Conference rusure::math

180.0. "Hopf's Conjecture solved?!" by PIXEL::PWONG () Fri Nov 16 1984 20:16

    Associated Press Fri 16-NOV-1984 02:52 		     Math Puzzle

         Professor's Study Solves Math Mystery

         TOLEDO, Ohio (AP) - A mathematical mystery four decades old has
    been solved by a University of Toledo professor, who has found a new
    geometric "shape" that no one has seen.

         Henry Wente,  like  many  of  his  colleagues,  long  has  been
    fascinated  with Hopf's Conjecture, a puzzle named after Heinz Hopf,
    the Swiss who presented it in the late 1940s.  It involves  geometry
    and the curvature of shapes, or surfaces.

         Wente, 48, said he has been aware of the puzzle since he was  a
    graduate  student  at  Harvard  University  in the 1960s, but didn't
    begin to tackle it seriously until about five years ago.

         He was reluctant to devote much time to what  he  considered  a
    possible  "wasted  effort,"  but  his  studies increased to up to 20
    hours a week, squeezed between teaching calculus classes,  plus  all
    his vacation time.

         In the last couple of years, he worried about rumors that other
    mathematicians might beat him to a solution.

         "I was afraid to say anything.  I was afraid to  get  scooped,"
    said  Wente, who has taught at Toledo since 1971.  "I guess I wanted
    to be the first one to get it."

         To understand  the  puzzle,  one  must  imagine  a  bubble.   A
    mathematician  could  measure the bending of the bubble's surface at
    any two points and would get the same measurement, called a constant
    mean curvature.

         Hopf believed that a sphere was the only surface that had  such
    a  property.   But  Wente  proved him wrong by discovering - through
    mathematics - another surface that no one ever had come up with  and
    which even mathematicians have a difficult time visualizing.

         He began with a mathematical object called a torus -  something
    he  described  as  a bulging donut shape, or a sphere with a hole in
    it.

         The torus' surface does not have a constant  curvature  like  a
    sphere,  but  through  drawings  and calculations, Wente was able to
    "stretch" the torus into a new surface with constant mean curvature.

         "I expect it will look something like a necklace  of  raindrops
    ..  It must be beautiful - that's an article of faith.  And I have a
    hunch that it will be useful," said David Hoffman, a math  professor
    at the University of Massachusetts, Amherst, who is trying to design
    a computer graphic of  Wente's  surface  so  it  can  be  understood
    visually.

         Hoffman and other colleagues say Wente's discovery will provide
    math researchers and scientists with new ways to solve problems.

         "Ten years from now, a lot  of  things  will  be  very  clear,"
    Hoffman  said.   "It  might  lead  to  dozens  of  other interesting
    things."

         Wente's  65-page  solution  to  Hopf's  Conjecture  is  to   be
    published next year in the Pacific Journal of Mathematics.

         "I don't know if there was a  certain  day  when  I  could  say
    `Eureka!'  but certain calculations fell into place perfectly and at
    the time I felt, `Gee, it should work,' " he said.