way the (Stanford). Place a few small balls in a using to study colloids have connections to spheres can be arranged, or packed. Miranda circular container and move the container other systems that are wonderfully broad— took an algorithmic approach to this question: around in a circle. At first, the balls move in from protein folding and virus assembly to What are all the ways to form a rigid cluster? the same direction as the container. But as robotics and origami, each with the essential Starting with a single cluster, the algorithm more balls are added, at some critical point feature of being a system with objects that breaks one point of contact (i.e. one side they begin to go in the opposite direction. move stochastically, subject to certain of a bar in Figure A), making the cluster “The only way this can happen is constraints. “floppy” instead of “rigid.” A rigid cluster can’t because of the friction between the balls Much of the work is also linked by rearrange itself—it can only jiggle its bonds and some collective property of how they’re randomness and geometry.