96 DS11 Abstracts

96 DS11 Abstracts IP0 IP3 Juergen Moser Lecture: The Many Facets of Chaos How Can We Model the Regulation of Stress Hor- mones? Chaos reveals itself differently in different situations. Un- derstanding its many aspects or facets will help in creating Daily and monthly rhythms of hormones are well recog- innovative models. My talk will illustrate how different nised. Less well known are the more rapid ultradian facets of chaos lead us in different directions in my recent changes which are a characteristic of most biologically ac- works on: HIV population dynamics; determining the tive hormone systems. We have looked at the regula- current state of the atmosphere (for weather prediction); tion of the stress hormones glucocorticoids secreted by genome assembly (determining the sequence of ACGT’s for the adrenal glands. It has always been assumed that the a species); partial control of chaos . episodic release of these hormones was a result of some form of pulse generator in the brain. A dispassionate look James A. Yorke at this system however, revealed that there was a feed- University of Maryland forward:feedback relationship between the pituitary gland Departments of Math and Physics and IPST and the adrenal gland providing scope for a peripheral os- [email protected] cillating hormonal system. The background to this system and the biological testing of our mathematical predictions will be described. IP1 Will the Climate Change Mathematics? Stafford Lightman University of Bristol, United Kingdom Computational models of the Earth system lie at the heart Staff[email protected] of modern climate science. Concerns about their predictions have been illegitimately used to undercut the case that the climate is changing and this has put dynamical IP4 systems in an awkward position. It is important that we Climate Sensitivity, Feedback and Bifurcation: extricate ourselves from this situation as climate science, From Snowball Earths to the Runaway Greenhouse whose true objective is to build an understanding of how the climate works, badly needs our expertise. I will discuss The concept of climate sensitivity lays at the heart of as- ways that we, as a community, can contribute by highlight- sessment of the magnitude of the imprint of human activi- ing some of the major outstanding questions that drive ties on the Earth’s climate. Most commonly, the ”climate” climate science, and I will outline their mathematical di- is represented by a simple projection such as a global mean mensions. I will put a particular focus on the issue of si- temperature, and we wish to know how this changes in re- multaneously handling the information coming from data sponse to changes in a single control parameter – usually and models. I will argue that this balancing act will im- atmospheric CO2 concentration. This problem is an in- pact the way in whch we formulate problems in dynamical stance of a broad class of related problems in parameter systems. dependence of dynamical systems. I will discuss the short- comings of the traditional linear approach to this problem, Christopher Jones particularly in light of the spurious ”runaway” states pro- University of North Carolina at Chapel Hill, USA & duced when feedback becomes large. The extension to in- University of Warwick, United Kingdom clude nonlinear effects relates in a straightforward way to [email protected] bifurcation theory. I will discuss explicit examples arising from ice-albedo, water vapor, and cloud feedbacks. Finally, drawing on the logistic map as an example, I will discuss IP2 the problem of defining climate sensitivity for problems ex- From Newton’s Cradle to New Materials hibiting structural instability. The bouncing beads of Newton’s cradle fascinate children Raymond T. Pierrehumbert and executives alike, but their symmetric dance hides com- The University of Chicago plex nonlinear dynamic behavior. Lift a bead on one side Dept. of the Geophysical Sciences off a chain of a few suspended beads, let it swing back: one [email protected] bead bounces off on the other side. Do the same with a long chain of beads: several beads bounce off on the other side. This represents an example of nonlinear wave dy- IP5 namics, which can be exploited for a variety of engineering Robust and Generic Dynamics: A applications. By assembling grains in crystals or layers in Phenomenon/mechanism Correspondence composites such that they support nonlinear waves, we are developing new materials and devices with unique proper- If we consider that the mathematical formulation of natural ties. We have constructed acoustic lenses that allow sound phenomena always involves simplifications of the physical to travel as compact bullets that can be used in medical laws, real significance of a model may be accorded only to