# Applied Mathematics

## From Euler to Born and Infeld, Fluids and Electromagnetism

As the Euler theory of hydrodynamics (1757), the Born-Infeld theory of electromagnetism (1934) enjoys a simple and beautiful geometric structure. Quite surprisingly, the BI model which is of relativistic nature, shares many features with classical hydro- and magnetohydro-dynamics. In particular, I will discuss its very close connection with Moffatt’s topological approach to Euler equations, through the concept of magnetic relaxation.

The Marsden Memorial Lecture Series is dedicated to the memory of Jerrold E Marsden (1942-2010), a world-renowned Canadian applied mathematician. Marsden was the Carl F Braun Professor of Control and Dynamical Systems at Caltech, and prior to that he was at the University of California (Berkeley) for many years. He did extensive research in the areas of geometric mechanics, dynamical systems and control theory. He was one of the original founders in the early 1970’s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.

This lecture is part of the Centre Interfacultaire Bernoulli Workshop on Classic and Stochastic Geometric Mechanics, June 8-12, 2015, which in turn is a part of the CIB program on

Geometric Mechanics, Variational and Stochastic Methods, 1 January to 30 June 2015.

## 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.

## Economies with Financial Frictions: A Continuous Time Approach 3

The recent financial crisis has made obvious the need for models of financial stability. These three lectures will cover recent advancements in the modeling of crisis episodes, with particular emphasis on the use of continuous-time methods which make these models more tractable. Useful background reading includes the following

## Economies with Financial Frictions: A Continuous Time Approach 2

The recent financial crisis has made obvious the need for models of financial stability. These three lectures will cover recent advancements in the modeling of crisis episodes, with particular emphasis on the use of continuous-time methods which make these models more tractable. Useful background reading includes the following

## Economies with Financial Frictions: A Continuous Time Approach

The recent financial crisis has made obvious the need for models of financial stability. These three lectures will cover recent advancements in the modeling of crisis episodes, with particular emphasis on the use of continuous-time methods which make these models more tractable. Useful background reading includes the following

## Contingent Capital and FInancial Networks 2

These lectures will cover two topics. The first is contingent capital in the form of debt that converts to equity when a bank

nears financial distress. These instruments offer a potential solution to the problem of banks that are too big to fail by

providing a credible alternative to a government bail-out. Their properties are, however, complex. I will discuss models for the analysis of contingent capital with particular emphasis on their incentive effects and the design of the conversion trigger. The second topic in these lectures is the problem of quantifying contagion and amplification in financial networks. In particular, I will focus on bounding the potential impact of network effects under the realistic condition that detailed information on the structure of the network is unavailable

## Financial Stability 3

The lender of last resort: An analysis of the economics and politics of banking crises, and episodes of bail-outs of

failing financial institutions.

- Rochet Vives (2004) “The Lender of last Resort: was Bagehot right after all?” JEEA, 6, 1116-1147, reprinted in Rochet J.C. (2008) “Why are there so many banking crises?, Princeton University Press, chapter 2

## Risk Sharing in Over the Counter Markets 3

## Channels of Contagion in Financial Systems 3

Understanding the mechanisms underlying systemic risk requires to change the traditional focus of risk modelling and examine the link between the structure of the financial system and its stability, with a focus on contagion mechanisms which may lead to large scale instabilities in the financial system. Some channels of contagion which have played an important role in past crises are: insolvency contagion through counterparty exposures, withdrawal of liquidity in funding channels and price-mediated contagion through fire sales of assets.

We review some recent work on the mechanisms underlying these channels of contagion, with a focus on the nature of the 'network' underlying each contagion mechanism and the implications of these results for the monitoring and regulation of systemic risk. In particular, we will attempt to illustrate the importance of the ineraction between these various channels and how this interaction may undermine regulatory efforts focussed only on a single mechanism.