Baker Named Eminent Scholar

Gregory R. Baker has been named Ohio Eminent Scholar in Numerical Analysis and Scientific Computation. He begins his appointment in the department Autumn Quarter, 1988. He comes to Ohio State from the position of Director of the Applied Mathematics program at the University of Arizona. During the past year, he has been a Research Mathematician at the Exxon Research and Engineering Company and a Visiting Fellow in the Department of Mechanical and Aerospace Engineering at Princeton University.

Baker was born in South Africa and received his initial scientific education there, earning a B.S. in 1970 and an M.S in 1973 (both in Physics) from the University of Natal. He earned his Ph.D. in Applied Mathematics from the California Institute of Technology in 1977. In 1984, Baker was selected as Presidential Young Investigator by the National Science Foundation, one of the first mathematical scientists to receive this prestigious award.

Baker's research is in the area of computational fluid dynamics-the numerical simulation of solutions of the equations which describe the behavior of fluids. His particular interest is in the development, and implementation for use on supercomputers, of sophisticated algorithms which allow researchers to detect and analyze the development of "singularities" in a fluid flow. Real life singularities appear, for example, as tornados or as whirlpools in a fast moving stream. His research findings have appeared in twenty-four papers in international journals or refereed conference proceedings. Six additional papers and two books are currently in the final stages of preparation. Baker also presented nine invited talks on this work at major universities and conferences during 1986-87, including talks at Oxford and Cambridge universities in the United Kingdom, and at the Courant Institute of Mathematical Sciences and Princeton University in the U.S.

Since 1982, the Department of Mathematics has been expanding and upgrading its program in Applied Mathematics. A major emphasis has been the development of an excellent program in Scientific Computation and Numerical Analysis. Such a program will provide a critical interface between the Department, various science and engineering departments at Ohio State, and the Scientific Computation community throughout Ohio. With his excellent academic credentials, his experience in the private sector, and his experience in developing a Scientific Computation component in a major university's Applied Mathematics program, Baker will play a central role both within the Department's academic programs and in the wider context of Ohio's Scientific Computation community.

Rubin Receives Presidential Young Investigator Award

Karl Rubin received a Presidential Young Investigator award from the National Science Foundation. He was selected from a nationwide field of nominees to be awarded this honor in support of his research. He joined the faculty of Ohio State in 1984 after spending two years at the Institute of Advanced Study in Princeton. In 1985, he received an Alfred P. Sloan Foundation research fellowship and, in 1987, he received Ohio State's Distinguished Scholar Award.

Rubin's research area is arithmetic algebraic geometry. Much of his work deals with the arithmetic of elliptic curves. The main problem in this subject can be most simply stated as follows: Find all rational number solutions of a given equation of the form Y2 = f(x), where f(x) is a polynomial of degree three having rational numbers as coefficients. In this particular case, it is known that the set of solutions, which may be finite or infinite, has an algebraic structure. Two solutions may be combined by a geometric process to form another solution. Although this procedure is quite old, it was only in 1922 that Mordell proved that the solution set is always finitely generated.

Mordell's result initiated a long program of research leading, in the mid sixties, to the conjecture of Birch and Swinnerton-Dyer. This conjecture relates several different objects contalning information on the solution set of the curve. One of these objects is the Tate-Shafarevich group. The Birch-Swinnerton-Dyer conjecture presupposes that this group is finite and proceeds to relate the number of elements in it to other characteristics of the original curve.

Another consequence of Mordell's result was the development of a computational algorithm to produce the solution set of an elliptic curve. Although the algorithm did work in some instances, it could not be shown that it would ever terminate when applied to a given elliptic curve. The algorithm would be effective if the Tate-Shafarevich group were finite.

The Tate-Shafarevich group has been at the center of the theory for over thirty years, and had long been conjectured to be finite. However, it has not been possible to demonstrate that fact. Indeed, there had been no single example of a curve for which the Tate-Shafarevich group could be shown to be finite!

Rubin has recently proved that, for a large class of elliptic curves, the Tate-Shafarevich group is finite. He has also given an effective bound for the number of elements in this group and has shown that the only prime numbers which divide the order of the group are those predicted by the Birch-Swinnerton-Dyer conjecture.

