Wednesdays at 4 PM in Eckhart 206.
Convergence to steady or time-periodic solutions in Hamilton-Jacobi equations with convex hamiltonian is studied. The relevant tool, introduced by Fathi in this context, is the Aubry-Mather set, i.e, crudely speaking, an attracting set for the associated geodesics problem. This, together with the dynamic programming principle, allows to prove convergence to steady states, time-periodic solutions, pulsating waves... in a number of cases (strictly convex case, 1D time-periodic, 2D eikonal equations). This investigation was initially motivated by a solid combustion problem.
Regions of intense vorticity in fluids are made up of vortex tubes undergoing a continuous process of stretching, folding and reconnection. The stretching and folding are inviscid processes. The reconnection, which involves change of topology, is not. I will describe analytical and numerical work relating the vortex reconnection process to a singular limit of the Riemannian connection associated to the diffusive Lagrangian flow. The numerical work is due to Ohkitani.
The behavior of thin elastic sheets is important in a variety of situations ranging from thin film deposition on semiconductors, to the mechanisms determining the shapes of leaves and flowers. I will give an introductory talk describing some of the recent work in this area, emphasizing how little we know about this problem. I will also discuss some of the interesting analytic and geometric questions that arise in this context.
The stationary asymptotic states of "roll" pattern-forming systems far from threshold have a remarkably universal structure consisting of local striped regions mediated by a fairly restricted variety of defects. Phase-diffusion equations are often used to model the macroscopic behavior of these regimes and we study a regularization of one of these, the Cross-Newell (RCN) equation. This equation is variational and its asymptotic minimizers provide a good match for what is seen in patterns far from onset. Mathematically, the RCN equation fits into a natural hierarchy of variational PDE whose asymptotic minimizers support defects. In particular it can be described as a Ginzburg- Landau model restricted to locally gradient director fields. We employ a "self-dual" ansatz to get estimates on the free energy of asymptotic minimizers and in certain geometries these estimates can be shown to be optimal. We illustrate the relevance of our results by comparison with controlled numerical and physical experiments.
The exclusion process is a Markov process consisting of random walks on a lattice with jumps to occupied sites excluded. Despite its simplicity, it displays some of the complex behaviour one expects from more realistic systems. We will discuss the time scale in the asymmetric case. In dimensions three and larger the system is diffusive. In one and two it is superdiffusive. A method will be described which gives superdiffusive lower bounds in these cases.
We prove rigorously that if a solution of the three-dimensional incompressible Navier-Stokes equation develops a singularity, the (normalized) pressure must become unbounded from below. This is a joint work with G. Seregin.
Many spatially extended dynamical systems exhibit patterns that are chaotic in space and time. Various kinds of defects are often striking features of these patterns. A tantalizing question is to what extent the defects can be used to characterize and describe the patterns and their dynamics. I will discuss a few types of defect-dominated spatio-temporal chaos and how we characterize them in terms of their defects. For hexagon patterns in rotating non-Boussinesq convection we find a state of penta-hepta defect chaos that is characterized by the induced nucleation of defects, which leads to defect statistics that differ significantly from that commonly found in defect chaos. In a Ginzburg-Landau model for parametrically driven waves we find a transition from a disordered to a spatially ordered chaotic state. We characterize it in terms of the statistics of the defects' space-time trajectories. This approach provides a somewhat intuitive picture of the role of the defects in the break-down of the spatial order.
`Pinning' refers to the anomalous clustering of stock prices near option strike prices on the third Friday of each month. This phenomenon has been widely documented (Ni, Pearson & Poteshma 2003, and references therein) -- but remained unexplained until now. We propose a model to describe stock pinning on option expiration dates. We argue that if the open interest in a particular contract is unusually large, Delta-hedging in aggregate by floor market-makers can impact the stock price and drive it to the strike price of the option. We derive a stochastic differential equation for the stock price which has a singular drift that accounts for the price-impact of Delta-hedging. According to this model, the stock price has a finite probability of pinning at a strike. The model is amenable to analysis, using the WKB ansatz for solution of the associated Fokker-Planck equation. We calculate analytically and numerically the probability of pinning in terms of the volatility of the stock, the time-to-maturity, the open interest for the option under consideration, and a ``price-elasticity" constant that models price impact. Recent communications with econometricians indicate that, according to available data, the above model is able to explain to the pinning effect to a large degree. Reference: M. Avellaneda and M.D. Lipkin, Quantitative Finance, Vol 3 (2003), 417-425. Note: This talk is intended for a broad applied mathematics audience. Expertise in finance, option theory or mathematical finance is not needed to follow it.
During the talk I will describe emerging connection between Hamiltonian dynamics and solutions to Hamilton-Jacobi PDE's. On dynamical side we have Mather theory developped in recent years by Mather, Fathi, Xia. A few years ago Fathi has notice that it is closely Related to theory of visousity solutions of Hamilton-Jacobi PDE's developped by Crandall-Lions. These connections are to be explored. Our main focus is application of recent Mather's resolution of Arnold's conjecture (about diffusion in nearly integrable Hamiltonian system with 3 degrees of freedom) to partially resolve another Arnold's conjecture about instabilities of totally elliptic points. This is a joint work with J.Mather and E.Valdinoci.
The above method is an effective criterion used in the proof of Anderson localization for random Schroedinger operators. The method, developed in 90's by Aizenman and Molchanov, was initially only applicable to lattice models. We will present joint work with M. Aizenman, S. Naboko, J. Schenker and G. Stolz which allows to extend the fractional moment method and its consequences to continuum Anderson-type models.
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