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Professor Receives Praise for 40 Year Old Problem

Posted by ScuttleMonkey on 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.'"

42 comments

  1. Kernel bounds checking by Anonymous Coward · · Score: 0
    Now if only he had found a constant-time solution to the problem of bounds-checking in kernel and user space.

    That could be coded into silicon and would solve so many problems with viruses and other malware.

  2. A nice little article by Starker_Kull · · Score: 4, Interesting

    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?

    1. Re:A nice little article by alicenextdoor · · Score: 3, Informative
      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 always in two dimensions."

      No, I don't understand it, either! Something tells me this is one of those classic problems that you just can't explain in words of one syllable...

      --
      of course, biting monkeys is not to everyone's taste - Konrad Lorenz
    2. Re:A nice little article by kabocox · · Score: 1

      "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.)

    3. Re:A nice little article by Anonymous Coward · · Score: 0

      "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?"

      Well, seeing since he's at the University of Missouri - Columbia (which I am currently attending), my I would hazard a guess that it has something to do with beer.

    4. Re:A nice little article by Nykon · · Score: 1

      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?

      --
      "It's better to be a pirate then join the Navy"
    5. Re:A nice little article by kabocox · · Score: 1

      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 one. I was the hardest math class that I had ever taken. There was discrete. It was supposed to be a prereq. to abstract alegbra according to the CS professors. Abstract alegbra was fun, but a ton of memorizing of proofs. I wouldn't have gotten through discrete without first having abstract. Discrete was all about relations. I remember taking Cal I as required, but Cal II and Abstract weren't for some reason.

      I went to www.uca.edu if you want to look up what they are currently requiring. I think that CS has always required too much math because half the CS professors just happened to have math degrees. Math in and of itself isn't required for CS. It is if you want a Masters or PHD in CS, but for an undergrad. Nope shouldn't really need much of what they require.

  3. Nobel prize for physics! by Gravis+Zero · · Score: 0

    i gotta say, with all the modern crazy math/physics stuff, i guess this one was a doozie. while there is no Nobel prize for mathematics, there is for physics, which this kinda integrates with... heh... integrates. for a 40 year old problem solved, i think he should get some award. if nothing else, he gets a giant gold star, the size of a football field from me. impressive. now if we can add this info into our 3D video games... :)

    --
    Anons need not reply. Questions end with a question mark.
    1. Re:Nobel prize for physics! by name773 · · Score: 1

      "now if we can add this info into our 3D video games... :)" ...then i'll finally have an excuse to buy new hardware ;)

    2. Re:Nobel prize for physics! by Alarash · · Score: 1
      while there is no Nobel prize for mathematics
      I wouldn't worry too much. Mathematicians have the Fields Medal don't they? That's the equivalent of the Nobel Prize, right?
    3. Re:Nobel prize for physics! by 0xC0FFEE · · Score: 3, Informative

      That's effectively the nearest equivalent except for a few differences. First among them is that the price is given to people _under_ 40. Second the price is given every 4 years. So the Fields is way more difficult to get because of those additional constraints.

    4. Re:Nobel prize for physics! by Anonymous Coward · · Score: 0

      There is instituted a fairly new prize (although the idea for the prize is a hundred years old) called the Abelprize in memory of the Norwegian mathematical genious Nils Henrik Abel. It's awarded by the Abel committee, consisting of an international group of renowned scientists. The prize comes with a large money-award and presented to the prize-winner with royalty and state present just like the Nobel peace-prize.

      Read more:
      http://www.abelprisen.no/en/

      Regards,
      Staalorm

  4. Why applications? by siwelwerd · · Score: 4, Insightful

    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?

    1. Re:Why applications? by ThyPiGuy · · Score: 0

      Because the breakthrough probably comes on the heels of some kind of grant which in one way or another the public pays for, and thus the public would like to know what they're spending money on.

    2. Re:Why applications? by jcwren · · Score: 1

      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.

    3. Re:Why applications? by foniksonik · · Score: 1

      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????

      --
      A fool throws a stone into a well and a thousand sages can not remove it.
    4. Re:Why applications? by mooingyak · · Score: 2, Insightful

      What applications are there for this question?

      --
      William of Ockham had no beard. The most likely explanation is that it was chewed off by squirrels every morning.
    5. Re:Why applications? by Anonymous Coward · · Score: 0
      Is anything really conveyed about the discovery by knowing about a possible application?

      Saying that calculus can be used for predicting the motion of planets or determining volume of irregularly-shaped objects doesn't really convey anything about what it actually is, so how is this any different?

      I think that the desire to know its application is motivated by a desire to understand something of the discovery, but I just don't think that it really conveys any understanding at all.

    6. Re:Why applications? by drsquare · · Score: 1

      Because only the most boring person in the world would find maths interesting.

