• home

Abstract: Host-pathogen interactions underlying the development and spread of infectious diseases can be modelled at multiple scales - from pathogen genome analysis to epidemiological compartment modelling of the host populations. It is also a highly interdisciplinary area requiring data inputs and modelling methods from all Sciences. I will summarize our work on HIV-1 genome analysis using the Chaos Game Representation, modelling HIV-1 Reverse Transcriptase protein (a drug-target) using Graph Theory, and elaborate on mathematical and statistical modelling of Malaria with reference to prevalence data from India.

Abstract: In this talk two-dimensional buoyancy-driven flows are investigated. While usually the Navier-Stokes equations are equipped with no-slip boundary conditions, here we focus on the Navier-slip conditions that, depending on the system at hand, better reflect the physical behavior. In particular, we study two systems, Rayleigh-Bénard convection and a closely related problem without thermal diffusion. In the former, bounds on the vertical heat transfer, given by the Nusselt number, with respect to the strength of the buoyancy force, characterized by the Rayleigh number, are derived. These bounds hold for a broad range of applications, allowing for non-flat boundaries, any sufficiently smooth positive slip coefficient, and are valid over all ranges of the Prandtl number, a system parameter determined by the fluid. For the thermally non-diffusive system, regularity estimates are proven. Up to a certain order, these bounds hold uniformly in time, which, combined with estimates for their growth, provide insight into the long-time behavior. In particular, solutions converge to the hydrostatic equilibrium, where the fluid's velocity vanishes and the buoyancy force is balanced by the pressure gradient.

Zoom link.

Abstract: Relating brain dynamics to function—whether tied to a task or a neurological condition—is a major challenge in modern neuroscience. Although functional Magnetic Resonance Imaging (fMRI) captures brain activity over time, the complex geometries that emerge are difficult to interpret. We propose a topological extension to the standard concept of functional connectivity, called formigram connectivity, which enhances the ability to recognize subjects based on their fMRI features. This approach boosts recognition performance from 9% to 37% on a publicly available fMRI dataset. The seminar will include an overview of Topological Data Analysis (TDA) and demonstrate how persistent homology can extend this simple pipeline, enabling the extraction of higher-level features in a similar fashion. This is joint work with Dr. Chunyin Siu from Stanford’s Brain Dynamics Lab.

Abstract: Isolated skyrmion solutions to the 2D Landau-Lifshitz equation with the Dzyaloshinskii-Moriya interaction, Zeeman interaction, and easy-plane anisotropy are considered. In a wide range of parameters illustrating the various interaction strengths, we construct exact solutions and examine their monotonic- ity, exponential decay, and stability using a careful mathematical analysis. We also estimate the distance between the constructed solutions and the harmonic maps by exploiting the structure of the linearized equation and by proving a resolvent estimate for the linearized operator that is uniform in extra implicit potentials. This is joint work with Ikkei Shimizu (U. Kyoto).

Zoom link.

Abstract: We consider the problem of dynamical stability for the vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic "particle plus field'' description of interacting vortices. In this talk we focus on co-rotational perturbations of vortices and establish a variety of pointwise dispersive and decay estimates for their linearized evolution in the relativistic (or Klein-Gordon) case. One of the main ingredients is the construction of the distorted Fourier transform associated with the (two) linearized operators at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag in the context of stability of blow-up for wave maps and relies on the spectral analysis of Schrodinger operators with strongly singular potentials (see also Gezstesy-Zinchenko). However, since the vortex is not given by an explicit formula, and one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, the linear analysis requires some additional work. In particular, to construct the distorted Fourier basis and to control the spectral measure some additional arguments are needed, compared to previous works. This is joint work with Fabio Pusateri.

ZOOM link .

Abstract: A fundamental problem in the theory of the Navier-Stokes equations is the uniqueness of solutions of the Cauchy problem. After discussing some of the recent progress in this area, we will describe a new approach to constructing solutions that exhibit non-uniqueness. As an application, we will show an example of non-unique Leray-Hopf solutions in a dyadic model of the 3D Navier-Stokes, with initial data in a sharp regularity class. Then we will present recent work with M. Coiculescu that uses a similar mechanism to construct non-unique solutions to the full Navier-Stokes whose data lies in a critical space.

Zoom link.

Abstract: Driven by an outdated problem in aerodynamics, we discovered a new principle in fluid physics. The Euler equation does not possess a unique solution for the flow over a multiply connected domain. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a two-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable. In this talk, I will present a special variational principle in analytical mechanics: Hertz’ principle of least curvature. Using this principle, we developed a variational formulation of Euler’s dynamics of ideal fluids that is different from the previously developed variational formulations based on Hamilton’s principle of least action. Applying this new variational formulation to the century-old problem of the ideal flow over an airfoil, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted wisdom about the viscous nature of the Kutta condition.

We also generalized this variational principle to Navier-Stokes’ via Gauss’ principle of least constraint, thereby presenting the fundamental quantity that Nature minimizes in every incompressible flow. We proved that the magnitude of the pressure gradient over the field is minimum at every instant! We call it the Principle of Minimum Pressure Gradient (PMPG). It is straightforward to prove that the Navier-Stokes’ equation is the first-order necessary condition for minimizing the pressure gradient cost subject to the continuity constraint. Hence, the PMPG turns a fluid mechanics problem into a minimization one where fluid mechanicians need not to apply Navier-Stokes’ equations, but minimize the pressure gradient cost. We close by posing two conjectures: one on nonlinear hydrodynamic stability and another on the mathematical problem of the inviscid limit of Navier-Stokes.

ZOOM link: https://uvic.zoom.us/j/5826187847?pwd=RlVrb0RoU0xDWTlLUDVkZW54ZThyQT09

Abstract: The stability of shear flows in the fluid mechanics is an old problem dating back to the famous Reynolds experiments in 1883. The question is to quantify the size of the basin of attraction of equilibria of the Navier-Stokes equations depending on the viscosity parameters, giving rise to the so-called stability threshold. In the case of a three-dimensional homogeneous fluid, it is known that the Couette flow has a stability threshold proportional to the viscosity, and this is sharp in view of a linear instability mechanism known as the lift-up effect. In this talk, I will explain how to exploit certain physical mechanisms to improve this bound: these can be identified with stratification (i.e. non-homogeneity in the fluid density) or rotation (i.e. Coriolis force). Either mechanism gives rise to oscillations which suppress the lift-effect. This can be captured at the linear level in an explicit manner, and at the nonlinear level by combining sharp energy estimates with suitable dispersive estimates.