Sixty Year Old Problem Solved

Randall Dougherty (left) and Matthew Foreman standing before part of the construction used in the proof of their result.

Randall Dougherty and Matthew Foreman have recently solved a problem that has been outstanding for over sixty years. Both Dougherty and Foreman are current members of the Department of Mathematics.

One of the most counterintuitive results of mathematics is the "Banach-Tarski Paradox" which states that the unit ball in three dimensional space can be partitioned into 5 pieces that can be rearranged by isometries to produce two(!!) disjoint unit balls. The "paradox" of this result is usually explained away by saying that the pieces are non-measurable and the construction is suspect because it uses the Axiom of Choice.

Dougherty and Foreman have produced a "paradox" with open sets that does not use the Axiom of Choice. Specifically, they have proved that there are 6 disjoint open subsets of the unit ball that can be rearranged by isometries so that their closure is two disjoint unit balls. In fact, if B is any bounded subset of three dimensional space which is the closure of its interior (i.e. an arbitrary large cube), then the unit ball contains a finite collection of disjoint open sets that can be rearranged by isometries so that their closure is B. Their result solves the "Marczewski Problem" dating from about 1930.

The precise result they have achieved is that the unit sphere can be decomposed into 6 disjoint pieces with the property of Baire which can be rearranged by isometries to be two disjoint unit spheres. To say the pieces have the property of Baire means that the pieces are in the sigma algebra generated by the Borel sets and the first category set. This sigma algebra is in many ways analogous to the algebra of measurable sets.

[For more information on the Banach-Tarski paradox and related matters see: Wagon, The BanachTarski Paradox, Cambridge University Press, 1985.]

Carroll and Parson Receive ASC Awards

Frank Carroll and Alayne Parson were recognized for their contributions to undergraduate education through recent awards from the Colleges of the Arts and Sciences.

The 1989 Honors Faculty Service Award was given to Alayne Parson for her continued contributions to the department, ASC, and University Honors programs. The award recognizes her for excellence in honors advising, honors instruction, honors committee work, and her general initiatives which have advanced undergraduate honors education in the arts and sciences.

Frank Carroll has received the 1989 Rosalene Sedgwick Faculty Service Award. The award recognizes his work as an undergraduate advisor, his service to the department as Director of Upper Division Studies, and his broad based service to ASC and the University on many committees related to curriculum issues, admissions, and registration.

Casian and Silverberg Selected Sloan Research Fellows

Luis Casian and Alice Silverberg have been selected as Sloan Resarch Fellows. Selection under this program, sponsored by the Alfred P. Sloan Foundation, follows a well-developed procedure designed to Sloan Awards (continued from page 1) identify young scholars who show great promise for doing original work in their field. The Sloan Research Fellowship program was begun in 1955 as a means of encouraging basic research and, over the years, the winners of these awards have made major contributions in their disciplines. There were 90 young scientists selected to receive fellowships this year, twenty of these were awarded in mathematics.

Casian is currently studying certain relationships that exist between representations of Lie algebras and the geometry of flag manifolds. The idea is that the structure of some important Lie algebra manifolds is describable in terms of the intersection cohomology of some singular manifolds.

Silverberg works in the areas of arithmetic algebraic geometry, number theory, and automorphic forms. At present, her interests include the study of abelian varieties over fields of characteristic zero and canonical models of arithmetic quotients of bounded symmetric domains.

Comments from the Chair

As is noted elsewhere in this newsletter, the byline on this column will change in the next issue of the Math Matrix, as I return to my regular faculty duties and a new chairperson is appointed. Though I am anxious to return to a, for me, more normal life on the faculty, I must admit to having enjoyed the stimulation of four years as chairperson. The varied activities of 100 faculty members, 250 Graduate Teaching Associates, and thousands of students presents an ever varying panorama with daily (or hourly) high and low points. For my closing column, I want to mention some of the high points of the 1989-90 year. These fortunately, exceeded the low points by a comfortable margin. If you think back to your calculus, you'll note that this statement implies that we have here a function which must have successive maxima with no minima in between. Also a mathematical high point! Two of our undergraduates were awarded National Science Foundation Fellowships to provide support for continued study in graduate school; departmental faculty swept the two faculty service awards presented by the Colleges of Arts and Sciences; two faculty members were among the finalists in the university-wide competition for the Alumni Distinguished Teaching Award; nine mathematics GTA's are finalists for the Outstanding Teaching Associate Award; three faculty members have been invited to address the 1990 International Congress of Mathematicians (the remaining nine Big Ten institutions received a total of five invitations); and two young faculty members were awarded Sloan Research Fellowships. You can imagine, I think, that a string of recognitions like this makes a chairman's adrenaline flow a bit faster, particularly when it is clear that many others are also contributing at similar high levels to fulfill our teaching, research, and service missions in a uniformly outstanding manner. The groundwork leading to this year's strong performance was in place well before I became Chairman. I hope that the efforts of the past four years will bear similar fruit a few years hence.

