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“I think too many students reject mathematics before they know what it is,” says Todd Arbogast, associate director of the Institute for Computational Engineering and Sciences Center for Subsurface Modeling (CSM) and mathematics professor.

For many, says Arbogast, mathematics is a subject defined by cryptic, seemingly stand-alone problems solved by moving around numbers and variables without a clear idea why they should be solved in the first place. Solving math this way is missing the point.

“Basically, that’s so far from what mathematics is about,” says Arbogast, who is also the chair of the institute’s graduate studies committee. “It’s about ideas and relationships between different types of objects.”

A large part of research across ICES’ 16 research centers and groups is uncovering the relationship between mathematics and tangible problems, and in Arbogast’s case, includes both applied and exploratory work. As a member of the applied mathematics group, he studies mathematical models for non-linear flow. And at the CSM, he focuses mathematics on the questions at the forefront of subsurface flow—namely, petroleum production, groundwater contamination, and carbon sequestration.

But forging a formulation that relates in theory as well as in the real world is rarely a pretty process.

“What you have in these types of problems is a physical process and then we have a mathematical description of it; it’s a highly non-linear, coupled system of equations,” said Arbogast. “It’s a very nasty thing.”

Arbogast’s research includes Eulerian-Lagrangian schemes for transport, mixed finite element and mortar techniques for flow, and homogenization and modeling of flow through multi-scale fractured and vuggy geologic media. The overall strategy is to use mathematics to distill complex problems into their essential parts while maintaining accurate results, said Arbogast.

“Approximating flow and simulating it is difficult because you require an extremely fine resolution to do this in a direct way.” said Arbogast. “We use mathematical techniques to try and reduce the amount of information that you need to get an accurate solution.

A notable example of Arbogast using the distillation approach is in modeling the petroleum travelling through microscopic fractures across kilometers of reservoirs—a phenomena that results in significant oil transport, but that is impossible to compute if each and every fracture is taken into account. So, to simulate the fluid flow, Arbogast turned to focusing on the collective result.

“A careful modeling of the essential physics resolves the dilemma and allows us to simulate the flow and visualize the results,” said Arbogast in a Parallel Computing Research newsletter profile.

In addition to research, Arbogast teaches advanced mathematics courses at both the graduate and undergraduate level. The course descriptions and curricula all share a single homepage, with a special section at the foot dedicated to the importance of mathematics’ role in ventures ranging from elections, to the stock market, to space probe navigation. It’s not a trophy hall, however, but a collection of inaccuracies, crashes and explosions—a list of what can happen when mathematical oversight slackens.

“I find [those examples] fascinating because they show why it’s important to understand the mathematics,” said Arbogast. “Because when you don’t, these are the things that go wrong.”

Another important mathematical relationship happens outside of the equation completely: the connection between researchers. Currently, Arbogast is working closely with fellow ICES researcher Marc Hesse, an assistant professor in the Department of Geological Sciences, to understand and simulate the movement of molten rock called magma in the earth’s mantle, the earth’s middle region that separates the crust that form the surface of the planet from the planets core. It's subsurface flow perhaps at its most extreme.

Understanding how the magma forms, how it migrates to the surface, and the changes it undergoes while travelling upward, can help gain insight in major tectonic processes, such as the formation of the ocean floor or so-called hotspots like the one underlying Yellowstone National Park, said Hesse.

“It’s the process that gives planets their first order structure: the crust, the mantle, the core,” said Hesse. “You start at home but there are many fascinating planetary science problems that can be studies with the mathematical models we are building.”

Hesse and Arbogast have been collaborating on questions surrounding magma dynamics for two years. Arbogast and CSEM graduate student Abraham Taicher are working on approximating subsurface equations, and Hesse and Jake Jordan, a geosciences graduate student, are focusing on understanding the underlying physics and chemistry that leads to melting in the Earth’s interior.

“If I didn’t have Marc helping with the underlying physics I wouldn’t have been able to add it all up, at least not easily. And then there are numerical aspects that he would have taken a long time to figure out,” said Arbogast.

The goal is to have agreement between the physics, numerics, and mathematics of the problem, which respectively represent the ‘real world’, the computer simulation, and the mathematical model.

“Trying to understand these three aspects is a part of every problem and it certainly comes into this collaboration,” says Arbogast. The newness of their research can at times the process especially arduous says Hesse, with new methods and programming approaches being tried and tested along the way, but uniqueness of the research makes it a task worth doing.

“There are topics where you publish along the way and then there are problems where you put in a lot of work and at the end you have something that nobody else has done before,” said Hesse. “There are only a handful of research groups who are trying to tackle this problem and that's what I find exciting.”

But Arbogast is used to the “nasty” problems. They may not be as elegant to look at, but they can help understand the relationships that mathematics exists to illuminate in the first place.

“Mathematics as abstract concepts are beautiful in and of themselves,” said Arbogast. “But I think you miss so much of the richness if you don’t realize that it’s connected to the physics and the numerics--the real world.”

*Written by Monica Kortsha*