# Mathematics

## Compressed Sensing

Many problems in science and engineering require the reconstruction of an object - an image or signal, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected. Compressed sensing is a mathematical theory and set of techniques that aim to improve reconstruction quality from a given data set by leveraging the underlying structure of the unknown object; specifically, its sparsity.

In this talk I will commence with an overview of the fundamentals of compressed sensing and discuss some of its applications. However, I will next explain that, despite the large and growing body of literature on compressed sensing, many of these applications do not fit into the standard framework. I will then describe a more general framework for compressed sensing which aims to bridge this gap. Finally, I will show that this new framework is not just useful in explaining existing applications of compressed sensing. The new insight it brings leads to substantially better compressed sensing-based approaches than the current state-of-the-art in a number of applications.

## Wavelets and Directional Complex Framelets with Applications to Image Processing

Wavelets have been successfully applied to many areas. For high-dimensional problems such as image/video processing, separable wavelets are widely used but are known to have some shortcomings such as lack of directionality and translation invariance. These shortcomings limit the full potential of wavelets. In this talk, we first present a brief introduction to orthonormal wavelets and tight framelets as well as their fast transforms using filter banks. Next we discuss recent exciting developments on directional tensor product complex tight framelets (TP-CTFs) for problems in more than one dimension. For image/video denoising and inpainting, we show that directional complex tight framelets have superior performance compared with current state-of-the-art methods. Such TP-CTFs inherit almost all the advantages of traditional wavelets but with directionality for capturing edges, enjoy desired features of the popular discrete Fourier/Cosine transform for capturing oscillating textures, and are computationally efficient. Such TP-CTFs are also naturally linked to Gabor (or windowed Fourier) transform and can be further extended. We expect that our approach of TP-CTFs using directional complex framelets can be applied to many other high-dimensional problems.

## Projecting the Uncertainty of Sea Level Rise Using Climate Models and Statistical Downscaling

Most global climate models do not estimate sea level directly. A semi-empirical approach is to relate sea level change to temperature and then apply this relationship to climate model projections of temperature for different future scenarios. Another possibility is to estimate the relationship between global mean temperature in historical runs of a model and instead apply this relationship to future temperature projections. We compare these two methods to estimate global annual mean sea level and assess the resulting uncertainty. Of more practical importance is to estimate local sea level. We exemplify this by developing models for projected sea level rise in Vancouver and Washington State and illustrate different sources of uncertainty in the projections.

**BIO**: Peter Guttorp is a Professor of Statistics, Guest Professor at the Norwegian Computing Center, Project Leader for SARMA, the Nordic Network on Statistical Approaches to Regional Climate Models for Adaptation, Co-director of STATMOS, the Research Network on Statistical Methods for Atmospheric and Ocean Sciences, Adjunct Professor of Statistics at Simon Fraser University and member of the interdisciplinary faculties in Quantitative Ecology and Resource Management and Urban Design and Planning. He obtained a degree from the Stockholm School of Journalism in 1969, a B.S. in mathematics, mathematical statistics and musicology from Lund University, Sweden, in 1974, a Ph.D. in statistics from the University of California at Berkeley in 1980 and a Tech.D. h.c. from Lund University in 2009. He joined the University of Washington faculty in September 1980.

Dr. Guttorp’s research interests include uses of stochastic models in scientific applications in hydrology, atmospheric science, geophysics, environmental science, and hematology. He is a fellow of the American Statistical Association and an elected member of the International Statistical Institute. During 2004-2005 he was the Environmental Research Professor of the Swedish Institute of Graduate Engineers, and in 2014 he was one of the Chalmers Jubilee Professors.

## The Mathematics of Lattice-based Cryptography

TBA

## Adam Clay Lecture 2 of 2

This lecture is part of a course organized by Dale Rolfsen.