96 DS13 Abstracts

96 DS13 Abstracts IP1 ity of earthquakes may be dynamically triggered. Our work Tire Tracks, the Stationary Schrödinger’s Equation indicates that the granular physics of the fault core, fault and Forced Vibrations gouge, plays a key role in triggering. Because direct access to the fault is not possible, we are characterizing the granu- I will describe a newly discovered equivalence between the lar physics of triggering at laboratory scales using physical first two objects mentioned in the title. The stationary experiments and numerical simulations. Schrödinger’s equation, a.k.a. Hills equation, is ubiqui- tous in mathematics, physics, engineering and chemistry. Paul Johnson Just to mention one application, the main idea of the Paul Los Alamos National Laboratory, USA trap (for which W. Paul earned the 1989 Nobel Prize in [email protected] physics) amounts to a certain property of Hill’s equation. Surprisingly, Hill’s equation is equivalent to a seemingly completely unrelated problem of “tire tracks”. As a fur- IP5 ther surprise, there is a yet another connection between Pattern Recognition with Weakly Coupled Oscilla- the “tire tracks” problem and the high frequency forced tory Networks vibrations. Traditional neural networks consist of many interconnected Mark Levi units and are thus inherently difficult to construct. In the Department of Mathematics lecture, we focus on neural network models of weakly cou- Pennsylvania State University pled oscillators with time-dependent coupling. In these [email protected] models, each oscillator has only one connection to a com- mon support, which makes them predestinated for hard- ware implementation. Two coupling strategies are consid- IP2 ered. The first network was proposed by Hoppensteadt and Linear and Nonlinear Problems of Mathematical Izhikevich [F.C: Hoppensteadt and E.M. Izhikevich, Phys. Finance Rev. Lett. 82, 2983 (1999)] and possesses a global coupling function. The second one is a novel architecture with indi- We start with a general overview of financial markets as vidual coupling functions. We present experimental real- dynamical systems and introduce primary financial assets izations of both networks and demonstrate that the devices including bonds, equities, currencies, and commodities. can reliably perform pattern recognition tasks. However, Next, we give a broad outline of market microstructure, the scalability of the novel network architecture is much market impact, and algorithmic trading, with a particular superior to the one of the globally coupled oscillators. emphasis of limit order books and their dynamics. We also discuss technical trading and time-series analysis. After Robert Hölzel, Kathrin Kostorz, Katharina Krischer that, we introduce financial derivatives and describe their Technische Universität München risk-neutral and real-world valuation. We discuss possi- [email protected], [email protected], ble choices of stochastic processes used for modeling pri- [email protected] mary assets; derive the corresponding pricing equations for derivatives, and demonstrate how these equations can be solved. We conclude our presentation by formulating some IP6 open problems of mathematical finance. The Topology of Fluid Mixing Alexander Lipton Topological chaos is a type of chaotic behavior that is Bank of America Merrill Lynch & forced by the motion of obstacles in some domain. I will Imperial College London, United Kingdom review two approaches to topological chaos, with applica- [email protected] tions in particular to stirring and mixing in fluid dynamics. The first approach involves constructing devices where the fluid motion is topologically complex, usually by impos- IP3 ing a specific motion of stirring rods. I will then discuss Particle Trajectories Beneath Irrotational Travel- optimization strategies that can be implemented. The sec- ling Water Waves ond approach is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits We describe the pattern of the particle trajectories beneath and their topological properties. Many tools and concepts a travelling wave moving at the surface of water in irrota- from topological surface dynamics have direct applications: tional flow and with a flat bed, with no underlying current, mapping class groups, braids, the Thurston-Nielsen classifi- both in the setting of periodic waves and in the setting of cation theorem, topological entropy, coordinates for equiv- solitary waves. alence classes of loops, and the Bestvina-Handel algorithm for train tracks. Adrian Constantin King’s College London, United Kingdom Jean-Luc Thiffeault [email protected] Dept. of Mathematics University of Wisconsin - Madison [email protected] IP4 Triggered Slip Processes in Earth IP7 In 1992, a magnitude 7.3 Landers earthquake occurred in