Georgia Institute of Technology College of Sciences

The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all angles \leq 120 degrees and all new angles \geq 60 degrees (small angles in the original polygon must remain).

In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space variables. Both of linear and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, which is an exact complement to the above well-posedness result for a special flow. This is a joint work with Chengjie Liu and Tong Yang.

Find the abstract at the link: http://people.math.gatech.edu/~gchen73/research/GA_Tech.pdf

A fourientation of a graph is a choice for each edge of whether to orient it in either direction, bidirect it, or leave it unoriented. I will present joint work with Sam Hopkins where we describe classes of fourientations defined by properties of cuts and cycles whose cardinalities are given by generalized Tutte polynomial evaluations of the form: (k+l)^{n-1}(k+m)^g T (\frac{\alpha k + \beta l +m}{k+l}, \frac{\gamma k +l + \delta m}{k+m}) for \alpha,\gamma \in {0,1,2} and \beta, \delta \in {0,1}. We also investigate classes of 4-edge colorings defined via generalized notions of internal and external activity, and we show that their enumerations agree with those of the fourientation classes. We put forth the problem of finding a bijection between fourientations and 4-edge-colorings which respects all of the given classes. Our work unifies and extends earlier results for fourientations due to myself, Gessel and Sagan, and Hopkins and Perkinson, as well as classical results for full orientations due to Stanley, Las Vergnas, Greene and Zaslavsky, Gioan, Bernardi and others.