In this talk we consider the limiting behaviour of the strong solution of the Navier--Stokes equation as the viscosity goes to zero, on a three--dimensional region with curved boundary. Under the Navier and kinematic boundary conditions, we show that the solution converges to that of the Euler equation (in suitable topologies). The proof is based on energy estimates and differential--geometric considerations. This is a joint work with Profs. Gui-Qiang Chen and Zhongmin Qian, both at Oxford. 

Let X be a smooth, complete geometrically connected curve defined over a one variable function field K over a finite field. Let G be a subgroup of the points of the Jacobian variety J of X defined over a separable closure of K with the property that G/p is finite, where p is the characteristic of K. Buium and Voloch, under the hypothesis that X is not defined over K^p, give an explicit bound for the number of points of X which lie in G (related to a conjecture of Lang, in the case of curves). In this joint work with Pazuki, we extend their result by requiring just that X is non isotrivial.

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

Understanding the structure of hadrons in terms of their fundamental constituents requires an understanding of QCD at large distances, a vastly complex and unsolved dynamical problem. I will discuss in this talk a new approach to hadron structure based on superconformal quantum mechanics in the light-front and its holographic embedding in a higher dimensional gravity theory. This approach captures essential aspects of the confinement dynamics which are not apparent from the QCD Lagrangian, such as the emergence of a mass scale and confinement, the occurrence of a zero mode: the pion, universal Regge trajectories for mesons and baryons and precise connections between the light meson and nucleon spectra. This effective semiclassical approach to relativistic bound-state equations in QCD can be extended to heavy-light hadrons where heavy quark masses break the conformal invariance but the underlying dynamical supersymmetry holds.
 

The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium exponentially fast.  This is joint with with Yan Guo.

Spline curves represent a simple and efficient tool for data interpolation in Euclidean space. During the past decades, however, more and more applications have emerged that require interpolation in (often high-dimensional) nonlinear spaces such as Riemannian manifolds. An example is the generation of motion sequences in computer graphics, where the animated figure represents a curve in a Riemannian space of shapes. Two particularly useful spline interpolation methods derive from a variational principle: linear splines minimize the average squared velocity and cubic splines minimize the average squared acceleration among all interpolating curves. Those variational principles and their discrete analogues can be used to define continuous and discretized spline curves on (possibly infinite-dimensional) Riemannian manifolds. However, it turns out that well-posedness of cubic splines is much more intricate on nonlinear and high-dimensional spaces and requires quite strong conditions on the underlying manifold. We will analyse and discuss linear and cubic splines as well as their discrete counterparts on Riemannian manifolds and show a few applications.

Because of atmospheric turbulence, images of objects in outer space acquired via ground-based telescopes are usually blurry.  One way to estimate the blurring kernel or point spread function (PSF) is to make use of the aberration of wavefront received at the telescope, i.e., the phase. However only the low-resolution wavefront gradients can be collected by wavefront sensors. In this talk, I will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and the PSF in high accuracy. I will end by relating the problem to high-resolution image reconstruction and methods for solving it.
Joint work with Rui Zhao and research supported by HKRGC.