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									Reporter’s Notebook:
A Seattle Sampler

                The Seattle Mathfest, held August 10–12, 1996,           ceeded. Frank Morgan of Williams College mod-
                on the campus of the University of Washington,           erated the panel consisting of Jenny Kelley and
                provided an occasion to sample some fine math-           Jean Taylor of Rutgers University and Helen
                ematics while taking in spectacular views of             Moore of Bowdoin College. The program said that
                Mount Rainier. And if three days of talks and            the panel would add their “two cents’ worth”
                panel discussions were not enough, there was a           after Hass spoke, but in fact the panel spoke be-
                symposium immediately after the Mathfest, en-            fore the lecture, during a break midway through,
                titled “In Celebration of the Centenary of the           and also at the end. Such a panel could be dis-
                Prime Number Theorem: A Symposium on the                 ruptive, but as it turned out the different view-
                Riemann Hypothesis”. The symposium, spon-                points and the questions of the panelists helped
                sored by the American Institute of Mathematics           to illuminate the topic.
                (AIM), featured a rare public lecture by Fields             The topic of Hass’s lecture was a variant on
                Medalist Atle Selberg, one of the major figures          the isoperimetric problem, which goes back to
                in the field. (An article about AIM and the sym-         the time of the ancient Greeks. As Virgil re-
                posium is planned for an upcoming issue of the           counts in the Aeneid, Queen Dido cut a deal
                Notices.)                                                with chieftains in North Africa in which she
                    The AMS has decided to get out of the busi-          would be allowed to keep as much land as she
                ness of this kind of summer meeting, at least till       could enclose inside a fence of a given length.
                the year 2000; starting in 1997, the Mathemati-          That her choice of a circular fence was the one
                cal Association of America will be the sole spon-        that gave her the most land was not established
                sor of the Summer Mathfest. The Society will con-        mathematically until the nineteenth century,
                tinue to offer specialized summer seminars and           when it was proven by Weierstrass. The 3-di-
                institutes, but has ended its participation in this      mensional isoperimetric problem poses an anal-
                kind of general summer mathematics meeting.              ogous question: Among all shapes of a given sur-
                The fact that the Seattle Mathfest was quite suc-        face area, which encloses the maximum volume?
                cessful, attracting about 1,200 attendees, must          Or, said differently, given a certain volume, what
                have left a few AMS hearts wistful. What fol-            is the smallest-area surface enclosing the vol-
                lows is a small sampler of the offerings in Seat-        ume? In 1882 Schwarz proved it is a sphere.
                tle.                                                        The question Hass explored is, What is the
                                                                         smallest-area surface enclosing two given vol-
                A Bit of Bubbly                                          umes? Intuitively, it seems it would be a double-
                The Seattle meeting featured a number of ses-            bubble: two identical spheres that meet at an
                sions that broke with traditional formats. A talk        angle of 120 degrees, with a flat disk separat-
                on the “double-bubble conjecture” by Joel Hass           ing their interiors. That this is the case was
                of the University of California, Davis, included         proven last year by Hass and Roger Schlafly of
                a panel that kibitzed on the lecture as it pro-          Real Software of Santa Cruz. Their proof relied

NOVEMBER 1996                                   NOTICES   OF THE   AMS                                               1363
       on work of Michael Hutchings, a graduate stu-           Schlafly’s proof was adaptable to this problem.
       dent at Harvard who participated in the Research        But he did mention that this result should apply
       Experiences for Undergraduates program at               to spaces with metrics that are “close to” the Eu-
       Williams. Hutchings showed that each of the             clidean metric. Hass, Hutchings, and Schlafly
       two pieces of the bubble have to be connected.          are working to generalize their result to bubbles
       Another important piece of information came             enclosing unequal volumes and bubbles in man-
       from joint work of Frank Morgan of Williams and         ifolds.
