Wednesdays at 4:15 PM in 384H.
We consider an integro-PDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We show that in the limit of small mutation rate, the solution concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings and Dockery et al. showed that for two competing species, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for a continuum of traits. This talk is based on joint work with King-Yeung Lam.
Abstract: Ray mappings are the fundamental objects of geometrical optics. We shall consider canonical problems in optics, such as phase retrieval and beam shaping, and show that their solutions are characterized by certain ray mappings. Existence of solutions and some properties of them can be established through a variational method - the Weighted Least Action Principle - which is a natural generalization of the Fermat principle of least time.
This is a joint work with D.Burago and S.Ivanov. As avid anglers we were always interested in the survival chances of fish in turbulent oceans. In this talk I will address this question mathematically, and discuss some of its consequences. I will show that a fish with bounded aquatic locomotion speed can reach any point in the ocean if the fluid velocity is incompressible, bounded, and has small mean drift.
We begin with the elementary observation that the $n$-step descendant distribution of any Galton-Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we study certain CSBPs (continuous state branching processes), which arise as scaling limits of Galton-Watson processes. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain CSBPs satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall in 2006.) We also prove the existence of Galton-Watson processes that are universal, in the sense that all possible (sub)critical CSBPs can be obtained as a sub-sequential scaling limit of this process.
Fluid-structure interaction problems with composite structures arise in many applications. One example is the interaction between blood flow and arterial walls. Arterial walls are composed of several layers, each with different mechanical characteristics and thickness. No mathematical results exist so far that analyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structure is composed of several layers. In this talk we will summarize the main difficulties in studying this class of problems, and present an existence proof and a computational scheme based on which the proof of the existence of a weak solution was obtained. Our results reveal a new physical regularizing mechanism in FSI problems with multi-layered structures: inertia of the thin fluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoretical results on modeling the human cardiovascular system will be discussed. This is a joint work with Boris Muha (University of Zagreb, Croatia), Martina Bukac (U of Notre Dame, US) and Roland Glowinski (UH). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL, Switzerland). Collaboration with medical doctors Dr. S. Little (Methodist Hospital Houston) and Dr. Z. Krajcer (Texas Heart Institute) is also acknowledged.
We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering an open problem posed by De Lellis and Szekelyhidi Jr. Moreover, we show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. This talk is based on a joint work with T. Buckmaster and S. Shkoller.
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) in 1d with ignition reactions f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of X_1 is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of X_1 is infinite.
We discuss two models of random walk in random environment, one from stochastic homogenization of composite materials, and the other from interacting particle systems. The goal is to explore the quantitative aspects of the invariance principle, i.e., to quantify the convergence of the properly rescaled random walk to a Brownian motion. The idea is to borrow PDE/analytic tools from stochastic homogenization of divergence form operator and apply them in the context of interacting particle systems. In particular, we will explain the proof of a diffusive heat kernel upper bound on the tagged particle in a symmetric simple exclusion process.
In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space for the 1D periodic case. This is demonstrated by constructing arbitrarily small solutions with a sequence of nonlinear oscillations, known as plasma echoes, which damp at a rate arbitrarily slow compared to the linearized Vlasov equations. Some connections with hydrodynamic stability problems will be discussed if time permits.
Nuclear magnetic resonance (NMR) spectroscopy is the most-used technique for protein structure determination besides X-ray crystallography. In this talk, the computational problem of protein structuring from residual dipolar coupling (RDC) will be discussed. Typically the 3D structure of a protein is obtained through finding the coordinates of atoms subject to pairwise distance constraints. RDC measurements provide additional geometric information on the angles between bond directions and the principal-axis-frame. The optimization problem involving RDC is non-convex and we present a novel convex programming relaxation to it by incorporating quaternion algebra. In simulations we attain the Cramer-Rao lower bound with relatively efficient running time. From real data, we obtain the protein backbone structure for ubiquitin with 1 Angstrom resolution. This is joint work with Amit Singer and David Cowburn.
The question addressed here is how fast a front will propagate when a line, having a strong diffusion of its own, exchanges mass with a reactive medium. More precisely,we wish to know how much the diffusion on the line will affect the overall front propagation. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by transportation networks. In a previous series of works, we were able to show that the line could speed up propagation indefinitely with its diffusivity. For that, we used a special type of nonlinearity that allowed the reduction of the problem to explicit computations. In the work presented here, the reactive medium is governed by nonlinearity that does not allow explicit computations anymore. We will explain how propagation speed-up still holds. In doing so, we will discuss a new transition phenomenon between two speeds of different orders of magnitude. Joint work with L. Dietrich.
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