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On the theory and applications of the longtime dynamics of 3-dimensional fluid flows on thin domains

Series: CDSNS Colloquium

The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are "good" in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions

Series: CDSNS Colloquium

Motivated by Smale's work on smooth dynamical systems, David Ruelle introduced the notion of Smale spaces. These are topological dynamical systems which are hyperbolic in the sense of having local coordinates of contracting and expending directions. These include hyperbolic automorphisms of tori, but typically, the spaces involved have a fractal nature. An important subclass are the shifts of finite type which are symbolic systems described by combinatorial data. These are also precisely the Smale spaces which are totally disconnected. Rufus Bowen showed that every Smale space is the image of shift of finite type under a finite-to-one factor map. In the 1970's, Wolfgang Krieger introduced a beautiful invariant for shifts of finite type. The aim of this talk is to show how a refined version of Bowen's theorem may be used to extend Krieger's invariant to all (irreducible) Smale spaces. The talk will assume no prior knowledge of these topics - we begin with a discussion of Smale spaces and shifts of finite type and then move on to Krieger's invariant and its extension.

Series: CDSNS Colloquium

The Cohen-Gallavotti Fluctuation theorem is a result describing the behaviour of simple hyperbolic dynamical systems. It was introduced to illustrate, in a somewhat simpler context, anomalies in the second law of thermodynamics. I will describe the mathematical formulation of this Fluctuation Theorem, and some variations on it.

Series: CDSNS Colloquium

A plasma is a completed ionized gas. In many applications such as in nuclear fusion or astrophysical phenomena, the plasma has very high temperature and low density, thus collisions can be ignored. The standard kinetic models for a collisionless plasma are the Vlasov- Maxwell and Vlasov-Poisson systems. The Vlasov-Poisson system is also used to model galaxy dynamics, where a star plays the role of a particle. There exists infinitely many equilibria for Vlasov models and their stability is a very important issue in physics. I will describe some of my works on stability and instability of various Vlasov equilibria.

Series: CDSNS Colloquium

Permutation entropy was introduced as a complexity measure of time series. Formally, it replaces the symbol blocks in the definition of Shannon entropy by the so-called ordinal patterns –a digest of the ups-and-downs along a finite orbit in a totally ordered state space. Later, this concept was extended to self maps of n-dimensional intervals, in metric and topological versions. It can be proven that, under some assumptions, the metric and topological permutation entropy coincide with their corresponding conventional counterparts. Besides its use as an entropy estimator, permutation entropy has found some interesting applications. We will talk about the detection of determinism in noisy time series, and the recovery of the control parameter from the symbolic sequences of a unimodal map (which allows to cryptanalize some chaotic ciphers).

Series: CDSNS Colloquium

Mathematical models are used to study possible impact of drug treatment of infections with the human immunodeficiency virus type 1 (HIV-1) on the evolution of the pathogen. Treating HIV-infected patients with a combination of several antiretroviral drugs usually contributes to a substantial decline in viral load and an increase in CD4+ T cells. However, continuing viral replication in the presence of drug therapy can lead to the emergence of drug-resistant virus variants, which subsequently results in incomplete viral suppression and a greater risk of disease progression. As different types of drugs (e.g., reverse transcriptase inhibitors,protease inhibitors and entry inhibitors) help to reduce the HIV replication at different stages of the cell infection, infection-age-structured models are useful to more realistically model the effect of these drugs. The model analysis will be presented and the results are linked to the biological questions under investigation. By demonstrating how drug therapy may influence the within host viral fitness we show that while a higher treatment efficacy reduces the fitness of the drug-sensitive virus, it may provide a stronger selection force for drug-resistant viruses which allows for a wider range of resistant strains to invade.

Series: CDSNS Colloquium

I will review results from binary black hole simulations and the role that these simulations have in astrophysics and gravitational wave observations. I will then focus on the mathematical and computational aspects of the recent breakthroughs in numerical relativity that have made finding binary black hole solutions to the Einstein field equations an almost routine exercise.

Series: CDSNS Colloquium

I will present a generalization of a classical within-host model of a viral infection that includes multiple strains of the virus. The strains are allowed to mutate into each other. In the absence of mutations, the fittest strain drives all other strains to extinction. Treating mutations as a small perturbation, I will present a global stability result of the perturbed equilibrium. Whether a particular strain survives is determined by the connectivity of the graph describing all possible mutations.

Series: CDSNS Colloquium

Nonlinear wave phenomena are of great importance in the physical world and have been for a long time a challenging topic of research for both pure and applied mathematicians. There are numerous nonlinear evolution equations for which we need to analyze the properties of the solutions for time evolution of the system. As the first step, we should understand the dynamics of their traveling wave solutions. There exists an enormous literature on the study of nonlinear wave equations, in which exact explicit solitary wave, kink wave, periodic wave solutions and their dynamical stabilities are discussed. To find exact traveling wave solutions for a given nonlinear wave system, a lot of methods have been developed. What is the dynamical behavior of these exact traveling wave solutions? How do the travelling wave solutions depend on the parameters of the system? What is the reason of the smoothness change of traveling wave solutions? How to understand the dynamics of the so-called compacton and peakon solutions? These are very interesting and important problems. The aim of this talk is to give a more systematic account for the bifurcation theory method of dynamical systems to find traveling wave solutions and understand their dynamics for two classes of singular nonlinear traveling systems.

Series: CDSNS Colloquium

Consider the classical Newtonian three-body problem. Call motions oscillatory if as times tends to infinity limsup of maximal distance among the bodies is infinite, while liminf it finite. In the '50s Sitnitkov gave the first rigorous example of oscillatory motions for the so-called restricted three-body problem. Later in the '60s Alexeev extended this example to the three-body. A long-standing conjecture, probably going back to Kolmogorov, is that oscillatory motions have measure zero. We show that for the Sitnitkov example and for the so-called restricted planar circular three-body problem these motions have maximal Hausdorff dimension. This is a joint work with Anton Gorodetski.