Modeling Reactive Events in Complex Systems Southern California. As seismic waves were radiated, other earthquakes were dynamically triggered both nearby and Reactive events such as conformation change of macro- far away, and elevated seismicity, termed delayed trigger- molecules, chemical reactions in solution, nucleation events ing, lasted for several months. Recent observations based during phase transitions, thermally induced magnetization on rapidly improving instrumentation show that a major- reversal in micromagnets, etc. pose challenges both for DS13 Abstracts 97 computations and modeling. At the simplest level, these lations are used to demonstrate collective synchronization events can be characterized as the hopping over a free en- properties along with spatiotemporal waves occurring on ergy barrier associated with the motion of the system along millimeter scales. While quorum sensing proves to be a some reaction coordinate. Indeed this is the picture un- promising design strategy for reducing variability through derlying classical tools such as transition state theory or coordination across a cellular population, the length scales Kramers reaction rate theory, and it has been successful to are limited by the diffusion time of the small molecule gov- explain reactive events in a wide variety of context. How- erning the intercellular communication. I will conclude ever this picture presupposes that we know or can guess with recent progress engineering the synchronization of beforehand what the reaction coordinate of the event is. thousands of oscillating colony biopixels over centimeter In many systems of interest – protein folding, enzyme ki- length scales. netics, protein-protein interactions, etc. – making such educated guesses is hard if not impossible. The question Jeff Hasty then arises whether we can develop a more general frame- University of California, San Diego work to describe reactive events, elucidate their pathway [email protected] and mechanism, and give a precise meaning to a concept such as the reaction coordinate. In this talk I will discuss such a framework, termed transition path theory, and in- SP1 dicate how it can be used to develop efficient algorithms to Juergen Moser Lecture: Title to Be Announced accelerate the calculations and analysis of reactive events. To be posted when available. Eric Vanden-Eijnden Nancy Kopell Courant Institute Boston University New York University Department of Mathematics [email protected] [email protected] IP8 CP1 Predicting Epidemic Rare Events: A Dynami- A New, Braid-Theoretic Approach to Uncovering cal Systems Perspective of Disease Extinction and Transport Barriers Control In oceanic settings, it is desirable to identify key transport As seen by the many new vaccination campaigns across the barriers that organize the material transport. In many world, disease control is of paramount importance in pub- cases the only velocity fields that are available for process- lic health with eradication as the ultimate goal. Without ing to find these barriers are geophysical models, which intervention, disease extinction in a large well mixed pop- may not reliably represent the surface flow or the motion ulation would be a rare event. In this talk, I will review of particles on the ocean surface. In situations where the some of the mathematical models and machinery used to surface velocity field is not reliably known, however, there describe the underlying dynamics of rare events in finite is the still the possibility of making direct measurements of population disease models. I will show how to derive a trajectories through tracking floats. We look to develop a new dynamical model that includes a dynamical systems method that has the potential to harness this trajectory in- description of the effective fluctuations of the noise that formation to identify the existence of transport barriers by drives the disease to extinction. In analyzing the dynamic using tools from topology, in particular braid theory. We topology of the new expanded model, we can understand will present our braid theory based method and procedures extinction from a dynamical systems perspective, thus dis- for improving the accuracy of the resulting approximation covering how to best use disease controlling resources. for the transport barrier location. Lora Billings Michael Allshouse Montclair State University Dept. of Mechanical Engineering Dept. of Mathematical Sciences Massachusetts Institute of Technology [email protected] [email protected] Thomas Peacock IP9 Massachusetts Institute of Technology Engineered Gene Circuits: From Oscillators to [email protected] Synchronized Clocks and Biopixels Jean-Luc Thiffeault This talk will focus on the development

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