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Math Science

Professor Receives Praise for 40 Year Old Problem 42

Posted by ScuttleMonkey from the obsessive-compulsive-professors dept.
An anonymous reader writes "The Kansas City Star is reporting that Steven Hofmann is in line to receive accolades from his peers this coming year in Madrid, Spain for solving a mathematical problem that has baffled mathematicians for over 40 years. Hofmann, a professor at the University of Missouri-Columbia, solved the 3 dimensional version of the 'Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds' (say that 10 times fast!). From the article: 'For three years, starting in 1996, Hofmann worked on the problem for two to eight hours every day [...] Hofmann said the solution could allow mathematicians to better describe the behavior of waves traveling through a medium that changes over time. But beyond that, he said, it is impossible for him to explain all the real-world applications.'"
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Professor Receives Praise for 40 Year Old Problem More

Professor Receives Praise for 40 Year Old Problem

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  • A nice little article (Score:5, Interesting)

    by Starker_Kull ( 896770 ) on Wednesday December 28, 2005 @03:17AM (#14350486)
    It really doesn't explain much about the problem, but it does do a nice job of explaining how some people wind up in mathematics:

    "Hofmann majored in math, he said, "because it was the path of least resistance." While his friends were writing history papers that were many pages long or spending hours in a computer lab, "all I had to do was solve math problems, and it was something that came to me naturally," he said.

    "By the time you get to graduate school, even if it comes naturally, it gets hard, and that is when you begin to develop a skill to go with the ability.""

    It's nice to see an article about a mathematician that isn't a "look at the freaky math guy" or "look at the useless thing we're paying people to do" kind of writeup, but just about someone who was enjoyed playing with mathematics, and has done well by it.

    Anyone have a better explanation of what he did or where it fits in? Is it more theoretical or applied? What stuff is it related to?
    • The abstract of the paper in question: "We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly elliptic operator L = div(A) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate Lf2 f2. We note, in particular, that for such operators, the Gaussian hypothesis holds alway
    • But if the summary of "waves in a changing medium" sums it up, then here are a few ideas. (Please note that if the summary I got is inaccurate or incomplete, then none of these examples would apply):
      • Supersonic and hypersonic aircraft design: The shockwave is a wave (duh!) and the medium it travels through (the air) is certainly changing. This applies to the shock going through the air into the surroundings, so could modify models of aircraft noise.
      • Vibrations within any aircraft: The vibrations are also a wa
      • Those seem valid to me.

        Some more:
            Understanding the effects of earthquakes as the shockwaves travel through rock.
            Better design of submarines (water density changes with temperature, salinity and depth).
            Higher-resolution ultrasonic medical scanners (humans vary in (body) density).

        • Explosions of all types - man-made as the explosive material deforms in response to the shockwave or natural as a star rips apart in a supernova.

          Not just earthquakes, but Earth internals by studying the signals earthquakes produce

          Other thinsg in a dynamic medium - perhaps most importnatly for the future - plasma - natural (eg solar wind) or man-made. On ethe problems in fusion development is that plasmas intense enough to be self-sustaining also want to rip apart, and getting a theoretical handle on this
      • Some others:

        - Earthquakes and Tsunamis (understanding them better not predicting them though that could improve as well)
        - Any kind of scanning microscope tech (they use waves of energy and interference patterns for imaging)
        - Radio telescopes (the corollary to scanning microscopes for viewing distant images where the waves of energy generated by the object being imaged)
        - Ultrasound and Sonar devices

        - Most anything that could be improved with more accurate analysis of wave signals since there's virtually no m
    • "Hofmann majored in math, he said, "because it was the path of least resistance."

      It sounds like the same reason that I minored in math. I only had to pick up Cal II and Abstract Algebra and boom I had a math minor. (Well, I did have to take all those other math classes required for a CS major.)
      • How did you have to "pick up" Calc II to minor in math if you were already a CS major? We had ot do through Calc III just to complete first semester of my second year as a CS major?
        • How did you have to "pick up" Calc II to minor in math if you were already a CS major? We had ot do through Calc III just to complete first semester of my second year as a CS major?