Although the complete proof of the Birch-Swinnerton-Dyer Conjecture may still be a long way off, Rubin is in the forefront of activity in this direction and is expected to be a rich source of further progress in attaining the goal.

Comments from the Chair

The relative calm of Summer Quarter provides us with a welcome moment to contemplate the high points of the year gone by and to anticipate the year to come. A look at raw numerical data (and what else should we mathematicians begin with) reveals two heartening trends. At the remedial level, enrollment plummeted nearly 20% as stricter Ohio State admission standards combined with strengthened high school graduation requirements to provide us with the best Autumn Quarter entering class in many years. Now, if we can only keep the progress derivative positive. The second interesting and welcome trend reveals that the number of mathematics majors has doubled in the past two years. Our Spring Graduation breakfast overflowed the Mathematics Lounge, as graduating seniors, parents, and faculty/staff guests began the day with coffee, fruit and fresh croissants. We wish those graduates much success, and look forward to seeing their children as students when that generation reaches college age.

As for the coming year, we enter it with some confidence that our numbers will continue to grow with the quality of our faculty. Several notable faculty successes during 1987-88: Harvey Friedman, 1984 Waterman Award Winner, was named to one of the two inaugural University Professorships at Ohio State; Karl Rubin was named a Presidential Young Investigator by The National Science Foundation and a Distinguished Scholar by Ohio State; L. Alayne Parson was selected to receive an Alumni Distinguished Teaching Award; and Gregory Baker, of Exxon Research Laboratories, was appointed to the faculty as Ohio Eminent Scholar in Numerical Analysis and Scientific Computing. Quite a star studded cast of participants in our ongoing quest for excellence in the classroom and in the research arena.

As I close this brief overview of what I feel was a most successful year for the department, I wish to thank those alumni who have donated gifts fo the Development Fund, specifically designated for the Department of Mathematics. These have provided the means to financially assist some outstanding students of mathematics and to fund interesting and challenging activities for our majors. These efforts, while relatively small in scale, have contributed directly to the growth of our program, both qualitative and quantitative. Indirectly, they have also contributed to the quality of faculty efforts which has led to the recognitions noted earlier.

INTERNATIONAL PROBABILITY CONFERENCE

An International Conference on Almost Everywhere Convergence in Probability and Ergodic Theory took place at the Holiday Inn on the Lane, June 11-14, 1988. Financial support was provided by the Institute for Mathematics and its Applications at the University of Minnesota, the National Science Foundation, the United States Army, and The Ohio State University. There were approximately 60 participants, of which 36 were invited participants, from the United States, Canada, Great Britain, France, Spain, Federal Republic of Germany, Austria, Israel, Hungary, and the Soviet Union. Among the speakers were Donald Ornstein (Stanford University), Alexandra Bellow (Northwestern University), Lester Dubins (University of California, Berkeley), Karl Peterson (University of North Carolina), Paul Rvsz (Technische Universitt, Vienna), Antoine Brunel (University of Paris), Donald Burkholder (University of Illinois), Steven Orey (University of Minnesota), Jean Bourgain (IHES, France), Michael Klass (University of California, Berkeley), Y. S. Chow (Columbia University), Ray Chacon (University of British Columbia), and Ulrich Krengel (Gottingen). Invited speakers from Ohio State were Vitaly Bergelson, Gerald Edgar, and Joseph Rosenblatt. The presence of the Soviet mathematician Dr. Arcadii Tempelman from Vilnius, Lithuania, produced considerable interest. This was Dr. Tempelman's first contact with many Western colleagues, since he has never before been allowed to leave the USSR. Proceedings of the conference will be published by Academic Press.

The conference organizer was Louis Sucheston (Ohio State). Other members of the organizing committee were Mustafa Akcoglu (University of Toronto), Alexandra Bellow (Northwestern University), Donald Burkholder (University of Illinois), and Gerald Edgar (Ohio State). A follow up conference will take place in October, 1989, at Northwestern University, under the direction of Alexandra Bellow.

Upcoming Conferences at Ohio State

October 27-29, 1988: Conference on Technology in Collegiate Mathematics. The focus of the conference will be on the use of technology in the teaching of collegiate mathematics, from precalculus and calculus through advanced topics. Franklin Demana, Thomas Ralley, Bert Waits, and Alan Osborne (Department of Education Theory and Practice) will serve as the Program Committee for the conference.