Actuarial Science Students Form Club

The newest undergraduate club is the Actuarial Club at Ohio State. The group formed this Spring Quarter as a result of increased interest, and increasing numbers, in the major in Actuarial Science. At present, there are about 111 undergraduates majoring in the Actuarial Science program in the department. The purpose of the organization is to develop in members an understanding of the actuarial profession and influence the conditions under which members are being prepared for their professional role in society. The organizational meeting of the Club attracted 50 attendees. The officers are Charles Bremer, President, John Bilancini, Vice President, David McLeroy, Treasurer, and Carol Ziccardi, Secretary. The Club's advisor is Robert B. Brown.

Gary Barhorst MA '87, currently on the faculty at Springfield South High School, recently received an "Excellence in Teaching Award" cosponsored by the Springfield/ Clark County Rotary Club, The Springfield News and Sun, and the Springfield Foundation. Gary received the award upon nomination of his students.

Marbue Brown, MA '85, Master of Applied Statistics '86, currently with the Measurement Research Group at Belicore, Piscataway, NJ.

Mark Rosenstein, B.A. '80, M.A. '83 (Economics) from UCSD, currently a member of the technical staff at Bell Communications Research.

Douglas T. Yost, B.S. '86, currently regional loss control manager at Wendy's International.

Campus Campaign

There is a continuing need for your support of the mathematics profession. Not only do we ask your help in identifying good students, undergraduate or graduate, to come to Ohio State for study, but also your financial support of various programs in the department is needed as well. There are several scholarship funds that provide opportunities to encourage students in their study of mathematics. The Undergraduate Support Fund and the Graduate Student Support Fund give small grants to students for attending national, regional, or local professional meetings. These funds are also used for social achvities like the Summer picnic and various receptions during the year. The Undergraduate Recognition Program each spring honors undergraduates who have received various awards from the department of the University in the past year. This program, which supports and encourages undergraduates to continue in the profession, receives its financing through alumni contributions.

The University is now starting its Campus Campaign. As Ohio State moves toward greater distinction, becoming a dominant public university, contributions from faculty, staff, and alumni will play a vital role in providing a margin of excellence. When you are contacted by the University (or even before) we hope you will recall your time here in the Department of Mathematics and choose to give generously. You can designate your gift directly to one of the Department of Mathematics accounts. In fact, there are two new funds which have just been inaugurated -- Arnold E. Ross Mathematics Teaching Award and Scholarship Fund and the Hans and Lieselotte Zassenhaus Mathematics Dissertation Fellowship Fund.

If you wish to support the department in this more visible way, your contributions can be made payable to The Ohio State University Development Fund and sent to Editor, Math Matrix, Department of Mathematics, 231 West 18th Ave., Columbus, Ohio 43210-1174. If you wish to designate a particular use, just note that on the check.

Recognitions for Graduate Associates

To accomplish the variety of teaching obligations of the department, the work of Graduate Teaching Associates is vitally important. The 180 graduate students serving in classroom assigned duties perform their tasks with a great deal of professional competence. Each year undergraduate students nominate several of our graduate students for the honor of the Graduate Associate Teaching Award. This year the department is pleased to recognize James Braselton, Diane Boring, Philip Green, Sarnya Johnson, Fong Chung Lee, Russell May, Barry Spieler, and Ivan Weisz who were so honored by their students. In addition, two other students who teach for the department, György Emese and Jeffrey Smith, who received MA degrees from the department and are now studying for PhD's in mathematics education, were also among the nominees. The department offers particular congratulations to Philip Green and György Emese who were named among this years' winners of the Graduate Associate Teaching Award. There are ten such awards made each year and the above named students were selected from among 80 students so nominated.

