# Applied Mathematics

## Financial System Architecture

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

## Oceans and Multiplicative Ergodic Theorems

In many physical processes, one is interested in mixing and obstructions to mixing: warm air currents mixing with cold air; pollutant dispersal etc. Analogous questions arise in pure mathematics in dynamical systems and Markov chains. In this talk, I will describe the relationship between obstructions to mixing and eigenvectors of transition operators; in particular I will focus on recent work on the non-stationary case: when the Markov chain or dynamical system is non-homogeneous, or when the physical process is driven by external factors.

I will illustrate my talk with analysis of and data from ocean mixing.

## Mathematics and the Planet Earth: a Long Life Together II

When Colombus left Spain in 1492, sailing West, he knew that the Earth was round and was expecting to land in Japan. Seventeen centuries earlier, around 200 BC, Eratosthenes had shown that its circumference was 40,000 km, just by a smart use of mathematics, without leaving his home town of Alexandria. Since then, we have learned much more about Earth: it is a planet, it has an inner structure, it carries life , and at every step mathematics have been a crucial tool of discovery and understanding. Nowadays, concerns about the human footprint and climate change force us to bring all this knowledge to bear on the global problems facing us. This is the last challenge for mathematics: can we control change?

This is a two-part lecture, investigating how our idea of the world has influenced the development of mathematics. In the first lecture on July 15, I will describe the situation up to the twentieth century, in the second one on July 17 I will follow up to the present time and the global challenges humanity and the planet are facing today.

## Mathematics and the Planet Earth: a Long Life Together I

When Colombus left Spain in 1492, sailing West, he knew that the Earth was round and was expecting to land in Japan. Seventeen centuries earlier, around 200 BC, Eratosthenes had shown that its circumference was 40,000 km, just by a smart use of mathematics, without leaving his home town of Alexandria. Since then, we have learned much more about Earth: it is a planet, it has an inner structure, it carries life , and at every step mathematics have been a crucial tool of discovery and understanding. Nowadays, concerns about the human footprint and climate change force us to bring all this knowledge to bear on the global problems facing us. This is the last challenge for mathematics: can we control change?

This is a two-part lecture, investigating how our idea of the world has influenced the development of mathematics. In the first lecture (July 15), I will describe the situation up to the twentieth century, in the second one (July 17) I will follow up to the present time and the global challenges humanity and the planet are facing today.

## Strong Oracle Optimality of Folded Concave Penalized Estimation

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimal solution computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution using the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely produces the same estimator in the next iteration. We show that the LASSO is a good initial estimator, which produces the oracle estimator using the one-step LLA algorithm for folded concave penalization methods. This is demonstrated by using three classical sparse estimation problems, namely, the sparse linear regression, the sparse logistic regression and the sparse precision matrix estimation, and illustrates the power of combining the LASSO and SCAD to solve sparse inartistical estimation problem. (joint work with Lingzhou Xue and Hui Zou)

## High Dimensional Data Analysis

Event page at: http://www.pims.math.ca/scientific-event/130523-iwpohddai

## A Computational Mathematician Combusts

Large scale production of very heavy oil is gaining momentum because of the decline of easy to produce reservoirs, the increasing oil demand and subsequent rising oil price, which makes such resources more economical. Considering the pressure on the oil market and our still very heavy dependence on oil, this move to heavy oil production seems inevitable. Typically, heavy oil reservoirs are stimulated thermally. Injecting steam that is generated at the surface is not always viable or desirable. An alternative technique for production is In-Situ Combution (ISC) where a steam drive is generated in the reservoir itself. In this process, (enriched) air is injected in the reservoir. After ignition a combustion front develops in-situ that burns a small percentage of the oil in place and slowly moves through the reservoir producing steam along the way. A side benefit of this process is that the heat thus generated often cracks the oil into heavy, undesirable components (the "guck") that stay behind in the reservoir and lighter, more valuable components that can be brought up to the surface. Performance prediction of ISC projects is rather tricky and poses many computational challenges. In this talk I'll discuss our work in ISC simulation, which is centered around the design of upscaling methods for kinetics and critical reservoir heterogeneities supported by laboratory experimentation.

## Pumps, Maps and Pea Soup: Spatio-temporal methods in environmental epidemiology

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*Further information about the Constance van Eeden Invited Speaker Program*

This talk provides an introduction to epidemiological analysis where the distribution of health outcomes and related exposures are measured over both space and time. Developments in this field have been driven by public interest in the effects of environmental pollution, increased availability of data and increases in computing power. These factors, together with recent advances in the field of spatio-temporal statistics, have led to the development of models which can consider relationships between adverse health outcomes and environmental exposures over both time and space simultaneously.

