Professor Dresden's Home Page

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Department of Mathematics

The College (of Arts & Sciences)

Washington & Lee University

Welcome to my web page. For academic info about me, be sure to check out my somewhat-up-to-date Curriculum Vitae [pdf file]. For class or research info, read on!

Here's the course you want to take this spring: Math 171, Cryptography and Number Theory.

How To Reach Me

Office Location: Robinson Hall R-1.
Office Phone: (540) 458-8806.
FAX: (540) 458-8239.
Mailing Address: Robinson Hall R-1 (Math Dept), Washington & Lee University, 204 West Washington Street, Lexington VA 24450-2116.
e-mail: dresdeng "at" wlu "dot" edu. If you don't hear from me after a few days, it's probably because your e-mail got deleted by my automatic spam filter. Try to send it again as a plain-text message (without attatchments), or just call me on the phone.

Office Hours

NEW for fall of 2009-10. (These are subject to change, and you can always make an appointment to see me.)

Monday 12:20 - 1:15 p.m.
Tuesday 9:05 - 10:00 a.m.
Wednesday 12:20 - 1:15 p.m.
Friday 9:05 - 10:00 a.m.
...and one "floating" office hour, per week, by appointment

Mathematical Research

I work in an area of mathematics called number theory, and in particular on the subject of the Mahler measure of a polynomial. I also study other topics from both number theory and abstract algebra (on polynomials, groups, algebraic extensions, etc). Here are some of the articles I have written:

Orbits of Algebraic Numbers with Low Heights, Math. Comp. 67 (April 1998), 815--820. (Download the pdf file.)

Two Irrational Numbers From the Last Non-Zero Digits of n! and nⁿ, Math. Mag. 74 (October 2001), 316--320. (pdf.)

Sums of Heights of Algebraic Numbers, Math. Comp., 72 (2003), 1487--1499. (pdf.)

On the Middle Coefficient of the Cyclotomic Polynomial, MAA Monthly 111 (June-July 2004), 531--533. (pdf.)

There Are Only Nine Finite Groups of Linear Fractional Transforms with Integer Coefficients, Math. Mag. 77 (June 2004), 211--218. (pdf.)

Finding Factors of Factor Rings over the Gaussian Integers, with Wayne Dymacek, MAA Monthly 112 (Aug-Sep, 2005), 602--611. (pdf.)

A Combinatorial Proof of Vandermonde's Determinant, with Art Benjamin, MAA Monthly 114 (Apr, 2007), 338--341. (pdf)

Three Transcendental Numbers From the Last Non-Zero Digits of nⁿ, F_n, and n!, Math. Mag. 81 (Apr, 2008), 96--105. (pdf).

Resultants of Cyclotomic Polynomials, to appear, Rocky Mountain Journal of Mathematics. (pdf).

Look, There's More to Say about Conway's Look and Say Sequence, with Jacob Siehler, submitted to Mathematics Magazine (Nov. 2009).

Binet-type formulas for r-generalized Fibonacci numbers, preprint.

A New Approach to Rational Values of Trigonometric Functions, preprint.

On the Mahler Measure of P(f/g), preprint. (pdf.)

I'm rather fond of my three articles that were published in the MAA Monthly, partly because two of them were jointly written with other mathematicians (which is a tremendous amount of fun), and partly because the MAA Monthly (as seen here) is the most widely read mathematics journal in the world.

Thanks to my joint articles, I have an Erdös number of 4. This means that I'm only four co-authors away from Paul Erdös, the most prolific mathematician in history (biographical links to Wikipedia here and to the MacTutor history of mathematics site here). The MathSciNet database gives the chain as Dresden -- Wayne Dymacek -- Craig Bailey -- Edgar Palmer -- Paul Erdös, but another chain is Dresden -- Dymacek -- Gerard Chang -- Derbiau Hsu -- Paul Erdös, and then there's also the link through my article with Art Benjamin, which gives Dresden -- Art Benjamin -- Gregory Levin -- Michael Jacobson -- Paul Erdös. How cool is that! (Some famous people with Erdös number 4 include Grigori Perelman, Edward Teller, and Bill Gates. Huh.)

Some of my articles have been referenced in AT&T's On-Line Encyclopedia of Integer Sequences, which is always nice to see. For example, check out 0, 0, 1, 0, 1, -1, ... (reference number A094754) and 1, 1, 2, 6, 4, 2, ... (reference number A008904), among others. Thanks to Jacob Siehler for providing the Mathematica code!

Along with Professor Siehler, my Abstract Algebra students and I worked on finding natural representations for finite rings. Please visit our Small Rings page for examples.

For information on the Mahler measure, please check out this excellent web site prepared by Mike Mossinghoff, a fellow Texas graduate (now at Davidson).

I co-hosted the SERMON 2003 math meeting at UNC-Greensboro (SERMON stands for South-East Regional Meeting on Numbers).