Congratulations are also extended to Fernando Rodriguez-Villegas, a PhD candidate from Argentina, who has been awarded a Presidential Fellowship. This extremely competitive award recognizes Mr. Rodriguez' outstanding research ability and promise. His area of research interest is algebraic number theory, and he is studying with Warren Sinnott.

A New Preprint Series

The Research Insitute is producing a preprint series, the OSU Mathematical Research Institute Preprint Series, for the distribution of research connected with the department and the Institute. The series has already produced the following titles.

1. Andrzej Derdzinski, Geometry of the Standard Model of Elementary Particles, Part I: Particle Models, Seminar Notes, May-August 1989.
2. Andrzej Derdzinski, Geometry of the Standard Model of Elementary Particles, Part II, Unifications, Seminar Notes, May-August 1989.
3. Boris N. Apanasov, Deformations of Conformal Structures on Hyperbolic Manifolds
4. R. Meyerhoff and Waler D. Neumann, An Asymptotic Formula fo the Eta invariants of Hyperbolic 3-Manifolds.
5. Walter D. Neumann and Alan W. Reid, Arithmetic of Hyperbolic 3-Manifolds.
6. Alice Silverberg, Representations of Galois Groups Associated to Canonical Models.
7. Alice Silverberg, Canonical Models and Adelic Representations.
8. Karl Rubin, Stark Units and Kolyvagin's Euler Systems.

If you are interested in any of the above titles, please contact Walter D. Neumann at the department's address.

AMS/MAA Joint Summer Meetings

The Joint Summer Meetings of the American Mathematical Society and the Mathematical Association of America will be held on Ohio State's campus August 8 - 11, 1990. The meetings will celebrate the 75th Anniversary of the founding of the Mathematical Association of America with a special day of activity on Wednesday, August 8. The Association was founded, and held its first meetings, on Ohio State's campus December 30-31, 1915. Commemorative plaques will be dedicated at Page Hall and the Department of Mathematics. All alumni of Ohio State are encouraged to make a special effort to attend these meetings for a grand reunion.

Problem Corner

(Material for this column is prepared by Gerald Edgar)

Another Chance to try your problem solving skills! Send your solutions to the Editor. The best (or most interesting) solutions will be published in a future issue of Math Matrix. You are encouraged to submit problems for inclusion in this corner. Problems with an applied flavor are especially welcome.

Problem 15. Two people have 8 quarts of wine which is in an 8 quart jar. They also have an empty 5 quart jar and an empty 3 quart jar. How can they divide the wine equally?

Solutions to Previous Problems

Problem 13. In a certain multiple- choice test, one of the questions was illegible, but the choice of answers was clearly printed. Determine the true answer(s), if the choice was:

(a) All of the below.

(b) None of the below.

(c) All of the above.

(d) One of the'above.

(e) None of the above.

(f) None of the above.

Solution by Jennifer Hoyt (Hilliard, Ohio). (a) is clearly negated by (b), which implies that (c) is untrue. (b) is also negaed by (d), since if (b) is true (d) would nullify it. Since (a), (b) and (c) are false, (d) is also false. Therefore (e) is true, which makes (f) false. The solution is (e).

Problem 14. Disect the hexagon ABCDEF pictured into three pieces which can be reassembled to form a square. (From the Pi Mu Epsilon Journal.)

We received the same solution (shown below) from Jennifer Hoyt (Hilliard, Ohio) and Harry Zantopulos (M. Sc. '60).

Department News

Ruth Charney has been elected to the Board of Directors of the Association for Women in Mathematics.

Frank Demana and Carolyn Mahoney have received Exemplary Service Awards from the Ohio State Young Scholars Program. Both Frank and Carolyn have been deeply involved with the Young Scholars Program since its beginnings. They have worked in the summer program activities and the year-round educational programs.

G.A. Edgar's book on Measure, Topology, and Fractal Geometry has just been published as one of Springer's Undergraduate Texts in Mathematics.