December 15-18, 1988: Interdisciplinary Conference on Axiomatic Systems. The conference will provide a forum for a thorough discussion on the development and use of axiomatics systems. Axiomatic systems are of fundamental importance for a wide variety of disciplines. This third interdisciplinary conference is being held under the auspices of the newly formed Program for Foundational Studies at The Ohio State University. Harvey Friedman is the founding director of this Program.

May 22 - June 3, 1989: NATO Advanced Study Institute on Orthogonal Polynomials and their Applications. It is expected that about 150 people will attend this specialized conference. Attendance is by invitation only. Paul Nevai is serving as Director.

Department News

Thomas Gregory of the Mansfield Regional Campus was on leave during the academic year 1987-88. He did study and research in Lie Algebras at the University of Wisconsin, Madison.

James R. C. Leitzel has been appointed as Vice Chairman. His responsibilities include graduate student recruitment, monitoring the academic progress of graduate students, supervising departmental publications, monitoring departmental affirmative action efforts, and coordinating the activities of the staff. Phil Huneke continues as the other Vice Chairman.

Dijen K. Ray-Chaudhuri has been elected as an active member of the New York Academy of Sciences

John 0. Riedl, Dean and Director of Ohio State's Regional Campus at Mansfield, received an award from the National University Continuing Education Association. He was recognized for his efforts to provide lifelong learning to mathematics and physical science educators, professionals and researchers. He was instrumental in establishing the Ohio Science Olympiad, which also received an award. The Olympiad, an annual tournament held at Ohio State, is a competition where students participate in a variety of events which require knowledge and application of science facts, concepts and skills.

Michael Davis, Warren Sinnott, and Bert Waits were promoted to professor.

Vitaly Bergelson, Andrej Derdzinski, Matthew Foreman, and Cary Rader (Newark Regional Campus) were promoted to associate professor with tenure.

Ed Overman, George Majda, and David Terman, associate professors, were awarded tenure as was Tariq Rizvi, assistant professor at the Lima Regional Campus

Arno Cronheim and Charles Saltzer retired this year. Arno, with 31 years of service, received the title Associate Professor Emeritus. Charles, with 26 years of service, retired as Professor Emeritus.

Development Funds Provide Scholarship Aid

During this past year various development funds provided a total of $10,800 in financial aid to 14 students. These students are just a few of the more than 250 students majoring in mathematics. As the cost of higher education continues to escalate, scholarship awards such as these do provide a source of much needed income. The awards also give recognition to the strong effort and high quality of current undergraduates. If you would like to help in this endeavor to support students, make a contribution to the Development Fund. Checks should be made payable to The Ohio State University Development Fund, and mailed to Editor, Math Matrix, Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210-1174.

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Remember these folks? Bill (PhD, '75) and Penny Higgins (MSc '71) Dunham at the Department of Mathematics Summer picnic. Both are at Hanover College, Hanover, IN. Bill is visiting the department and Penny is completing her studies for a PhD in mathematics education.

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 expecially welcome.

Problem 10. Let the function f(x) be defined by f(x)=log₁₀(log₁₀(exp(exp x))). Using a scientific calculator, it seems that if we start with any number and apply the function f over and over, the results converge to a number -0.64029. Evaluate this limit in closed form. [Suggested by Brian R. Scalley, Mayfield Heights, Ohio].

Problem 11. How many different ways are there to make change for a dollar, using only coins with 1,5,10, and 25 cents?

Solutions to previous problems

Problem 9. Provide an elementary proof of this fact: Suppose a sequence (p_n) has the property that whenever (X_n) is a bounded sequence, the sequence (y_n) given by

We have a solution by Daniel P. Giesy (B.A., M.A. 1960).

The proof is by contradiction: we show that if diverges, then there is a bounded sequence (x_n) for which the sequence (y_n) of the problem statement is unbounded.

Case I: (p_n) is unbounded. Then take x₀ = 1 and x_n = 0 for n != l. then y_n = p_n, so is unbounded.

Case II: (p_n) is bounded, say |p_n| <= M for all n. Choose u_n such that |u_n| = 1 and u_np_n = |p_n|. The sequence (x_n) will be constructed using numbers from (u_n), so will be bounded. Since diverges, the series has arbitrarily large partial sums.

