October 1995

Duke University

Math News

Department of Mathematics, Box 90320, Durham, North Carolina 27708-0320


Duke Press Honors John Nash

The Duke University Press will publish a special volume of the Duke Mathematical Journal in honor of Nobel Laureate, John Nash. Nash was awarded the 1994 Nobel Prize in Economics for his seminal work in game theory. He also made fundamental contributions in more classical areas of mathematics such as the Nash blowups and the Nash embedding theorem. The volume of the Duke Mathematical Journal will contain a previously unpublished paper of Nash still of research interest, a transcript of the Nobel prize seminar on Nash's work in game theory, as well as contributed papers in mathematics and economics of top research caliber.

Throughout October, the display cabinet in the lobby of the Physics building will feature an exhibit on Nash and his work.

New Course

Beginning in the spring '96 semester, the Mathematics Department will be offering a new course in partial differential equations (PDE's), oriented toward math majors. To a remarkable degree, phenomena in the world around us are described by PDE's. These include flow of heat, wave propagation, motion of fluids, and the structure of an atom. The new course, Math 133, will emphasize how such equations arise in science, and qualitative properties of solutions, as well as techniques for finding solutions. Often in realistic problem solving, qualitative understanding and numerical solutions can be used to complement each other. The course will be taught this spring by Tom Beale. The prerequisites are Math 111 or Math 131 and a good working knowledge of Math 103.

Actuarial Exams

Through a grant from the Cigna Foundation, the department will reimburse students the registration fees for actuarial exams. Please bring your receipt to the Administrative Assistant, Carolyn Sessoms.

Class of '95

So, what do you do with a degree in mathematics? A survey was conducted that asked graduating math majors last year what plans they had for the future. Here are some of the results:

A math degree gives a good background for many different occupations and areas of study. For instance, several math graduates are planning to go on to law school and medical school, where their reasoning and analytical skills will give them an edge. Many other are going on to graduate school in such areas as physics, computer science, economics, engineering, biology, and of course mathematics.

Banks and other businesses are always looking for financial advisors, and who better to advise them than math students? The second largest group of responses to the survey came from people anticipating jobs with such companies as Nations Bank and Merrill-Lynch. Companies that do extensive computation, such as AT&T, often hire math majors, too.

Other students are planning to teach, at least for a year or two, travel, or do research.


The Dirichlet-Neumann Problem
by Tung Tran

Tung Tran has been working on this problem through the Research Experience for Undergraduates (REU) program, which is open to any undergraduate student interested in a mathematical career. The program offers paid summer research opportunities at various locations around the country. The applications are usually due in February.

Tung presented his paper at the joint MAA/AMS meeting in August.

The problem consists of determining the flow in the interior of a network by making measurements on the outside edge. Examples of this include finding currents and resistances inside a network of resistors but measuring only the outermost wires, and finding out how much oil is in an area from information on its boundary.

On the n-extent of Lens Spaces
by Jason Samuels

Jason Samuels' research team has been working on a paper about lens spaces which will appear in the Furman University Electronic Journal for Undergraduate Mathematics, which is available on the Web page: http://math.furman.edu/~ mwoodard/fuejum

Here is the abstract:

The paper attempts to find the n-extent of certain lens spaces. The paper includes a conjecture as to the actual formula of the n-extent, the maximum average distance of n points for a given compact metric space, of these spaces, as well as a proof of a lower bound. The evidence provided by running a computer program supported the conjecture. The value of the n-extent has a correlation with the Euler characteristic of 4-dimensional manifolds of positive sectional curvature with nontrivial isometric actions. The lower bound for the n-extent places certain bounds on the Euler characteristic. The latter bounds would be stricter if our conjecture of equality for the n-extent holds.

The best way to have a good idea is to have a lot of ideas.

-- Dr. Linus Pauling



The 17th Annual Virginia Tech Regional Mathematics Contest will be held on Saturday morning, October 28, 1995. The hours are from 9 AM to 11:30 AM.

Universities and colleges from throughout the southeast take part--each participating school, under the direction of a faculty member, gives the test to its own entrants. Duke has done very well in the contest, producing first, second and third prizes in 1993.

If you would like to enter the contest please contact Dick Scoville, 225 Physics before-hand to reserve a place. You may also simply show up around 8:30 on October 28th in Room 120 Physics Building. However, the number of Duke entries is limited to 20, so don't get left out.

The fifty-sixth William Lowell Putnam Mathematical Competition will be held on Saturday, December 2, 1995. The Eastern Time Zone hours for the two sessions are 10 AM to 1 PM and 3 PM to 6 PM.

A list of all students taking part in the competition must be mailed in by October 8!

Please let Dick Scoville know before then if you plan to participate. You can reach him at ras@math.duke.edu or at his office, 225 Physics.

In the past ten years, only two schools have taken first prizes: Harvard nine times and Duke once. Wouldn't it be great for Duke to do it again?

