For some, spending more than three years working to solve a more than 40-year-old math problem sounds like a nightmare. For University of Missouri-Columbia mathematics professor Steve Hofmann, solving a problem posed by one of the most famous mathematicians in the second half of the 20th Century has been a dream since his college days.

The major mathematical accomplishment is earning him significant recognition. Hofmann has a speaking invitation at this summer鈥檚 meeting of the International Congress of Mathematicians in Madrid, Spain, which takes place only once every four years.

鈥淚t鈥檚 a problem that has interested me since I was a graduate student,鈥� Hofmann said. 鈥淚t was one of the biggest open problems in my field and everybody thought it was too hard and wouldn鈥檛 be solved. I had toyed with it for years and then put in three years of very serious work before hitting the key breakthrough.鈥�

The problem goes back to two papers written by Tosio Kato, University of California-Berkley, in 1953 and 1961. It turned out to be quite difficult and became known as the 鈥淜ato Conjecture鈥� in mathematical circles. The one dimensional version of the problem was solved 20 years later.

鈥淚 think I was the last person working on it,鈥� Hofmann joked. 鈥淚 think everyone else had given up.鈥�

Hofmann admits explaining the problem is difficult because it is rather technical. Its solution applies to the theory of waves propagating through different media, such as a seismic wave traveling through different types of rock. Hofmann said the solution allows mathematicians to better describe the behavior of waves traveling through a medium which itself changes over time.

鈥淭o work on a problem for three years and finally crack it open feels fantastic!鈥� Hofmann said. 鈥淚t鈥檚 the reason mathematicians work on problems 鈥� for moments like that.鈥�

Hofmann鈥檚 work is funded by the National Science Foundation. The solution to the 鈥淜ato problem鈥� is detailed in a series of papers published with his research collaborators Pascal Auscher, Michael Lacey, John Lewis, Alan McIntosh and Philippe Tchamitchian.

Source: University of Missouri-Columbia