# Applied Mathematics

## Channels of Contagion in Financial Systems 2

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.

## Channels of Contagion in Financial Systems

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.

## Risk Sharing in Over the Counter Markets 2

These lecture notes are part of a series on "*Risk Sharing in Over-the-Counter Markets"*

## Reconstructing carbon dioxide for the last 2000 years: a hierarchical success story

Knowledge of atmospheric carbon dioxide (CO2) concentrations in the past are important to provide an understanding of how the Earth's carbon cycle varies over time. This project combines ice core CO2 concentrations, from Law Dome, Antarctica and a physically based forward model to infer CO2 concentrations on an annual basis. Here the forward model connects concentrations at given time to their depth in the ice core sample and an interesting feature of this analysis is a more complete characterization of the uncertainty in "inverting" this relationship. In particular, Monte Carlo based ensembles are particularly useful for assessing the size of the decrease in CO2 around 1600 AD. This reconstruction problem, also known as an inverse problem, is used to illustrate a general statistical approach where observational information is limited and characterizing the uncertainty in the results is important. These methods, known as Bayesian hierarchical models, have become a mainstay of data analysis for complex problems and have wide application in the geosciences. This work is in collaboration with Eugene Wahl (NOAA), David Anderson (NOAA) and Catherine Truding.

## Imaging with Waves in Complex Environments

The talk is concerned with the application of sensor array imaging in complex environments. The goal of imaging is to estimate the support of remote sources or strong reflectors using time resolved measurements of waves at a collection of sensors (the array). This is a challenging problem when the imaging environment is complex, due to numerous small scale inhomogeneities and/or rough boundaries that scatter the waves. Mathematically we model such complexity (which is necessarily uncertain in applications) using random processes, and thus study imaging in random media. I will focus attention on the application of imaging in random waveguides, which exhibits all the challenges of imaging in random media. I will present a quantitative study of cumulative scattering effects in such waveguides and then explain how we can use such a study to design high fidelity imaging methods.

## The Mathematics of Bats

2010 Fields Medal recipient, Cédric Villani, Director of the Institut Henri Poincaré in Paris, France, will give a Friday evening talk entitled The Mathematics of Bats.

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## Math Modeling in Indudustry Team 7 - Final Report

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## Math Modeling in Indudustry Team 6 - Final Report

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## Math Modeling in Indudustry Team 4 - Final Report

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## Math Modeling in Indudustry Team 3 - Final Report

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