Wednesdays at 3:30 PM in Eckhart 202.
Consider the planar 3 body problem consisting of large, small, and very small body,e.g. a Sun-Jupiter-Asteroid system. Assume that Asteroid has mass zero. During the talk we discuss analyze variety of unstable motions for this problem. Motions is called oscillatory if as time tends to infinity limsup (resp. liminf) of distance to the origin is infinite (resp. finite). A long standing conjecture by Kolmogorov: oscillatory motions form a set of measure zero. Jointly with A. Gorodetski we show that these motions form a set of the full Hausdorff dimension. Then we discuss phenomenon of Arnold diffusion for this problem, proved jointly with A. Delshams and T. Seara, and existence of nonlocal instabilities proved using Aubry-Mather theory. The latter is related to Herman's oldest open question in dynamical systems.
We present some new results on the existence, blow up and regularity properties of solutions of Burgers' equation with fractional dissipation. We prove the existence of finite time blow up for the powers of Laplacian $\alpha < 1/2,$ and global existence as well as analyticity of solution for $\alpha \geq 1/2.$ We also discuss solutions with very rough initial data.
We consider the formal and analytic aspects of the strong asymptotics of pseudo-orthogonal polynomials (i.e. "orthogonal" with respect to a complex measure). On a formal level the asymptotics is intimately related to the classification of commuting pairs of difference operators one of which is symmetric and tridiagonal (a la Krichever/Novikov). This in turn realizes the common solutions of these difference operators in terms of sections of line bundles on a Riemann surface (hyperelliptic). From geometrical considerations stemming from the Riemann-Hilbert analysis we show that the line bundle which is relevant to the asymptotics is a spinor bundle. In a second stage we show that the heuristics is sound and gives the actual strong asymptotics using the nonlinear steepest-descent method developed Deift-Zhou using the Riemann--Hilbert formulation introduced by Fokas-Its-Kitaev; the main ingredient is the ability of constructing hyperelliptic Riemann surfaces with a prescribed harmonic real function satisfying certain inequalities. Such construction impinges on the notion of quadratic differentials and their theory, developed by K. Strebel and used in a different context for associating Riemann surfaces to decorated ribbon graphs. Based on a work with M. Y. Mo.
We propose a new proof of the partial regularity of solutions to the incompressible Navier-Stokes equation in dimension 3 first proved by Caffarelli, Kohn and Nirenberg. The proof relies on a method introduced by De Giorgi for elliptic equations.
In this talk, I will review several integrators that have been introduced to simulate efficiently systems with widely disparate time scales: projective integration methods, HMM-like multiscale integrators, and boosting methods. I will discuss the theoretical background of these integrators, show how they differ and explain what are their strengths and weaknesses. I will also illustrate HMM-like multiscale integrators in the context of the simulation of Markov jump processes arising e.g. in chemical kinetics.
Simple mean-field stochastic models of epidemics exhibit a peculiar threshold effect at criticality: when the size of the infected set exceeds a certain threshold value, the evolution changes abruptly from branching process behavior to damped branching. In this talk I will discuss an analogous threshold effect for a simple class of stochastic spatial epidemics, and exhibit measure-valued scaling laws for the large-density limit.
I. Monday, April 23 4PM in Ryerson 251: An Overview of Diffusion Geometries
We use diffusion time as a scaling mechanism for structural, multiscale harmonic analysis on Riemannian manifolds or, more generally, on subsets of R^N, and on graphs. We augment approaches to data analysis by showing that the diffusion distances are key, intrinsic geometric quantities linking spectral theory of a Markov process to the corresponding geometry of the data, and give examples. II. Tuesday, April 24 4:30PM in Ryerson 251: Multiscle Analysis I The ideas of Lecture I form part of a well-understood theory relating wavelets to Fourier analysis, in which the "geometry of a linear transformation" is directly linked to its spectral properties. We explain this theory in general and give various applications to classification, to the analysis of data matrices and operators and to signal processing on data functions. III. Wednesday, April 25 4PM in Ryerson 251: Multiscale Analysis II
A continuation of the preceding lecture.
Thermal or stochastic effects are prevalent in physical, chemical, and biological systems. Particularly in small systems, noise can overpower the deterministic dynamics and lead to ``rare events'' that would never be seen in the absence of noise. One example is the thermally-driven switching of the magnetization in small memory elements. We use Wentzell-Freidlin large deviation theory and concepts from stochastic resonance to analyze magnetic switching. A surprising and physically relevant result is that in multiple-pulse experiments, unconventional ``short-time switching pathways'' can dominate. One advantage of the method is that it generalizes to systems with spatial variation. To discuss the complications and richness that emerge in the PDE setting, we consider the (simpler) Allen-Cahn equation. The associated action functional and its sharp-interface limit represent a new problem in the calculus of variations. We present some results, including a $\Gamma$-convergence result for the problem in one space dimension. This talk includes joint work with Bob Kohn, Felix Otto, Yoshihiro Tonegawa, and Eric Vanden--Eijnden.