Using illustrative examples, from outbreaks of cholera in London in the 1850s, episodes of smog in the 1950s to present day epidemiological studies, we discuss a variety of issues commonly associated with analyses of this type including modelling auto-correlation, preferential sampling of exposures and ecological bias. The precise choice of statistical model may be based on whether we are explicitly interested in the spatio-temporal pattern of disease incidence, e.g. disease mapping and cluster detection, or whether clustering is a nuisance quantity that we need to acknowledge, e.g. spatio-temporal regression. Throughout we consider the practical implementation of models with specific focus on inference within a Bayesian framework using computational methods such as Markov Chain Monte Carlo and Integrated Nested Laplace Approximations.

The talk also serves as a precursor to a graduate level course on spatio-temporal methods in epidemiology. This course will cover the basic concepts of epidemiology, methods for temporal and spatial analysis and the practical application of such methods using commonly available computer packages. It will have an applied focus with both lectures and practical computer sessions in which participants will be guided through analyses of epidemiological data.

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**BACKGROUND INFORMATION:** The Statistics Department, with the support of the Constance van Eeden Fund, is honoured to host Dr Gavin Shaddick during term 2 2012-13. Dr Shaddick, a Reader in Statistics in the Department of Mathematical Sciences at the University of Bath, has achieved international prominence for his contributions to the theory and application of Bayesian statistics to the areas of spatial epidemiology, environmental health risk and the modelling of spatio-temporal fields of environmental hazards.

Dr Shaddick will begin his visit to the Department, by giving the 2012-13 van Eeden lecture. That lecture will inaugurate a one term special topics graduate course in statistics, which the Department of Statistics is offering next term. It will be given by Dr Shaddick and Dr James Zidek (Statistics, UBC) on the subject of spatial epidemiology. This course, which is aimed primarily at a statistical audience, will provide an introduction to environmental epidemiology and spatio-temporal process modeling, as it applies to the assessment of risk to human health and welfare due to random fields of hazards such as air pollution. Please see the course outline for more information.

## An Octahedral Gem Hidden in Newton's Three Body Problem (2012 Marsden Memorial Lecture)

Richard Montgomery, University of California, Santa Cruz will deliver a talk entitled, "An Octahedral Gem Hidden in Newton's Three Body Problem." The lecture will take place on July 25, 2012 at the Fields Institute, as part of the conference on "Geometry, Symmetry, Dynamics, and Control: The Legacy of Jerry Marsden."

Richard Montgomery received undergraduate degrees in both mathematics and physics from Sonoma State in Northern California. He completed his PhD under Jerry Marsden at Berkeley in 1986, after which he held a Moore Instructorship at MIT for two years, followed by two years of postdoctoral studies at University of California, Berkeley.

Montgomery's research fields are geometric mechanics, celestial mechanics, control theory and differential geometry and is perhaps best known for his rediscovery - with Alain Chenciner - of Cris Moore's figure eight solution to the three-body problem, which led to numerous new 'choreography' solutions. He also established the existence of the first-known abnormal minimizer in sub-Riemannian geometry, and is known for investigations using gauge-theoretic ideas of how a falling cat lands on its feet. He has written one book on sub-Riemannian geometry.

The PIMS 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 1970s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.

The inaugural Marsden Memorial Lecture was given by Alan Weinstein (University of California, Berkeley) in July of 2011 at ICIAM in Vancouver.

## Predicting Criminal Incidents Using Geographic, Demographic, and Twitter-derived Information

Predictive policing seeks to anticipate the times and locations of crimes to better allocate law enforcement resources to combat these crimes. The key to predictive policing is modeling that

combines available data to forecast or estimate the areas most threatened by crimes at different times. We have developed models that integrate geographic, demographic, and social media information from a specific area of interest to produce the needed predictions. In this presentation, I describe our approach to this predictive modeling, which combines spatial-temporal generalized additive models (STGAM) with a new approach to text mining. We use the STGAM to predict the probability of criminal activity at a given location and time within the area of interest. Our new approach to text mining combines Latent Dirichlet Allocation (LDA) with Latent Semantic Indexing (LSI) to identify and use key topics in social media relevant to criminal activity. We use social media since these data provide a rich, event-based context for criminal incidents. I present our application of this approach to actual criminal incidents in Charlottesville, Virginia. Our results indicate that this combined modeling approach outperforms models that only use geographic and demographic data.