Joan Leitzel has been named Director of the Division of Materials Development, Research, and Informal Science Education in the National Science Foundation's Directorate for Science and Engineering Education. The division's programs include development and assessment of instructional materials, research on teaching and learning, and informal science education through media such as television and museums. She will be the NSF's chief spokesperson in the areas of materials development and research for precollege science and mathematics education.

Henri Moscovici, Steve Rallis, and Michele Talagrand have been invited to give one hour addresses at the International Congress of Mathematicians to be held in Kyoto, Japan, in August 1990.

Joseph Rosenblatt was the organizer of a Special Session on Ergodic Theory held at the regional meeting of the American Mathematical Society in March in Manhattan, Kansas.

Mary W. Scott has been appointed as the new Head of the Mathematics Library. She comes to Ohio State with a degree from the University of Pittsburgh. While there, she worked in the Science Technology Department of the Carnegie Library in Pittsburgh. Previously she was the Science/Engineering Librarian at the University of North Dakota.

Monson Selected for Professional Development Program

Judie Monson has been selected to participate in Ohio State's 1990-91 University Internship Program. This special program is part of the University sponsored Staff Development Program. Judie has been a counselor in the Mathematics Advising Office since 1980. The program provides thoese selected with an opportunity to further develop their interpersonal and professional skills, obtain new knowledge and skills, and work with staff and faculty in a different academic or administrative unit of Ohio State. Judie, already an extremely valuable member of the Mathematics Advising Office staff, will spend 6 months enhancing her skills and abilities in the area of Honors recruitment and admissions.

Research Semester in Low-Dimensional Topology

As noted in the Spring 1989 issue of Math Matrix, the department has received an Academic Challenge Award. As part of this award, a Research Semester in LowDimensional Topology has been conducted over the past several months. The program began in early February and will continue through the end of June, 1990. Thus far, approximately 50 visitors have participated in the activities of the Research Semester. In addition to mathematicians from various parts of the United States, visiting mathematicians from Australia, France, Japan, the Soviet Union, and the United Kingdom have spent varying amounts of time at Ohio State.

Most of the visitors come for periods of one to three weeks, but the program is also funding five visitors for stays of up to three months. These are L. Siebenmann (Paris), T. Yoshida (Tokyo), B. Apanasov (Novosibirsk), V. Turaev (Leningrad), and S. Orvkov (Moscow). Typical activity during a given week will include three lectures, given by the visitors, and then extended periods of time for collaboration among the participants.

In March, a conference was held at which 12 talks were given and more than 60 guests were present. All of the lectures highlighted new developments in low-dimensional topology. In particular, there were several talks on new 3-manifold invariants and their connection with conformal field theory and recent work on hyperbolic 3manifolds and geometric group theory.

Demana Directs Planning for State Mathematics Coalition Project

Frank Demana has been named Director for the planning of a state mathematics coalition in Ohio. The formation of state mathematics coalitions to reinvigorate school mathematics nationwide is a priority of the Mathematical Sciences Education Board (MSEB). The coalitions are designed to promote state and local actions that will move curriculum, instruction, and assessment toward the goals proposed earlier in Everybody Counts, published by the National Research Council, and Curriculum and Evaluation Standards for School Mathematics, published by the National Council of Teachers of Mathematics. Over the next decade, the coalitions will use these national standards to guide state and local programs to improve mathematics education. The coalition will bring together educators, business representatives, and developers of public policy to set objectives for mathematics programs that meet state and local needs. Among the objectives of the coalition are to communicate the specifics of proposed changes in mathematics education, to stimulate development of state goals compatible with national ones, and to promote the adoption of policies which foster long term improvement. Ohio was one of 25 states to receive initial planning grants.

NEH Seminar at Ohio State

William Dunham, PhD '74, on the faculty at Hanover College, Hanover, Indiana, will return to Ohio State this summer to conduct a program funded by the National Foundation on the Arts and Humanities. Professor Dunham conducted a similar seminar at Ohio State in 1988. The response to that offering was extremely positive and many applicants were unable to attend. He has received funding to host the program, "The Great Theorems of Mathematics in their Historical Context," during July 1990. It should also be noted, that Professor Dunham's book, Journey Through Genius: The Great Theorems of Mathematics, has just been published by John Wiley and Sons, Inc.

