Articles from the Institute Letter

Additional articles from new and past issues of the Institute Letter will continue to be posted over time and as they become available.

by Shiraz Minwalla

Shiraz Minwalla has uncovered an unexpected connection between the equations of fluid and superfluid dynamics and  Einstein’s equations of general relativity.
Shiraz Minwalla has uncovered an unexpected connection between the equations of fluid and superfluid dynamics and Einstein’s equations of general relativity. (Photo: McKay Savage)

How the Movement of Water Molecules Corresponds to Ripples in Spacetime

There is an interesting connection between two of the best-studied nonlinear partial differential equations in physics: the equations of hydrodynamics and the field equations of gravity.

Let’s start with a brief review of hydrodynamics. At the microscopic level a tank of water is a collection of, say, 1025 molecules that constantly collide with one another. The methods of physics may be used to model this collection of water molecules as follows: we set up equations that track the position and momentum of each of the water molecules and predict their time evolution. These conceptually complete equations have of order 1025 variables and so are clearly too difficult to handle in practice.

Does it then follow that tanks of water cannot be usefully studied using the methods of physics? As every plumber knows, this conclusion is false: a useful description of water is obtained by keeping track of average properties of water molecules, rather than each individual molecule.

Think of a tank of water as a union of non-overlapping lumps of water. Each lump is big enough to contain a large number of molecules but small enough so that gross macroscopic properties of the water (energy density, number density, momentum density) are approximately uniform. The fundamental assumption of hydrodynamics is that under appropriate conditions, all the “average” properties of any lump are completely determined by its conserved charge densities (in the case of water, molecule number density, energy density, and momentum density). In particular, the conserved current for molecule number jµ and the conserved current for energy and momentum Tµν are themselves dynamically determined functionals of local thermodynamical densities in a locally equilibrated system (fluctuations away from these dynamically determined values are suppressed by a factor proportional to the square root of the number of molecules in each lump). The equations that express conserved currents as functionals of conserved densities are difficult to compute theoretically but are easily measured experimentally and are known as constitutive relations.

When supplemented with constitutive relations, the conservation equations ∂µ jµ =0, and ∂µ Tµν=0(2) turn into a well-posed initial value problem for the dynamic of conserved densities. They are the equations of hydrodynamics. Let me reemphasize that the effect of the ignored degrees on the evolution of conserved densities is inversely proportional to the square root of the number of molecules in a lump, and so is negligible in an appropriate thermodynamic limit, allowing the formulation of a closed dynamical system for conserved densities.

My research concerns how the equations of hydrodynamics pop up in an apparently completely unrelated setting: in the study of the long wavelength dynamics of black holes governed by Einstein’s equations with a negative cosmological constant.

Einstein’s gravitational equations describe the dynamics of the geometry of spacetime. The ripples of spacetime (gravitational waves) have interesting dynamics even in the absence of any matter. For most of this article, I will be referring to Einstein’s equations in the absence of matter.


By Olga Holtz

Olga Holtz lecturing at the Women and Mathematics Program at the Institute for Advanced Study (2014)
Olga Holtz (Photo: Dan Komoda)

The Making of a Mathematician

My love affair with George Pólya began when I was seventeen. It was in Chelyabinsk, Russia, and my first year at the university was coming to an end. I had come across a tiny local library with an even tinier math section, which nobody ever seemed to visit, and had taken out most of those math books one by one before I came across The Book. It was George Pólya’s Mathematics and Plausible Reasoning.

By that time I was a total bookworm, having devoured almost a thousand volumes of my parents’ home library, mostly fiction. My familiarity with math books was much poorer although, growing up, I had enjoyed Yakov Perelman’s popular books for children on math and physics. I was a proud graduate of a specialized math and physics school, the only one in town, and had had a few wins at local olympiads in math and science. A top kid in class as far back as I could remember, I was arrogant as hell.

