Research
My research interest lies in an intersection of differential geometry, computational mathematics, physical and geometric modeling, and computer graphics.
A common theme in applied math, computational physics and computer graphics is to model, analyze and simulate phenomena in the physical world efficiently and insightfully. The subjects may include not only dynamical systems but also the geometry emerging from them. Differential geometry is a language allowing one to describe the problem through the structures hidden in the system. These description often lead to simple formulations and natural generalizations of the problem. Formalism it may seem to be, one often discovers surprising connections to different areas of research. For example, the sharp features in a fluid flow data, such as vortex tubes and vortex filaments, can be described as topological defects of manifold-valued functions (sections of bundles), which can be found through a Ginzburg-Landau equation that models superconductivity in condensed matter physics. These insights are then translated to new algorithms which are competitive in terms of structure capturing and phenomenological reproduction.
Another research that plays an important role here is discrete differential geometry (DDG). DDG is an emerging field studying discrete versions of smooth theories (usually from differential geometry). Instead of discretizing a problem based on approximations, one seeks for discrete definitions of mathematical objects, discusses their structural properties, and finds theorems analogous to their smooth counterparts. These discrete theories not only are bases for numerical methods, but also provide new insight in the smooth theory in return.
Geometric Fluid Dynamics
In geometric fluid dynamics, we study the state of a fluid and its dynamics in a coordinate-free language. Most considerations are incompressible and inviscid fluids, but compressible fluids, gas dynamics and magnetohydrodynamics are also often discussed. Specifically we are interested in using the geometric and topological method to study the following subjects:
- Interface problems in terms of currents (geometric measure theory), long exact sequences (algebraic topology), Steklov problems and boundary element methods.
- Vorticity and helicity configurations. A generalization of Helmholtz's vortex theory to manifolds with arbitrary dimensions using homology and cohomology theory.
- Dynamics of geometric Clebsch variables.
Geometric Clebsch variables
Clebsch variables, proposed by Alfred Clebsch in 1859, are a set of auxiliary variables \((\lambda_1,\ldots,\lambda_m,\mu_1,\ldots,\mu_m,\varphi)\) that represents the fluid velocity (1-form) via a Pfaffian form \[u^\flat = \sum_{i=1}^m\lambda_i d\mu_i + d\varphi.\] Formulations for fluid dynamics in terms of Clebsch variables have both Eulerian and Lagrangian nature. In a branch of differential geometry, Pfaffian forms are studied as contact forms in contact geometry. These contact forms are nowhere integrable hyperplane fields (shown in the figure "impossible staircase") that connects to an underlying symplectic geometry. Such a contact structure over a symplectic structure is known as a prequantization known in geometric quantization for quantum mechanics.
In our study, we generalize Clebsch's fluid representation using the notion of prequantization. The formalism leads to a clearer view how classical fluid dynamics is related to, for example, Schrödinger flow in quantum mechanics and superfluids.
Incompressible Schrödinger Equation
In 1926, Erwin Madelung discovered that the linear Schrödinger's equation is equivalent to a compressible quantum Euler equation, similar to that in hydrodynamics. This is done by a straightforward change of variables called Madelung transform, where the squared norm of the wavefunction is the density (which is in fact the probability density) and the spatial derivative of the phase is the velocity (momentum). Due to the conservation of probability and momentum encoded in Schrödinger's equation, probability flows like a fluid.
Classical fluids at the scale of interest, on the other hand, is incompressible. In this project we impose an incompressible constraint to the Schrödinger's equation, resulting in decent simulations of classical fluids at low computation cost. Specifically, no nonlinear advection is ever needed.
Moreover, vortices, which are important sharp features in fluids, are encoded as topological defects in the wavefunctions. They tend to persist stably in an incompressible Schrödinger simulation comparing to most direct numerical simulations of Euler equations.
In this research, we are interested in using incompressible Schrödinger equation as a new way of looking at fluid dynamics problems. This new approach may give new insights to not only fluid modeling (by applying to general fluids such as complex fluids, magnetohydrodynamics and gas dynamics) but also long-standing PDE problems in fluid equations.
