Applied Mathematics

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.
 

• Lecture 1 (of 2)
• Lecture 2 (of 2)

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)

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.

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.

In this paper we study the dynamics of a population where the individuals can either be contributors (tax payers) or no contributors (tax evaders or cheaters). We introduce a 2D cellular automaton on which the probability of transition from one of the above states to the other is the sum of the local effect and of the global field effect. The model also includes the policy that allocates a fraction of the budget to fight tax evasion. This scheme allowed us to simulate the cases in which inhomogeneous strategies in contrasting tax evasion is applied in a region and the case in which cooperative policies are adopted by neighbor societies.