Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3

Calculating Vision by Bruce Lilly

illustration of eye

“You give me anything, any area, from the stock market to biology, and I’ll show you where partial differential equations appear.”

The voice of Jacob “Koby” Rubinstein, professor of mathematics at Indiana University Bloomington, bursts with enthusiasm as he points at his office door and launches into an explanation of how math figures in door manufacturing. And that’s just the beginning. Farming, emotions, food, clothing—there seems to be no end to Rubinstein’s examples of how mathematics affects our world.

After explaining how math has helped makers of garage doors understand why a certain bar tended to break in the same place over and over again, Rubinstein moves on to economics. “In the stock market, the main tool for the options market is partial differential equations,” he says. “Now, every main brokerage firm is employing mathematicians and physicists to solve partial differential equations arising in the stock market.”

Rubinstein, who recently came to IUB from Technion Israeli Institute of Technology in Haifa, Israel, has made a career out of connecting the ethereal world of higher math to the concrete world in which we live. An applied mathematician, Rubinstein has analyzed problems ranging from the behavior of superconductors at extremely low temperatures (work he has conducted with IUB mathematics colleague Peter Sternberg) to the behavior of human beings in highly complex situations. (One example of his work in the latter area concerns auction theory, a subset of a field known as game theory.) His primary area of research is optics, including the creation of eyeglass lenses.

Intuition might suggest that designs of lenses for microscopes, telescopes, and cameras involve much more complex calculations than creating lenses for a pair of spectacles. But, Rubinstein counters, “even though they look very simple, modern eyeglasses are a very complicated object, because the eye scans in many different directions. With a camera, there’s one lens, and you look straight through it.”

Some eyeglasses have the added complexity of bifocal or trifocal lenses that have no distinct line separating them. Designing and manufacturing such lenses requires the use of partial differential equations. In the case of lenses, solving partial differential equations reveals the complex curves needed in the surface of the lens to produce the optimal refraction of light. Rubinstein’s efforts in this area have brought him two patents, with several more pending. One patent concerns measuring lenses; the other is related to the design of multifocal progressive lenses.

Optics research today remains largely within the domain of industry, but Rubinstein points out that the IU School of Optometry in Bloomington is an exception to the rule. “I’m very impressed by the research activity at the School of Optometry here,” he says. “These people are undoubtedly one of the leading groups in the world.”

He is equally impressed by the Department of Mathematics at IUB, which he joined in 2001. “It’s a really good group of people,” he says, “and they have a very strong tradition of work in applied math. In fluid mechanics, for example, which is one of the most difficult areas in science, Indiana University is one of the strongest places on earth.”

For Rubinstein, the distinction between applied and theoretical research often gets blurry. “It is very difficult to predict what kind of mathematical work is going to be applied,” he says. But the appeal of having a tangible impact drives his research, as it did in his initial work on eyeglasses.

“It was, for me, very attractive,” he says. “I do some mathematics, maybe write some software, and then I have something I can feel, that I can hold in my hand, that will provide a remedy for some eyesight problems.”

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