Ginzburg-Landau Models
Ginzburg-Landau models describe emerging pattern in the form of topological defects in fields minimizing a Ginzburg-Landau energy. These fields, called order parameters, are usually used to represent the coherent phase of a soup of quantum particles with very low temperature (Bose-Einstein condensation). But Ginzburg-Landau energies are also useful, in our research, to be objective functions in optimizations where the minimizers give desired geometric objects.
Ginzburg-Landau energy is essentially the Dirichlet energy of a section of a fiber bundle. The connection in the fiber bundle, which may have nonzero curvature, leads to nucleation of topological defects in the same way magnetic fields gives rise to vortices in (type-II) superconductors. The metric in each fiber (for taking the norm square for the Dirichlet energy) can also be modified to adjust the "sharpness" in these patterns.
For example, given a velocity field (like the delta-wing flow dataset shown in the figure), we can consider a complex vector bundle whose connection is given by the velocity. By adjusting the sharpness through rearranging the metric in the complex vector spaces, the emerging patterns in a Ginzburg-Landau-energy-minimizing field converge to the vortex tubes hidden in the given flow data.
In this research, we are interested in applying Ginzburg-Landau models to a broader type of problems, for example in geometry processing.
Perfectly Matched Layers
There is a large demand in computational waves (such as scattering problems) for non-reflecting boundary condition. Perfectly matched layer (PML) is a state of the art technique for such an absorbing boundary. In the layer, one modifies the wave equation in such a way that the waves are damped while no reflection is created at its interface with the physical domain.
There is however a long standing problem: The non-reflecting property of PML fails after discretization. For more than 20 years researchers have been optimizing the parameters in discrete PMLs to reduce the numerical reflections.
In our approach to this problem, we mimic the derivation of the continuous PML but purely in the discrete setting. It exploits the discrete complex analysis, a subject in discrete differential geometry, and yields the first discrete PML that gives no numerical reflection at all.
Spin Geometry Processing
In math visualization and geometry processing, a desired property for surfaces realized in \(\Bbb R^3\) is immersion, a.k.a. locally embedded surfaces. Unlike curves in \(\Bbb R^2\) or \(\Bbb R^3\), surface immersion can not always be reached by perturbation or smoothing from a generic surface. The genuine non-immersed configurations are pinch points, which one can find in surfaces such as the Roman surface. Pinch points can not be removed without a large deformation or surgery.
In a conventional description of surface one describes a frame over the abstract surface as the instruction how the tangent space should be placed in 3D. This frame is a section over a frame bundle that is associated with \({\rm SO}(3)\) gauge group. Unfortunately, pinch points are invisible in this framework.
Fortunately, the universal cover (a double cover) \({\rm SU}(2)\) over \({\rm SO}(3)\) allows one to distinguish immersed disk and pinch points. This information is hidden in the liftability of an \({\rm SO}(3)\) frame to an \({\rm SU}(2)\) frame. Since the deck transformation group in this cover is \({\Bbb Z}_2\), the \(\Bbb Z_2\)-coefficient homology and cohomology plays an important role in the \({\rm SU}(2)\)-based description. Due to the insight in quantum mechanics, \({\rm SU}(2)\) objects usually have the word "spin" attached to the names.
In practice, one study the regular homotopy class \({\frak q}(\gamma)\) of each closed strip given as closed curve \(\gamma\) in the surface. Incidentally there are only two regular homotopy classes for closed strip: Figure-8 (\({\frak q}=1\)) and Figure-0 (\({\frak q}=0\)). This \(\Bbb Z_2\)-valued function \({\frak q}\) on closed \(\Bbb Z_2\)-coefficient 1-chains are known as an Arf invariant. A surface is immersed if and only if its \({\frak q}\) descends to a well-defined \(\Bbb Z_2\)-valued function on the 1-st \(\Bbb Z_2\) homology. The dual object to the Figure-8/0 properties of closed chains are rims, which are \({\Bbb Z}_2\)-valued relative cochains (modulo boundary). A rimmed edge switches the figure-8/0 property of every closed strip that goes across it.
By including this immersion theory through spin geometry, one can set up optimization problems in geometry processing favors immersions (even with a prescribed regular homotopy class) rather than pinched surfaces. For a non-immersed surface, we also know the appropriate location for adding rims that guarantees immersion.