# Mathematics

## Fast and Somewhat Accurate Algorithms

In applications such as image processing, computer vision or image compression, often times accuracy and precision are less important than processing speed as the input data is noisy and the decision making process is robust against minor perturbations. For instance, the human visual system (HVS) makes pattern recognition decisions even though the data is blurry, noisy or incomplete and lossy image compression is based on the premise that we cannot distinguish minor differences in images. In this project we study the tradeoff between accuracy and system complexity as measured by processing speed and hardware complexity.

Knowledge of linear algebra, computer science, and familiarity with software tools such as Matlab or Python is desirable. Familiarity with image processing algorithms is not required.

Fig. 1: error diffusion halftoning using Shiau-Fan error diffusion

Fig. 2: error diffusion halftoning using a single lookup table

References:

1. Wu, C. W., "Locally connected processor arrays for matrix multiplication and linear transforms," Proceedings of 2011 IEEE International Symposium on Circuits and Systems (ISCAS), pp.2169,2172, 15-18 May 2011.

2. Wu, C. W., Stanich, M., Li, H., Qiao, Y., Ernst, L., "Fast Error Diffusion and Digital Halftoning Algorithms Using Look-up Tables," Proceedings of NIP22: International Conference on Digital Printing Technologies, Denver, Colorado, pp. 240-243, September 2006.

## Disconnecting the G_2 Moduli Space

Little is currently known about the global properties of the moduli space of a closed 7-manifold, ie the space of Riemannian metrics with holonomy modulo diffeomorphisms. A holonomy metric has an associated -structure, and I will define a Z/48 valued homotopy invariant of a -structure in terms of the signature and Euler characteristic of a Spin(7)-coboundary. I will describe examples of manifolds with holonomy metrics where the invariant is amenable to computation in terms of eta invariants, and which are candidates for having a disconnected moduli space. This is joint work in progress with Diarmuid Crowley and Sebastian Goette.

## Universal torsion, L^2-invariants, polytopes and the Thurston norm

We introduce universal torsion which is defined for -acyclic manifolds with torsion free fundamental group and takes values in certain -groups of a skew field associated to the integral group ring. It encompasses well-know invariants such as the Alexander polynomial and -torsion. We discuss also twisted -torsion and higher order Alexander polynomials which can also be derived from the universal invariant and assign certain polytopes to the universal torsion. This gives especially in dimension 3 interesting invariants which recover for instance the Thurston norm.