          I remember linear alegbra was a 3000 level math class, but all that stupid class was doing matrix math by hand. (It was taught by some 90 year old professor that believed that everyone should be able to muliplty 2 5x5 matrices together by hand without any errors doing basic math.) There was statatics. I could see the use in that

  • Why applications? (Score:4, Insightful)

    by siwelwerd ( 869956 ) on Wednesday December 28, 2005 @04:06AM (#14350594)
    Why is the first question about a mathematical breakthrough always "What are the applications?" Why can people not accept that mathematics is interesting in its own right?

    • Why? It's simple. (Score:4, Insightful)

      by John Nowak ( 872479 ) on Wednesday December 28, 2005 @04:50AM (#14350706)
      When people hear of something like this, oftentimes they can feel threatened that someone is so much more intelligent then they are. (If this is true or not, or if intelligence is even quantifiable doesn't matter -- That's how they're feeling.) As a defense, they pose the question "what is this actually good for". They take comfort in that the answer is "not much", hence allowing them to know that at least they're not wasting their time on such useless nonsense, and no matter how "intelligent" the discoverer is, he's still an "idiot" for "wasting his time" on it.
      • I ask "what is this actually good for" to get even a small understanding of the problem he solved. Sure, it's explained in the abstract of this paper http://www.math.sciences.univ-nantes.fr/edpa/2001/ pdf/tchami.pdf [univ-nantes.fr] but it doesn't help much.
      • To add to that, all too often it takes just as long to find uses for the solution as finding the solution itself. How long did we have Boolean mathematics before they wer put into use for digital compters? More than 70 years.

        If we find more uses for it, great. If not, we have a better collective grasp of pure mathematics.
      • When people hear of something like this, oftentimes they can feel threatened that someone is so much more intelligent then they are.
        Just because a person wishes to ask the relevance of solving a specific equation doesn't make them an idiot compared to the person who solved the problem. Take my numerical analysis class for example: one of my professors took an entire 50 min lecture discussing the error involved with Lagrange interpolation, his final result included an equation that required taking the nt
        • As someone who had a brief stint at Cooper Union, I understand exactly what you're saying. Obviously, my comment was a generalization. Also, I never stated that the person who asked is an idiot compared to the originator -- See the sentence directly after the one you quoted.

    • Maybe because the titles are so esoteric that by defining what it's good for, people can relate to it.

      Just based on the title of Mr. Hofmanns paper ('Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds'), I have *no* idea why it might be useful (I'm also not a mathematician), but when someone says "It could be useful for understanding the effects of earthquakes as the shockwaves travel through rock.", at least I have *some* idea of where it's going.

    • People respond to pure mathematics like they do to religion... it's a mystery that is mostly useless to them in their everyday lives, so you have to tell them how it's going to either make their lives easier or ensure they go to heaven... which one does this do????

    • What applications are there for this question?
    • Because only the most boring person in the world would find maths interesting.

  • Mathematician's are players (Score:5, Funny)

    by isthisorigional ( 527077 ) on Wednesday December 28, 2005 @04:30AM (#14350658)
    Hofmann, a professor at the University of Missouri-Columbia, solved the 3 dimensional version of the 'Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds'

    Now if that doesn't give him a good pickup line, I don't know what will.

  • I'd love it if he came up with a good CS use for this and called it "Hofmann codes".

  • Don't ask what it's good for. (Score:3, Insightful)

    by Metasquares ( 555685 ) <{moc.derauqsatem} {ta} {todhsals}> on Wednesday December 28, 2005 @01:05PM (#14352554) Homepage
    Math is related to itself in so many ways that even the most abstract of problems can have benefits in seemingly unrelated areas. For example, if you can prove a certain bound on the divisor function (lowercase sigma), you'll be able to prove the Riemann hypothesis. These are two seemingly unrelated problems, but solving one will yield a solution to the other.

    There's nothing too impressive about solving a 40 year-old problem, though: Some problems went unsolved for hundreds of years. Still, I can't even understand this problem, let alone attempt a solution at it (and I studied math), so bravo!

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