       Brian White of Stanford, who proved that what-
       ever the most efficient surface was, it had to be       The Ghost of Rochester
       a surface of revolution. The field of solution          Although the crisis at the University of Rochester
       candidates was then reduced to two: the double-         has been resolved, its ghost still haunts the
       bubble or one of a family of “torus bubbles”—a          mathematical community. Last fall the Rochester
       torus bubble is a sphere with a torus around its        administration announced it would cut its math-
       middle.                                                 ematics department faculty by half and eliminate
          The Hass-Schlafly proof was unusual in that          the mathematics graduate program. The move
       a computer carried out some of the critical steps.      produced a strong outpouring of protest, a good
       The torus bubbles can be classified according to        deal of it generated by the AMS Task Force on
       two parameters, an angle and a mean curvature.          Rochester, and by March the administration and
       Using geometric and other arguments, they were          the mathematics department reached an agree-
       able to reduce the possible values of each para-        ment in which the graduate program would be
       meter to a closed interval, so that the the para-       reinstated and the cuts in faculty made less se-
       meter space became a rectangle. All that is left        vere. Despite the relatively happy ending, the fact
       to do is to check the areas and volumes of all of       that a highly regarded research university was
       the candidates to see if any of them is more ef-        seriously questioning the value of having a math-
       ficient than the double-bubble. The problem is,         ematics graduate program left a deep impression
       of course, that there are infinitely many candi-        on the mathematical community.
       dates. How can the computer handle such a cal-             “How Can You Defend Your Graduate Program
       culation?                                               in Mathematics?” was the title of a panel dis-
          Hass explained that their program used what          cussion sponsored by the AMS Committee on Ed-
       is known as “interval arithmetic”, in which the         ucation and organized by committee member
       computer performs calculations not on float-            Harvey Keynes of the University of Minnesota.
       ing-point numbers but on intervals with end-            The three panelists took rather different tacks
       points that are floating-point numbers. For each        on the question. John B. Conway, head of the
       2-dimensional interval in the parameter space,          mathematics department at the University of
       the program produces an interval of values of           Tennessee at Knoxville, said he is not convinced
       the volumes of the torus bubbles represented by         that graduate programs are under attack. “If
       that interval. If the range of the volumes of each      Rochester is n = 1 , then what is n = 2 ?” he asked.
       component are unequal, the program tosses out           He pointed out that it is very difficult to ferret
       that interval, since only equal volume solutions        out why the Rochester crisis occurred; one would
       are being sought. Other tests are used if volume        have to examine the situation two, three, or even
       comparison does not apply. The remaining in-            ten years earlier to understand how it came to
       tervals are then subdivided again, and the same         pass. Nevertheless, he had a number of sugges-
       elimination is carried out. This process produces       tions to offer to protect one’s graduate pro-
       strict upper and lower bounds on the volumes            gram: teach calculus better, examine precalcu-
       of each component of the torus bubble. From             lus, improve the undergraduate major program,
       these and similar bounds, the proof follows.            and, finally, connect to engineering and science
       (For an expository presentation of this result, see     departments. “If the crunch comes and darkness
       “Bubbles and Double Bubbles”, by Joel Hass and          is at your door,” he said, “these are the only al-
       Roger Schlafly, American Scientist, September-Oc-       lies you can have.”
       tober 1996, pages 462–467. This and related pa-            Bus Jaco of Oklahoma State University saw a
       pers may be found on the World Wide Web at              number of threats to mathematics graduate pro-                       grams. One of them is the mathematical com-
       bubbles.html.)                                          munity itself: overproduction of mathematics
          The panel asked Hass if this method would            Ph.D.s has fueled comments that the lesser lights
       work for the analogous “triple-bubble” problem,         among graduate programs should close up shop.
       which is still open. Could it be three spheres          For Oklahoma State, which has been improving
       stuck together, or maybe a double-bubble with           in recent years but finds its backwater image
       a torus around its “waist”? It is not even known        hard to shed, such comments hit close to home.
