Mathematician Solves Problem After 30 Years

Newswise — When Dr. Claudio Morales presents his findings on the Leray-Schauder condition to a regional mathematics conference later this month, it will be the culmination of work that started more than 30 years ago.

Morales presents his findings on his home turf when the southeastern section of the American Mathematical Society meets Oct. 24-26 at The University of Alabama in Huntsville.

"Over 30 years ago I became aware of this problem when my Ph.D. advisor wrote about it," said Morales, a professor of mathematical sciences at UAHuntsville. He started work on the problem soon after, but really focused on it after he joined the UAHuntsville faculty.

"When I came here in 1982 I really worked hard to solve this problem," he said. "I never stopped working on it. I didn't work on it constantly, but I never put it completely aside."

Instead, Morales spent the last 26 years working on related problems in an area of mathematics known as fixed point theory on infinite dimensional vector spaces. It was the work in related areas that ultimately led to the solution of the bigger problem.

"Suddenly I was able to put together some ideas that came from my other work," he said. "I realized I could apply what I was working on to my old problem. I was extremely excited that day."

His solution, which was published in a September edition of Proceedings of the American Mathematical Society, is sending ripples across the mathematical world. Mathematicians from Canada, Greece and Japan may come to Huntsville to meet Morales and hear his presentation.

"This is very exciting," he said.

Morales' presentation is scheduled for Saturday, Oct. 25, in a session that begins at 2:30 p.m. in room 103 of UAHuntsville's Shelby Center for Science and Technology.

The problem posed by the University of Iowa's William Kirk in 1975 asked whether there is a fixed point for a nonexpansive mapping defined on a closed bounded and convex set, with nonempty interior under the Leray-Schauder condition. (The answer is: Yes.)

Although the Huntsville meeting is a regional AMS conference, the work behind the 237 scheduled presentations comes from mathematicians in 35 states, the District of Columbia and 21 countries.