Mathematics and Materials
The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.
The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.
The program in 2014 will bring together a diverse group of mathematicians and scientists with interests in fundamental questions in mathematics and the behavior of materials. The meeting addresses several themes including computational investigations of material properties, the emergence of long- range order in materials and self-assembly, the geometry of soft condensed matter and the calculus of variations, phase transitions and statistical mechanics. The program will cover several topics in discrete and differential geometry that are motivated by questions in materials science. Many central topics, such as the geometry of packings, problems in the calculus of variations and phase transitions, will be discussed from the complementary points of view of mathematicians and physicists.
Student preparation: We seek motivated students in the mathematical sciences with an interest in interdisciplinary work, and mathematically inclined chemists, physicists, and engineers. The lectures cover many distinct topics and formal preparation in all these areas is not expected. The dominant themes in the lectures are the geometry of packing and its effect on material properties (Brenner, Cohn, Frankel), statistical mechanics (Cohn, Elser, Frankel, Kotecky), soft condensed matter and geometry (Kohn, Palffy-Muhoray), and phase transitions and geometry (James, Kotecky, Palffy-Muhoray). While prior exposure to these topics will help, we expect that most of the material will be developed from scratch. The lectures will be complemented by tutorial sessions run by senior graduate students.
The 24th Annual PCMI Summer Session will be held June 29 – July 19, 2014.
Click HERE to apply to the Graduate Summer School program.
2014 Graduate Summer School Lecturers
Michael Brenner, Harvard University, Self-assembly of sphere packings
Henry Cohn, Microsoft, Packing, coding, and ground states: from information theory to physics
In these lectures, we'll study simple models of materials from three different perspectives: geometry (packing problems), information theory (error-correcting codes), and physics (ground states of interacting particle systems). These perspectives each shed light on some of the same problems and phenomena, while bringing different techniques to bear.
One noteworthy phenomenon is the exceptional symmetry that is found in certain special cases, and we'll examine when and why it occurs.
Veit Elser, Cornell University, Model building in statistical mechanics
According to Einstein, "Everything should be made as simple as possible, but not simpler." A similar, though not exactly equivalent principle applies to the study of statistical mechanics: to be useful, a model of many-particle behavior should be no more complex than it needs to be. In statistical mechanics the object is not so much to find ever more accurate (and at the same time simple) laws, but to distill simplified models that capture the behavior of interest in minimalist terms. Mathematicians tend to equate statistical mechanics models (and their solution) with the subject itself, and miss out on the equally important process of constructing models. These lectures address this second point, with examples drawn from models well known to mathematicians: tilings, sphere packings, and percolation.
Daan Frenkel, University of Cambridge, Entropy, probability and packing
The Second Law of Thermodynamics is exceptional because it distinguishes between the past and the future. It also allows us to define a quantity called Entropy that is maximal for a closed system in equilibrium. However, the Second Law does not give a physical interpretation of entropy. Over the past decades our understanding of entropy has increased substantially - partly due to our ability to perform numerical simulations. In my lectures I will discuss different aspects of entropy in Soft Matter and Granular Media. My conclusion is that Gibbs is always right.
Richard D. James, University of Minnesota, Phase transformations, hysteresis and energy conversion: the role of geometry in the discovery of materials
We identify a particular problem of geometry whose solution profoundly affects the reversibility of phase transformations. This problem has deep links to the study of the calculus of variations and partial differential equations. Solutions of this problem in special cases lend themselves to the discovery of new materials, by systematically changing the composition of known materials to satisfy certain nongeneric restrictions on the lattice parameters of the two phases. This procedure is being put into practice widely. Some materials found by this procedure can be used in interesting ways to convert heat to electricity (without the need of a separate electrical generator), and provide possible ways to recover the vast amounts of energy stored on earth at small temperature difference. For further background, see the video and references at www.aem.umn.edu/~james/research/home.html
Robert V. Kohn, Courant Institute, Wrinkling of thin elastic sheets
Why does the Mobius strip have its familiar shape? Why does a crumpled sheet of paper have sharp folds meeting at points? How is wrinkling different from folding? What determines the shape of a hanging drape? I'll address these and other questions, while explaining why the mechanics of thin sheets has become a research frontier in the Calculus of Variations.
The mathematical heart of the matter is the elastic energy of a thin sheet, which is very nonconvex. It is often fruitful to ask how the minimum energy scales with the film thickness and other physical parameters. Finding the answer requires proving upper bounds and lower bounds that scale the same way. The upper bounds are often easier, since nature gives us a hint. The lower bounds tend to be subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns.
Roman Kotecký, University of Warwick, Statistical mechanics of nonlinear elasticity
The aim of these lectures is to explore statistical physics foundations of nonlinear elasticity. The relevant microscopic models are random gradient vector fields with Gibbs probability distribution. Eventually, the variational characterization of nonlinear elasticity is obtained as a scaling limit in terms of large deviations of random gradient fields. This can be actually viewed as a microscopic justification of the Cauchy-Born rule. We will begin by studying statistical properties of gradient fields in a better understood scalar case. In particular, we will discuss various phase transitions for random interfaces modeled by random gradient scalar fields and related models.
Technical difficulties involved in studying random gradient fields stem from a slow decay of their correlations. This prevents us to treat perturbations by a straightforward cluster expansion and leads to a necessity of a multi-scale handling with a help of a renormalization group approach. All these technical tools will be carefully explained and implemented.
Peter Palffy-Muhoray, Kent State University, The effects of particle shape on soft condensed matter systems
Soft matter systems can show remarkably complex responses to excitations. Much of this rich diversity originates in interparticle interactions, which can be partitioned into long range attractive and short range repulsive contributions. Particle shape plays an important role in both of these. I will examine these contributions using density functional theory, and discuss relevant results from convex geometry. I will present some new results, and consider their implications on the behavior of soft condensed matter systems.