       whether these are the only possible candidates.         Jaco suggested that some sort of national ac-
       Hass said he did not know whether his and               creditation might be in order, though he op-

1364                                  NOTICES   OF THE   AMS                              VOLUME 43, NUMBER 11
poses “Ph.D. birth control” on a national scale.      specific skills such as Java programming—and
He also suggested that, rather than large changes     the salaries far exceed that of an assistant pro-
in the graduate program, what may be needed           fessor. Being willing to take advantage of such
is a change in faculty attitude, with more atten-     opportunities could give students a very differ-
tion paid to such things as exploring career op-      ent perspective on graduate school in math-
tions for new Ph.D.s, discussing professional         ematics. In his response Conway took a more tra-
and teaching issues with students, and insuring       ditional tack. A while back he realized that most
that students get early experiences in math-          of the Ph.D.s from his department were getting
ematics research.                                     jobs at four-year institutions. “Our graduate pro-
   Echoing this idea was William Rundell, chair       grams have ignored this reality,” he noted, as
of the mathematics department at Texas A&M            they tend to focus on preparing students for po-
University. Rundell’s vision is for mathematics       sitions in research institutions. To address this
departments to “own the boundary” of the dis-         problem, Conway is trying to set up a program
cipline so that, for example, mathematics majors      whereby his students can teach at local com-
are prepared to enter Ph.D. programs in a vari-       munity colleges to get some experience that
ety of areas. Unfortunately, many departments         might help them in landing jobs later on. For the
pay scant attention to undergraduates and sim-        mathematical community more generally, the
ply complain about their academic weaknesses.         difficulty is that no one wants to give up any part
Rundell pointed out that, judging by SAT or GRE       of the graduate program to do anything differ-
scores, other departments see mathematics ma-         ent. Said Conway, “We are sometimes our own
jors as “Rolls Royces”. “We complain about our        worst enemies.”
students as being miserable,” he said. “But if we
do that, we’ll look foolish.” One result is that      Whimsical Mathematics
mathematics loses many students to manage-            Mathematical objects usually have names like x ,
ment and engineering.                                 Q , or maybe ℵ, if you want to get fancy. For Colin
   One of the most pernicious threats to grad-
                                                      Adams and Edward Burger, these plain-Jane
uate programs in mathematics is simple eco-
                                                      names are not good enough. At their talk in
nomics. During his presentation Rundell quoted
                                                      Seattle, these two Williams College mathemati-
figures about the cost-per-student-credit-hour of
                                                      cians favored names like Bubba, Bosco, Olive, and
certain courses offered at Texas A&M. He im-
                                                      Carlo. One object was likened to a tumor, while
plored the audience not to write them down,
                                                      a deformed torus was called a “quasimodonut”.
lest the figures end up in the hands of his state
                                                      Just what were they doing with all this whimsy?
legislature. Suffice it to say that the difference
                                                         Believe it or not, they were proving theorems.
between the cost-per-student-credit-hour of cer-
                                                      For their AMS-MAA Joint Invited Address, Adams
tain business courses and that for graduate
                                                      and Burger wrote, produced, and starred in a play
mathematics courses is two orders of magnitude.
                                                      called “Casting About: About Casting”. The char-
With the popularity of the business major and
the employment troubles of mathematics Ph.D.s,        acters, Sam and Buddy, were workers at the
the mathematics graduate program can seem             Acme Casting Factory, a metal casting plant in
the right place to cut. “Despite the prevalence       Allentown, Pennsylvania. The show opened with
of overall departmental cost/student data, these      slides of Adams and Burger getting ready for
particular figures are rarely asked for,” says        work, complete with hardhats and plaid shirts,
Rundell. “They could be dangerous in the hands        while the song “Allentown” by Billy Joel played
of those looking for simplistic solutions.”           in the background. After this introduction, the
   During the question period, one member of          two appeared onstage in their workers’ getup.
the audience said that after hearing the pan-         As they chatted during their lunch break, the
elists he wasn’t sure he was at the right session;    mathematics crept in slowly: amid talk about
he thought he had come to hear about how to           work at the plant Sam mentions a new treat at
defend one’s graduate program against com-            the local donut shop called a “glazed handle-
plaints of the students. Many of his students         body”.
