We propose and study a new information transmission problem motivated by today’s internet. Suppose a real number needs to be transmitted in a network. This real number may represent data or control and pricing information of the network. We propose a new transmission model in which the real number is encoded using Bernoulli trials. This differs from the traditional framework of Shannon’s information theory. We propose a natural criterion for the quality of an encoding scheme. Choosing the best encoding reduces to a problem in the calculus of variations, which we solve rigorously. In particular, we show there is a unique optimal encoding, and give an explicit formula for it.

We also solve the problem in a more general setting in which there is prior information about the real number, or a desire to weight errors for different values non-uniformly.

Our tools come mainly from real analysis and measure-theoretic probability. We also explore a connection to classical mechanics.

Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. Holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. Curvature flow method is nonlinear and requires higher quality triangulations, but it is more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms are demonstrated by the experimental results.

Accurately simulating fluid dynamics on arbitrary surfaces is of significance in graph- ics, digital entertainment, and engineering applications. This paper aims to improve the efficiency and enhance interactivity of the simulation without sacrificing its accuracy. We develop a GPU-based fluid solver that is applicable for curved geometry. We resort to the conformal (i.e., angle-preserving) structure to parameterize a surface in order to simplify differential operators used in Navier-Stokes and other partial differential equations. Our conformal flow method integrates fluid dynamics with Riemannian metric over curved geometry. Another significant benefit is that a conformal parameter- ization naturally facilitates the automatic conversion of mesh geometry into a collection of regular geometry images well suited for modern graphics hardware pipeline. Our algorithm for mapping general genus zero meshes to conformal cubic maps is rigorous, efficient, and completely automatic. The proposed framework is very general and can be used to solve other types of PDEs on surfaces while taking advantage of GPU acceleration.

Teichmüller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmüller space, only if they can be conformally mapped to each other. Teichmüller shape space can be used for surface classification purpose in shape modeling.

This work focuses on the computation of the coordinates of high genus surfaces in the Teichmüller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to deform the surface conformally using surface Ricci flow, such that the Gaussian curvature is −1 everywhere. The surface is decomposed to several pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary.

The algorithms are based on Teichmüller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmüller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the practical value of the method for geometric database indexing, shape comparison and classification.