I read the introduction to Mathematics and Plausible Reasoning and its Chapter I standing up next to the bookshelf. It read like a novel. A cerebral one alright, which made you pay quick attention. Chapter I started out in the least orthodox way, comparing mathematical induction to a domino chain. The book endeavoured to explain not only what was mathematically true but how and why. I was hooked. Chapter I ended with a list of problems. I solved a couple of them still standing up but quickly came to a halt on Problem 3.

The arrogance kicked in––I had to solve those problems. I still remember carrying that book home after I checked it out. It was late spring, gorgeous weather, bird songs in the air, romantic couples––you get the picture. I was besotted with The Book.


By Lucy Colwell

How do proteins self-assemble into functional molecules?

Proteins are typically cited as the molecules that enable life; the word protein stems from the Greek proteois meaning “primary,” “in the lead,” or “standing in front.” Living systems are made up of a vast array of different proteins. There are around 50,000 different proteins encoded in the human genome, and in a single cell there may be as many as 20,000,000 copies of a single protein.1

Each protein provides a fas­cinating example of a self-organ­izing system. The molecule is assembled as a chain of amino acid building blocks, which are bonded together by peptide bonds to form a linear polymer. Once synthesized, this polymer spontaneously self-assembles into the correct and highly ordered three-dimensional structure required for function. This ability to self-assemble is remarkable—each linear polypeptide chain is highly disorganized, and has the potential to adopt an array of conformations so vast that we cannot enumerate them, yet within less than a second a typical protein spontaneously assumes the correct, highly ordered three-dimensional structure required for function. The identity and order of the amino acids that make up this polypeptide, that is the protein sequence, typically contain all the information necessary to specify the folded functional molecule.2


By Nicola Di Cosmo

Did an unusually favorable climate create conditions for a new political order under Chinggis Khan?

In his recent book Global Crisis: War, Climate Change & Catas­trophe in the Seventeenth Century, Geoffrey Parker states: “although climate change can and does produce human catastrophe, few historians include the weather in their analyses.” This is generally true, and the distance between historians and the weather may not have improved (indeed, may have been underscored) by the evolution of environmental history as a separate branch of historical research. Moreover, while the collection of historical climate data has never been more robust, instances of collaboration between scientists and historians are still very few and far between. In 2006, the National Science Foundation launched a program for research on Coupled Natural and Human Systems, capturing the need to model the interaction between societies and environments. Few of the projects funded so far, however, involve a long-term historical perspective or engage actual historical questions. One of these, funded last year, is titled “Pluvials, Droughts, Energetics, and the Mongol Empire” and is led by Neil Pederson, Amy Hessl, Nachin Baatarbileg, Kevin Anchukaitis, and myself.


By John Padgett

Do actors make relations or do relations make actors?

The encounter of historical and evolutionary perspectives within the intermediate trading zone of social science often has been unsatisfactory. Biological metaphors of social evolution were common among the original founders of the social sciences—in sociology and anthropology especially—but collectivist functionalism1 now is thoroughly discredited. Horrific misuses of biological and evolutionary “scientific theories” by nineteenth- and twentieth-century racist social movements need no recounting. More recently, sociobiology—the analysis of discrete social behaviors and cultural “memes” as if these were genes in evolutionary competition—has gained an enthusiastic following as a sect, but sociobiology is viewed as simplistic and naive by most contemporary social scientists. 

Less well known among social scientists, the reverse reception of historicist arguments in evolutionary biology also has been rocky. Stephen Jay Gould is widely known and praised outside of his own subfield, but his arguments are held at arm’s length if not in disdain by his evolutionist peers. Celebrating “historical contingencies” to them seems tantamount to giving up on scientific explanation altogether. Postmodernists in the social sciences and the humanities are willing to take that step, but contemporary evolutionary biologists (including the late Gould himself) have nightmares of creationists and intelligent designers exploiting indeterminacy in evolutionary theory for their own purposes.