have to moonlight during graduate school, he             The jokes flowed fast and corny, inspiring
said, and once they finish they cannot find jobs.     roars of laughter among the folks packing the
Some he hires on as instructors, and they take        800-seat auditorium. The humor was tailor-made
the jobs because they pay marginally better than      for this audience: where else would talk of down-
the community college across town.                    sizing at the “Rochester casting plant” have even
   Rundell had one question to ask: How much          caused a titter?
can these students program in Java? Currently,           BUDDY: (Stunned) What?! (Pause) Rochester
he notes, there is a real demand in major com-        downsized?
puter software industries for a combination of           SAM: Yeah. The company almost cut the en-
traditional mathematical sciences training and        tire casting division, but with some pressure

NOVEMBER 1996                                        NOTICES   OF THE   AMS                                 1365
                                                                           result, which follows from the theorem proved
                                                                           after the intermission, says that 3-space can be
                                                                           tiled with knotted tori. Saying that he read about
                                                                           the result in Better Homes and Gardens, Buddy
                                                                           claims that this method of tiling space is all the
                                                                           rage for decorating bathrooms. Indeed, why tile
                                                                           just the bathroom floor when you can tile the
                                                                           whole space?
                                                                               BUDDY: So now the entire bathroom is filled
                                                                           with these knotted doughnut shaped tiles.
                                                                               SAM: I love it. Seems totally pointless, but I
                                                                           love it. We should make and sell tiles that look
Williams College professors Edward Burger (left) and Colin                 like that.
Adams (right) in character and costume as Buddy and Sam from                   BUDDY: Hey, you know what we should do?
their math play “Casting About: About Casting”.                            The next time there’s a big math conference, we
                                                                           could go there and sell these knotted tiles! Heck,
                                                                           they went for those silly bronze tetrahedra;
                  from the MAA and the AMS, they decided to                they’d gobble these things up.
                  back off.                                                    SAM: Hey, yeah, but I got a better one. [Laugh-
                      BUDDY: The MAA?                                      ing] Maybe at their next conference, we could go
                      SAM: Yeah, the Metalworkers Association of           and give a presentation.
                  America.                                                     BUDDY: Yeah, right [laughing]; you and me
                      BUDDY: Oh, yeah, and the American Molders            talking to an auditorium filled with mathemati-
                  Society. Boy, those are powerful organizations,          cians about casting and tiling. That’d be a good
                  I’m telling you.                                         one. Come on, lunch is over; we’d better get
                      Well, you had to be there.                           back to work.
                      Before this sort of thing could wear too thin,
                  Sam and Buddy start discussing what kind of ob-                                            —Allyn Jackson
                  jects one can create from a two-piece casting
                  mold (with each mold deformable into a ball).
                  Buddy bets Sam $5 that the only possibilities are
                  a ball, a donut, a “glazed handlebody”, or “an
                  apple with wormholes”. The two stay in charac-
                  ter, with Sam naming the object to be cast after
                  himself and the two parts of the mold “Bubba”
                  and “Bosco”. He asks skeptical questions as
                  Buddy goes through the proof, and the result is
                  an intuitive explanation that is perhaps more un-
                  derstandable than a more straightforward lec-
                      After a 5-minute intermission, Sam bets Buddy
                  $5 that he can prove that one can cast anything
                  using a 3-piece mold (provided that the object
                  has only one boundary component). This time
                  Sam goes through the proof of this rather sur-
                  prising result, with Buddy asking the questions.
                  The two theorems proved in the play are new re-
                  sults by Adams and Burger. At one point one of
                  them says somewhat disappointedly that the re-
                  sults are all “theoretical”. “Imagine if that’s what
                  you did all day, just sitting around chewing the
                  cud on theoretical nonsense?” laughs the other.
                  “Yeah, and what if they actually paid you to do
                  it? Ah, we’re being silly!”
                      Buddy and Sam also discuss a third result by
                  Adams about tiling of 3-space. First they discuss
                  tiling space by tetrahedra (“Tetrawhatdra?” one
                  of them asks) and reminisce about when Acme
                  Casting made bronze tetrahedral mementoes
                  for a mathematics meeting. Adams’s surprising

1366                                              NOTICES   OF THE   AMS                             VOLUME 43, NUMBER 11

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