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# The Golden Age of Math

#### 18th August 2006

There is a legitimate question to ask, I think, with regards to last weeks post about the predictions of Mihail Rocco senior advisor for the nanotechnology to the National Science Foundation. Why is he so confident that fantastic things are going to happen in materials research in five year intervals or so over the next 20 years.

The brief answer is that he is not only confident in the current generation tools and methodologies but he is also confident in the steady improvement of the the current generation tools and methodologies. We’ve already discussed computing power in 5-10 years–and how the the NSF & DOE are funding research to make super computers 1000 times more powerful than today–within 10 years. The second equally important part of this is that the National Science Foundation is funding all kinds of exotic math projects. These math projects form the basis for algorithms which lie at the heart of computer software programs.

This past week press releases on two NSF funded math projects came out which have bearing on desalination membrane/catalyst work. The first math project may have some bearing on sensors that can act in real time to give a complete picture of acivity across the entire surface of membranes/catalysts–with only a limited number of sensors. As Mathematician Robert Ghrist at the University of Illinois at Urbana-Champaign — puts it “Using topological tools, however, we can more easily stitch together information from the sensors to find and fill any holes in the network and guarantee that the system is safe and secure.”

Anyhow here is the PR

## Mathematician uses topology to study abstract spaces, solve problems

CHAMPAIGN, Ill. — Studying complex systems, such as the movement of robots on a factory floor, the motion of air over a wing, or the effectiveness of a security network, can present huge challenges. Mathematician Robert Ghrist at the University of Illinois at Urbana-Champaign is developing advanced mathematical tools to simplify such tasks.

Ghrist uses a branch of mathematics called topology to study abstract spaces that possess many dimensions and solve problems that can’t be visualized normally. He will describe his technique in an invited talk at the International Congress of Mathematicians, to be held Aug. 23-30 in Madrid, Spain.

Ghrist, who also is a researcher at the university’s Coordinated Science Laboratory, takes a complex physical system – such as robots moving around a factory floor – and replaces it with an abstract space that has a specific geometric representation.

“To keep track of one robot, for example, we monitor its x and y coordinates in two-dimensional space,” Ghrist said. “Each additional robot requires two more pieces of information, or dimensions. So keeping track of three robots requires six dimensions. The problem is, we can’t visualize things that have six dimensions.”

Mathematicians nevertheless have spent the last 100 years developing tools for figuring out what abstract spaces of many dimensions look like.

“We use algebra and calculus to break these abstract spaces into pieces, figure out what the pieces look like, then put them back together and get a global picture of what the physical system is really doing,” Ghrist said.

Ghrist’s mathematical technique works on highly complex systems, such as roving sensor networks for security systems. Consisting of large numbers of stationary and mobile sensors, the networks must remain free of dead zones and security breaches.

Keeping track of the location and status of each sensor would be extremely difficult, Ghrist said. “Using topological tools, however, we can more easily stitch together information from the sensors to find and fill any holes in the network and guarantee that the system is safe and secure.”

While it may seem counterintuitive to initially translate such tasks into problems involving geometry, algebra or calculus, Ghrist said, that doing so ultimately produces a result that goes back to the physical system.

“That’s what applied mathematics has to offer,” Ghrist said. “As systems become increasingly complex, topological tools will become more and more relevant.”

A second PR on a NSF funded math project deals with the math of minimal surfaces. I reckon it would be used for modeling and simulation.

For most people, soap bubbles are little more than ethereal, ephemeral childhood amusements, or a bit of kitsch associated with the Lawrence Welk Show.

But for Johns Hopkins University mathematician William Minicozzi, the translucent film that automatically arranges itself into the least possible surface area on the bubble wand is an elegant and captivating illustration of a mathematical concept called “minimal surfaces.” A minimal surface is one with the smallest surface area that can span a boundary.

What does this have to do with membranes?

“Minimal surfaces come up in a lot of different physical problems, some more or less practical, but scientists have recently realized that they are extremely useful in nanotechnology,” he said. “They say that nanotechnology is the next Industrial Revolution and that it has the potential to alter many aspects of our lives, from how we are treated for illness to how we fulfill our energy needs and beyond. That’s why increasing numbers of material scientists and mathematicians are discovering minimal surfaces.”

Anyhow, the rest of the article is here.

A final note. In looking through — and considering — the flow of information on materials research for the last several weeks–it occurs to me that the feds are providing leadership — but not of a kind that’s generally recognized. So it may well be that the exasperation with federal leadership as expressed by the MIT official last week may have come as a result of not understanding what the feds are up to. Certainly it looks from here that the answers to water problems — like those of energy — will come from materials research.

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[…] Finally, it bears mentioning that computers for the first time have given mathematicians an ever more powerful tool. The consequence is that we have entered a golden age for math. […]

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