applications, have developed new materials for absorbing those properties that are robust under perturbations. In explosive blasts, and are exploring new ways to test air- loose terms, robustness means that some main features of a craft wings and bone implants nondestructively with the dynamical system are shared by all nearby systems. In the help of nonlinear waves. talk, we will explain the structures related to the presence of robust phenomena and the universal mechanisms that Chiara Daraio lead to lack of robustness. Providing a conceptual frame- Aeronautics and Applied Physics work, the goal is also to show how to provide a generic California Institute of Technology correspondence phenomenon/mechanism for all dynamical [email protected] systems. Enrique Pujals DS11 Abstracts 97 Instituto Nacional de Matemática Pura e Aplicada, Brazil viability and suitability of the resulting tissue constructs. [email protected] In this talk, I highlight some of our recent mathematical modelling work that aims to provide insights into tissue engineering applications. IP6 Models and Control of Collective Spatio-Temporal Sarah Waters Phenomena in Power Grids University of Oxford [email protected] We are asking modern power grids to serve under condi- tions it was not originally designed for. We also expect the grids to be smart, in how they function, how they with- IP9 stand contingencies, respond to fluctuations in generation Moving Pattern Formation from the Real World to and load, and how the grids are controlled. To meet these the Lab, and the Reverse ever increasing expectations requires extending power grid models beyond the scope of traditional power engineering. This talk will describe three pattern formation experiments In this talk aimed at applied mathematicians and physi- where natural systems were imported directly into the lab- cists I first review basics of power flows, and then outline oratory. The overall shape and subsequent rippling insta- a number of new problems in modeling, optimization and bility of icicles is a complex free-boundary growth prob- control theory for smart grids. In particular, I describe new lem. It has been linked theoretically to similar phenomena approaches to control of voltage and reactive flow in dis- in stalactites. We grew laboratory icicles determined the tribution networks, algorithms to study distance to failure, motion of their ripples. Washboard road is the result of and statistical analysis of cascading blackouts in transmis- the instability of a flat granular surface under the action sion networks. of rolling wheels. The rippling of the road sets in above a threshold speed and leads to waves which travel down the Michael Chertkov road. We studied these waves both in the laboratory and Los Alamos National Laboratory using 2D molecular dynamics simulation. Columnar joints [email protected] are uncanny formations of ordered cracks in certain lava flows. We studied these both in a lab analog system and in the field. Each of these three cases nicely illustrates the IP7 pleasures and pitfalls of such ”naturalistic” pattern forma- Pattern Formation and Partial Differential Equa- tion experiments. Collaborators: Antony Szu-Han Chen, tions Nicolas Taberlet, Jim McElwaine, Lucas Goehring and L. Mahadevan The research I present is motivated by specific, but ubiqui- tous pattern in models from physics: Domain and wall pat- Stephen Morris terns the magnetization forms in ferromagnets, the coars- University of Toronto, Canada ening of the phase distribution in demixing of polymer [email protected] blends, the roughening of a crystal surface under depo- sition. Dynamically speaking, the type of models ranges from variational formulations, over (driven) gradient flows CP1 to non-gradient systems. The challenge for a rigorous anal- Internal Lever Arm Model for Myosin II ysis lies in the fact that we are interested in generic behavior of solutions, as expressed by (experimentally and Myosin II is a special type of enzyme, called motor protein, numerically observed)scaling laws, that hold in the limit of capable of transforming chemical energy into mechanical large system sizes. We argue that methods from the theory work. Among the many different approaches of the dif- of partial differential equations can be used to provide at ferent disciplines, one of the most commonly used is the least one-sided, optimal bounds on these scaling laws. enzyme kinetic approach, that uses a set of (arbitrary) dis- crete states, with different transition rate constants be- Felix Otto tween them. Here, we present a purely mechanical model Max Planck Institute for Mathematics in the Sciences giving a more realistic continuous pathway between the lo- Leipzig, Germany cal equilibrium states of

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