Pick m_l so that

Try Your Hand at this One

"Age Problems" seem to be a staple in the mathematics literature. The following problem, gleaned from the Dover publication, Mathematical Bafflers by Angela Dunn, was suggested by Roy Meyers.

Lottie and Lucy Hill are each 90 years old. Mary Jones, on the other hand, is half again as old as she was when she was half again as old as she was when she lacked 5 years of being half as old as she is now. How old is Mary?
[Answer: 90; in other words, Mary is as old as the Hills]

Haase Wins Courant Institute Prize

M. Christopher Haase was awarded second prize in the 1986-87 Courant Institute Prize for Mathematical Talent. There were two prizes awarded in this competition, open to undergraduates at U.S. colleges and universities. Haase project, "Quadratic Reciprocity over Z_p[X]," was completed under the supervision of Professor Warren Sinnott. In commenting on the experience, Haase writes: It is not often that a young person can immediately identify an occasion or incident which has changed the way he or she views the surrounding world. For five summers, I have participated in a summer program founded and supervised by Dr. Arnold Ross, OSU Professor Emeritus of Mathematics. As a 14-year-old participant in this program, I discovered a whole world of ideas - more importantly questions - I never thought existed. Surrounded by students with impressive skills and diverse academic backgrounds, I was overwhelmed at first. The first night of the program, I found myself, in the company of other participants, struggling with the new task of posing my own questions and reasoning them out something that was totally alien to me. Today, this alien feeling that I experienced the first night of the program (and many nights that soon followed!) has matured into admiration and friendship toward the founder of the program. This newly acquired skill of conjecturing that Dr. Ross helped me develop has enabled me to pursue topics along unique avenues, such as in a recent paper entitled "Quadratic Reciprocity over Z_p[X]." In this paper, I derived and proved a reciprocity law disjoint from the abstract jungle of class field theory. Later, I presented this paper at Rose-Hulman Institute, the National Security Agency, and the Ohio Section of the MAA.

I certainly am grateful for the interest and individual attention I have received at Ohio State. The inspiration and encouragement of Dr. Ross and other outstanding OSU faculty members offer the means and direction for students like myself to make the most of their capabilities.

L. Alayne Parson Receives the 1988 Alumni Award for Distinguished Teaching

L. Alayne Parson has been recognized for her outstanding teaching by receiving the 1988 Alumni Award for Distinguished Teaching. Those named as Distinguished Teachers are selected by a committee of alumni, students, and past award recipients from nominations submitted by their students and colleagues. Parson has taught students at all levels. She has actively developed honors opportunities for students. She has chaired the department's Honors Committee and has served on the Colleges of the Arts and Sciences Committee on Honors Programs. In addition, she advises undergraduate majors and graduate students. Her mathematical area of interest is complex analysis and analytic number theory. Her current efforts are devoted to providing a classification for rational periodic functions for the modular group. Those students who know her, know well that she tries to share the excitement of discovery in her classes. As she says, "You have to be excited about what you're teaching in order to get the students excited."

Editorial Note

There has been a brief interlude in the publication of Math Matrix. We hope this issue will encourage your reading and response. The intention is to produce two issues a year. Look for another in your mail later in the year.

Harvey Friedman Named to University Professorship

Harvey Friedman was one of the two distinguished faculty named this year to a University Professorship. The other was Leo A. Paquette of Chemistry. The University Professorship program honors faculty members who have done outstanding research or other creative work. In addition to his appointment in mathematics, Friedman holds appointments in the departments of Philosophy and Computer and Information Sciences. He is internationally known for his research on the foundations of mathematics.

This year, he has been invited to give a one hour address surveying recent developments in logic and foundations at the 1988 Summer Meetings of the AMS-MAA in Providence. This is part of the special program celebrating 100 years of American Mathematics. In mid-April, he hosted a conference on foundations at Ohio State. The topic and papers presented at that conference centered on the theme of randomness.

In Memoriam

Rosemary Carroll, wife of Frank Carroll, died June 19, 1988 after an extended illness. Rosemary was a vibrant and positive force in various aspects of the department's activities.

Kuang-l Kurt Chang, Ph.D. '76, currently serving as a data communications project supervisor in Perkiomenville, PA.