The Euler equations, describing a potential flow of infinitely deep 2-D ideal incompressible fluid with free surface, takes a compact closed form after the conformal mapping of the domain filled with fluid up to the lower half-plane. The "conformal" evolution equations of surface dynamics are suitable both for analytic study and numerical simulation. The main tool of analytic investigation is the consideration of singularity dynamics in the upper half-plane. In a typical situation the singularities are the moving and broadening cuts. As far as the cuts are narrow, the problem can be solved analytically. It describes the formation of drops and shapes of surface, similar to the "Saffman fingers". A certain class of initial data can be described approximately by the famous Laplace Growth Equation (LGE). In this and even more general cases the conformal evolutionary equations have "extra" constants of motion, which are not connected with natural symmetries of the system. It leads to conjecture that the system in completely integrable but this question is still open.
The conformal equations could be efficiently solved numerically by the use of the spectral code. We elaborated a comfortable and stable numeric algorithm making possible to model the nonlinear wave propagation during a very long time (up to 100 000 periods). We performed long-time modeling of nonlinear stage of the Stokes wave modulational instability and found that the instability leads to formation of solitonic turbulence and finally, to the appearance of freak waves.
This talk will focus on a lower bound for the real part of eigenvalues of dissipative operators L = H + i g F with H the 1D Harmonic oscillator, F an analytic function on the real line, and g a coupling parameter. When the coupling g is large the eigenvalues lie in a half plane {z : Re z > g^v} with v an exponent that depends on F. This cannot be seen directly by variational arguments since the perturbation i g F is skew-adjoint, but can be seen using complex dilations. As a result of the bound, one sees that the dissipation of the semi-group e^{-t L} is greatly enhanced compared to e^{-t H} although the difference of the generators is skew-adjoint. This is a linear manifestation of the phenomenon of ``hypocoercivity" identified by C. Villani.
Smoluchowski derived his coagulation equations in 1917 to describe the accretion of colloids. His model has since been used to describe a variety of clustering processes (eg. the formation of smoke, dust and haze; the kinetics of polymerization; gravitational accretion). In the past fifteen years, this and other mean-field models of clustering have received considerable mathematical attention. An important feature is the formation of `universal' distributions as mass is transported from small to large scales by clustering. Burgers turbulence is the study of shocks in Burgers equation with random initial data or forcing. In the absence of forcing, shocks interact only by clustering. An elegant result of Carraro and Duchon (1994) and Bertoin (1998) reduces this problem to Smoluchowski's equation with additive kernel for a large class of random initial data. I will describe the two models, the solution procedure that links them, and the consequent characterization of universality. A general theme is the fruitful interplay between aspects of the problem that are natural from the analytic and probabilistic points of view. This is joint work with Bob Pego (Carnegie Mellon).
We consider parabolic equations on R^N which have solutions blowing-up in finite time as well as solutions decaying to zero. We examine the borderline solutions, that is, solutions that do not exhibit either of the two behaviors. An emphasize will be put on problems that cannot be treated by energy methods (because of time dependence in the equation, for example) and we shall discuss new techniques suitable for the study: a priori estimates based on parabolic Liouville theorems, asymptotic symmetrization, and exponential separation for the linearized equations.
Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the production of regularly spaced ''islands'' of cartilage similar to the cartilage primordia of the developing limb skeleton. In this talk we describe a discrete, stochastic model for the behavior of limb bud precartilage mesenchymal cells in vitro [1]. It is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects that can change their shape.
In the development of multiscale biological models it is crucial to establish a connection between discrete microscopic or mesoscopic stochastic models and macroscopic continuous descriptions based on cellular density. In this talk a continuous limit of a two-dimensional Cellular Potts Model (CPM) with excluded volume will be demonstrated, describing cells moving in a medium and reacting to each other through both direct contact and long range chemotaxis [2]. The continuous macroscopic model is obtained as a Fokker-Planck equation describing evolution of the cell probability density function. All coefficients of the general macroscopic model are derived from parameters of the CPM and a very good agreement is demonstrated between CPM Monte Carlo simulations and numerical solution of the macroscopic model.
1. Christley, S., Alber, M.S. , Newman, S.A. [2007], Patterns of Mesenchymal Condensation in a Multiscale, Discrete Stochastic Model, PloS Computational Biology, 3, 4, e76.
2. Alber, M., Chen, N., Glimm, T., and P. Lushnikov [2006], Multiscale dynamics of biological cells with chemotactic interactions: From a discrete stochastic model to a continuous description, Phys. Rev. E. 73 051901.
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