  5. Not sure what others would apply it to by jd · · Score: 2, Insightful
    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 wave. The aircraft changes (as a medium) with temperature, also because it uses up fuel (and therefore changes in composition and distortion) but also as a result of the waves (stress in the metal, for example). The same will apply to any vehicle, provided there is sufficient change to the vehicle to be significant.
    • Gravity waves: Gravity alters space, space alters the movement of the gravitational bodies, the gravitational bodies alter the waves.
    • Microwave ovens: The microwaves heat the food. In so doing, you change the composition of the atmosphere through which the microwaves travel, thus absorbing some and potentially causing refraction.

    --
    It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
    1. Re:Not sure what others would apply it to by B5Fan · · Score: 1

      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).

      --
      Borg:"Lawsuits are irrelevant. GPL3 is irrelevant. DRM is good. We understand security... Alert! MS are assimilating us!
    2. Re:Not sure what others would apply it to by foniksonik · · Score: 1

      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 medium that doesn't change over time

      --
      A fool throws a stone into a well and a thousand sages can not remove it.
    3. Re:Not sure what others would apply it to by tqft · · Score: 1


      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 is hard - coupled non-linear partial differential equations. if this gives the fusion peop0le a better handle on how to create a self-stabilising intense plasma - in "50 years" we may get our power from Hoffman generators.

      --
      The Singularity is closer than you think
      Quant
  6. Mathematician's are players by isthisorigional · · Score: 5, Funny
    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.

    1. Re:Mathematician's are players by fbjon · · Score: 4, Funny

      Hey baby, would you like to increase my 3-dimensional divergence by heating my kernel bounds?

      --
      True confidence comes not from realising you are as good as your peers, but that your peers are as bad as you are.
    2. Re:Mathematician's are players by Anonymous Coward · · Score: 0
      Now if that doesn't give him a good pickup line, I don't know what will.
      Of course you don't know what will give him a good pickup line. You are posting on Slashdot!
    3. Re:Mathematician's are players by foniksonik · · Score: 1

      Now if he can truly apply "the behavior of [orgasm] waves traveling through a medium that changes over time" to his sex life... word will seriously get around to all those hotties in lab coats.... and he'll be a real player, without having to say a word!

      --
      A fool throws a stone into a well and a thousand sages can not remove it.
  7. Why? It's simple. by John+Nowak · · Score: 3, Insightful

    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.

    1. Re:Why? It's simple. by Hangeron · · Score: 1

      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 but it doesn't help much.

    2. Re:Why? It's simple. by Anti_Climax · · Score: 2, Insightful

      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.

      --
      Even people that believe in pre-destiny look both ways before crossing the street.
    3. Re:Why? It's simple. by mj2k · · Score: 1

      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 nth derivative of what was being approximated. Valuable result? Not if you're interpolating data whose true behavior is unknown. If you knew the function you were approximating to begin with, why waste the enormous computational time to compute a lagrange interpolating polynomial? Point is, there are some mathematical results that while interesting, are not valuable in a real-world sense. I'm not saying this particular result is one of those cases, but questioning the practical significance of a particular find is a valid question. One of my now-retired professors put it this way "Engineers solve problems in order to resolve a physical phenomena, mathematicians and physicists solve problems because they are there". Do engineers have a debt to the mathematicians and physicists for what they have been able to accomplish? Sure, but do the mathematicians and physicists have engineers to thank as well? Well, everytime they drive their car, operate their AC, or turn on a computer, they should. Bottom line is there are two types of people: some can solve mathematical problems for the sake of solving them;some spend their lives applying them, I personally couldn't do the former. These mathematicians are like brick makers. The engineer comes along and looks at the bricks and says, "what can i do with these?". The engineer can do little without the mathematician, but the mathematician's work is not appreciated until someone does the equally difficult job of figuring out how to apply it to something useful.

    4. Re:Why? It's simple. by John+Nowak · · Score: 1

      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.

  8. Could be good by 77Punker · · Score: 1

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

    1. Re:Could be good by Tatarize · · Score: 1

      Yeah! That wouldn't be confusing in the least.

      --

      It is no longer uncommon to be uncommon.
    2. Re:Could be good by 77Punker · · Score: 1

      It's supposed to be confusing. If it wasn't, it just wouldn't be CS.

    3. Re:Could be good by Tatarize · · Score: 1

      Yeah, if it were simple we would all be out of jobs.

      BTW, anybody have a job?

      --

      It is no longer uncommon to be uncommon.
  9. Don't ask what it's good for. by Metasquares · · Score: 2, Insightful

    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!

    1. Re:Don't ask what it's good for. by subtropolis · · Score: 1

      love your sig

      --
      "Our interests are to see if we can't scale it up to something more exciting," he said.
  10. Perhaps he knows Pretty Polly Nomial... by slowhand · · Score: 1

    You can view her experience at http://www-users.cs.york.ac.uk/susan/joke/polly.ht m

    --
    Busy aligning my non-linear thoughts.
  11. parent link by subtropolis · · Score: 1

    naught safe for work

    --
    "Our interests are to see if we can't scale it up to something more exciting," he said.