Duke University Math Union

The math union, known as DUMU (``doo-moo''), is a club which meets once a month on Wednesdays from 5:30 to 6:00 in the Physics building. The club hosts guest speakers, coordinates contests, and hosts movies and parties. We also have a friendly Ultimate-Frisbee contest going with the Society of Physics Students, and we need all the people we can get to maintain our winning record! There are tentative plans for a study lounge, a trip to the wind symphony's Viennese Ball on November 10, and maybe even a ``Semi-formal'' with SPS.

The biggest undertaking is the all-day math contest we host for high schools. It's in the ARML format, which includes team, power, individual, and relay rounds, and requires problem writers, graders, and lots of people in general. Watch for more about this.

For more information, contact Tung Tran at trantt@acpub.duke.edu or Michael Rierson at rierson@math.duke.edu.

One cannot conceive anything so strange and so implausible that it has not already been said by one philosopher or another.

-- René Descartes

Le Discours de la Mé thode [1637], I


Joint Math Meeting

This past August, the MAA and AMS had a joint meeting in Burlington, Vermont. Several representatives of Duke University were there. Tung Tran and Paul Dryer both presented their research papers. Paul's was about knot theory and the human pretzel game. Tung's was about finding the values of resistors in a network using as few measurements as possible. Tung Tran is currently a junior, and Paul graduated last spring.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

-- Bertrand Earl Russell

Recent Work on the Principles of Mathematics [1901]. In International Mathematical Monthly, vol. 4, p. 84.

Here are some problems you might enjoy working. The solutions will appear in the next issue.

Problem 1: A Hole Problem

Define the ``hole'' operation to be a real number for all x and y not equal to zero which obeys the following rules:

  1. if and only if x + y = 0

Find a formula for which uses only addition, subtraction, multiplication, and division of real numbers.

Problem 2: A Little Number Theory

Find all solutions in natural numbers x and y:

Find a general method for solving:

for any given natural number a.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.

-- Roger Bacon

Opus Majus, bk.1, ch4.

Notes from the DUS

Greetings to all mathematics majors and minors from the Director of Undergraduate Studies!

Those of you who had declared your majors and minors as of early September should have received the 1995--1996 edition of the Handbook for Mathematics Majors and Minors. If you did not receive a Handbook, you may pick up a copy in Room 217, Physics Building, near where my office (217B) is located. The new version of the Handbook will soon appear on the department's web site (http://www.math.duke.edu), under the heading ``Information for Math Majors and Minors.''

As you know, registration for the spring semester begins on Wednesday, October 25. My predecessors have sometimes organized sessions in which those faculty teaching upper-level courses in the upcoming term provide a brief introduction to those courses. Following in that tradition, I have tentatively scheduled a one-hour meeting for Tuesday, 24 October, at 4:00 PM in Room 120, Physics. At recent meetings, student attendance has been low, so if you would like these sessions to continue, please come! Refreshments will be served.

Mathematics courses to be offered next semester, with the currently scheduled instructors, include: 120S, Scoville; 121, Lawler; 126, Trangenstein; 128, Pardon; 131, Zhou; 133, Beale; 135, Coyle & Schaeffer; 139, Moore; 196S, Layton; 201, Yang; 204, Reed; 206, Saper; 238, Schaeffer; and 240, Lawler.

I am currently assigning advisors for newly-declared first majors. Please let me know any specific requests and any information on interests or career plans that could help me in my assignments.

Finally, I want to do all I can---within the limits of the resources and time that I have available to me---to help you get the most out of your study of mathematics at Duke. I am delighted to receive suggestions on how to best accomplish that goal.

-- Harold Layton


This newsletter is open to contributions from students, both graduate and undergraduate, and faculty. Information about the Mathematics Department, classes, degrees, and projects would all make good contributions. We're also looking for quotes, and of course problems and puzzles. Please send any ideas you might have to the editor, Garrett Mitchener: wgm2@acpub.duke.edu or talk to Harold Layton or Joan McLaughlin in the Mathematics Department.

New Letters to the Editor Column

The newsletter would like to begin having a column of letters to the editor. These can be questions for the department, or opinions that you'd like to state. (Points of interest and project abstracts are printed elsewhere.) We reserve the right to edit letters for length and clarity. We may not be able to print all letters; the most approporiate and timely ones will be given priority.

Until we come up with a better way, just submit them to Harold Layton, or the editor wgm2@acpub.duke.edu.

New Problem Corner

We have started to include problems in Duke Math News. If you have any puzzles, brain-teasers, or any sort of problems, please send them to us. It would be really nice to include solutions, too, so we can print them in the following issue.

Read It on the Web

Current and previous issues of Duke Math News are available online through the World Wide Web at: http://www.math.duke.edu

Joan McLaughlin
Wed Oct 11 16:03:29 EDT 1995