Special Note

This is a special note to the friends of Art Bocian, MSc '78. Art's wife, Dena, passed away several months ago, leaving Art alone with his young son Russell. Dena's illness was long and difficult, but their many friends sustained them through caring and calling. Art has established a memorial fund in her memory at the Columbia University School of Social Work. Those of you who remember Art may wish to be in touch with him. His address is 63 Brookside Avenue, Livingston, New Jersey 07039. If you wish to make a contribution to the memorial fund, make the check payable to CUSSW Dena Fliegel-Bocian Memorial Fund and mail to Columbia University School of Social Work, c/o Alumni Office, 622 West 113th Street, New York, New York 10025.

Reflections on 25 Years in the Mathematics Library

"How long are you staying?" This was the most frequent question asked Lou Karlquist when she joined the staff of the Mathematics Library in March, 1965. Today, she could answer at least 25 years, for that is the time already spent in service and she plans to continue in her position. She reflects on those 25 years in the following remarks.

"When I arrived, we had a 50% Professional Head, 3 student assistants and my position was 100%, 40 hours a week. The students worked the evening and weekend hours. Mrs. Eunice Lovejoy was our Acting Head until Mr. John Demos came July 1, 1965 to become Head, Mathematics (50%) and Physics (50%) Libraries.

The library served the mathematics department only, as both statisticians and computer scientists were still groups within the department. There were subscriptions to 141 serials (today there are over 525) which were displayed on the top shelf of a book truck and 11,316 bound serials and books (today the count is over 47,000). Seating was available for 42 people and there were empty shelves in the alcove at the back of the library. When the library closed between Summer and Autumn Quarters. I was assigned to the Physics Library for those weeks.

The circulation of books and journals was handled manually. In order to find other available copies of an item, I had to call the Main Library. A staff member there checked the shelf list and told me where other copies might be located. The next step was to call each location and they then checked their shelves until an available copy could be found. Life is much easier and faster today, thanks to LCS! In the year I came, the Library Committee consisted of Professors Daniel Troy (Chair), Leroy Meyers, Ransom Whitney, Charles Saltzer, Hans Zassenhaus, and Stephen Drobot.

Fines for faculty were nonexistent and for students, $.10 a day for overdue material. The graduate students provided me with many gems of advice, lots of help, but most importantly made me feel at home, that I was needed, and they wanted someone who wanted to stay. The faculty helped greatly with my "Library Russian," at least to the point that I did not panic when someone had a Russian reference.

A highlight of those "good old days" was the Faculty/Graduate Student Lounge located in MA 052-054 (currently used for storage of pre-1975 journals). Here fresh brewed coffee was available amid comfortable furniture groupings and tables seating four. Many active mathematical discussions took place there, migrating between the lounge and the library, journals flowing back and forth until a point was proven or disproven. At the tables would be an occasional bridge game, game of Go, or lunch and conversation. It seemed then that everyone knew each other.

My hearty thanks to my "trainers" of 25 years ago, the Ohio State libraries for my continuous training throughout the 25 years -- my memories are unforgettable. I have had the pleasure of meeting people from around the world, an opportunity which I shall always treasure and, in an unusual way, I have been a part of the world of mathematics, statistics, and geodetic science."

Editor's Note

In a certain sense, this issue of Math Matrix is a time for reflection. The article about Lou Karlquist and her 25 years at Ohio State helps us recall that time. Both Joe Ferrar and I came to the department in that same time period. This issue marks a transition. Joe, who has served the past four years as Chairman, will leave that office on June 30. He will be succeeded by Dijen Ray-Chaudhuri. During Joe's time as chair, the department has seen several changes. These are detailed in his column. The continued growth and health of the department will be in good hands as Dijen takes over on July 1 .

Math Matrix, Volume 1/Issue 1, appeared in Spring, 1984. Alan Woods was chair of the department at that time. This issue marks my last as Editor. In September, I will be on a year's leave of absence. It has been fun, as well as challenging, to prepare this newsletter twice a year. I hope that you have found it interesting and informative. Please keep in touch with the department, we do enjoy hearing from you.