Daniel Perry Giesy, M.A. '60 Ph.D. '64 from the University of Wisconsin at Madison, currently a computer specialist with Planning Research Corporation, Hampton, VA. Previously he has taught at the University of Southern California, Western Michigan University, and Norfolk State University. Before developing his computer specialty, he did research on convexity in Banach spaces and probability theory for Banach space valued random variables.

Hank Heiberg, M.Sc. '68, Ph.D. '71, was recently promoted to Group Actuary, an officer of SafeCo Insurance Company, Seattle, WA.

Melanie Holzman, B.Sc. '83 in math and chem, currently writer, editor, and owner of a real estate investment newsletter.

Sr. Marlene Kline, M.Sc. '68, currently teaching mathematics at Mount Alverina High School in Pittsburgh, PA.

Jean W. MacCluer, B.Sc. '59, Ph.D. '68 (Human Genetics) from the University of Michigan, currently a scientist with the department of genetics at the Southwest Foundation for Biomedical Research and Professor in the Department of Cellular and Structural Biology at the University of Texas Health Science Center in San Antonio.

Jeffery T. McLean, M.Sc. '67, Ph.D. '73, currently an associate professor of mathematics and Director of the Aquinas Scholars Honors Program at the College of St. Thomas, St. Paul, MN. He and Neal Raber, Ph.D. '72, are collaborating on a college geometry text.

Richard E. Mercer, B.Sc. '73, Ph.D. '80 from University of Washington, currently an associate professor in the Department of Mathematics and Statistics at Wright State University, Dayton,Ohio.

Attila T. Relle, B.Sc. '85, D.D.S. '85, currently in private dental practice and working toward an M.B.A.

Joseph J. Rushanan, M.Sc. '82, Ph.D. '86 from CalTech, currently a member of the technical staff at the Mitre Corporation. Cdr.

Donald J. Santapaola, B.Sc. '71, commander in the United States Navy and Chief Staff Officer of Fighter Wing One, Oceana.

William Scott, M.A. '41, Ph.D. '47, currently a professor at the University of Utah, Salt Lake City.

Ralph C. Steinlage, M.Sc. '63, Ph.D. '66, currently a professor of mathematics at the University of Dayton, Dayton, Ohio.

Joseph S. Torok, M.Sc. '78, Ph.D. '85 (Engineering Mechanics), currently an assistant professor of mechanical engineering at Rochester Institute of Technology, Rochester, N.Y.. His research interests include dynamical systems and moving boundary problems.

Dave Vickers, B.Sc. '73, M.Sc. '76, currently employed at Stratus Computer Inc. in Phoenix, Arizona.

Christine Browning, MA '84, Ph.D. '88 (Mathematics Education), currently Assistant Professor at Western Michigan University.

Charles Vonder Embse, MA '84, Ph.D. '87 (Mathematics Education), currently Assistant Professor at Central Michigan University. At the time of his graduation, his research was recognized through the award of the Flesher Fellowship. The basic purpose of the Fellowship is to give public recognition to outstanding doctoral students in the Ohio State College of Education.

Recognitions for Graduate Associates

The department is heavily dependent on graduate students to assume much of its teaching obligations. The 182 graduate students in the various graduate programs offered by the department, for the most part, meet their teaching assignments in a professionally competent way. Each year undergraduate students nominate several of our graduate students for the honor of the Graduate Student Teaching Award. This year we extend hearty congratulations to Grzegorz Banaszak, Timothy Brown, Alan Cring, Dorian Edwards, James Ham, Ali Lari-Lavassani, Robert Pontius, Mark Reeder, Donna Stohr who were recognized by their students.

Mate Wierdl, a graduate student from Hungary studying with Vitaly Bergelson, was awarded a Presidential Fellowship. This prestigious award provides financial support for a full year so that the student can devote maximum time to research. This past year, Gabriel Villa-Salvador, a student from Mexico working with Manohar Madan in algebraic function fields, studied under the same award.

Mark Your Calendars!

The Summer Meetings of the American Mathematical Society and The Mathematical Association of America will be held on the Ohio State campus August 8-11, 1990. These meetings will celebrate the 75th anniversary of the founding of The Mathematical Association of America. The MAA was organized and held its first meeting in 101 Page Hall on the Ohio State campus December 30-31, 1915. The Department and the University are pleased to host this special event. We extend a cordial invitation to all alums to attend these meetings. Let us have a good turn out and a grand reunion.