I've given many presentations on mathematics and one on teaching mathematics, at local and national conferences. Also, Art Benjamin (mentioned above) gave a presentation on our joint work at MIT in December of 2004, and fellow W&L professor Wayne Dymacek gave a talk on our joint paper here at W&L. Recently, I gave a talk at JMU on the Mahler measure and again on the Look-and-Say sequence, and a student of mine gave a talk at Loyola (in Maryland) on her senior honors thesis (on the subject of factor rings).

Math Information for Washington & Lee Students (and others)

An extremely cool reference on the web is Eric Weisstein's World of Mathematics, hosted by the nice folks at Wolfram Research.
My Math Department actuary web page tells you everything you want to know about how to pursue this lucrative and rewarding career (and yes, we have math classes for that).
Anticipating having trouble with Calculus 101 or 102? You might want to check out the free nightly tutoring sessions in Robinson 6, Sunday through Thursday. Check with the department secretary for details.
Prof. Finch and I share teaching one of our favorite courses, Math 365. This is an introductory course in number theory, a very beautiful area of mathematics that dates back to the ancient Greeks, yet contains many modern applications ranging from deciphering German codes during WWII to verifying the account numbers on your ATM card. Students who took this course during Winter term really enjoyed it.
I'm very excited about a new spring-term course, Math 171: cryptography and number theory. This course covers many of the issues concerning the current encryption methods used by commercial programs and by the military: how the codes work, why they work, and how to break them. Surprisingly, one can create very secure (but not perfectly secure!) encryption systems using some fairly easy facts of basic number theory about prime numbers. An excellent example of an encryption system is the PGP system by Phil Zimmermann. So secure is Zimmermann's PGP program that the U.S. government classified it as a restricted munition, and tried to keep him from sharing it with folks in foreign (and potentially hostile) nations. Indeed, many fear that terrorists and criminals can use these systems to encrypt their files, thus making recovery nearly impossible.

The most exciting part of the course, I think, will be our field trips to the Marshall Museum here in Lexington and the NSA Museum in Washington, DC. Both museums have extensive collections of cryptographic materials and artifacts. In this picture from a few years back, Dr. Dymacek (on the right) explains to us (at the Marshall Museum) some of the details about the ENIGMA machine. You'll recall that the ENIGMA was used by the Germans to encode military messages during the war. British and American scientists (led by the brilliant Alan Turing) managed to break the code, and thus tremendously advanced the Allied war effort.

Did I mention that one of the exams in this course will itself be written in code? So, even before you can take the exam, you have to decode it. Sounds like fun!

Links

For general mathematical information, here are some of my favorite links:

The Association for Women in Mathematics has many wonderful programs to encourage women in pursuing a career in the mathematical sciences. AWM student membership is only $20, and yes, men can join too!
MAA Online, from the Mathematical Association of America, has many great links to career and grad school information. MAA student membership is $30, and well worth it.
- The MAA Careers page can help to answer the question, "What can I do with a math major?"
- The page Career Profiles demonstrates that there are many options beyond just teaching or graduate school. Check out the dozens of profiles in investment, medical science, sales and marketing, law, engineering, business management, actuarial science, and more... and all of them are about former math majors!
The AMS Home Page, from American Mathematical Society, contains valuable professional information, along with a searchable index to mathematical articles.
- Here is a direct link to the MathSciNet search page,
- ... and to the MR Citation database (to find out who's quoting whom).
- If you're thinking of going to graduate school, you might want to look into getting an AMS student membership, at around $40 or so.
- You should also look at the AMS/YMN math grad student blog, which includes links to some really funny math comics!
- Also, there's the AMS Early Careers and general career pages, both well worth checking out.
UCLA's Terrence Tao is widely considered to be the most brilliant mathematician alive. His blog is well worth reading.
What happens when two guys in Silicon Valley decide to build a place for mathematicians to hang out? Well, you get the American Institute of Mathematics, soon to be a major research center.
Want to find out who your advisor's advisor was? Check out the Mathematics Genealogy Project, hosted by the AMS amd the folks at North Dakota State University.
The Mathematics Information Server is a giant listing of just about every mathematical group you can imagine, including a world-wide list of on-line college and university mathematics departments. This is very helpful in providing a direct link to math professors and programs around the world. Here's a direct link to the Number Theory web. For a (rather rough) ranking of graduate math departments, visit this AMS web site.
Texas has a nice listing of US cc's, colleges, and universities.
Here are the W&L portals to the on-line resources JSTOR and MathSciNet.
Two free on-line search engines for math articles are arXiv.org and Google Scholar, but they are of lower quality than the AMS and MAA search engines (MathSciNet and JSTOR) above.
These are not math-related, but if you want to know what's happening in the area, follow these links:
- For events and news about campus activities, check out...
  - the text-based W&L master calendar,
  - the W&L Inclement Weather bulletin,
  - the Emergency Events Bulletin,
  - or the live but slightly blurry Colonnade Cam. On the far right of the picture, by the way, is Robinson Hall, home to the mathematics department!
- There's also the VMI event calendar for our sister school, the Virginia Military Institute, ...
- ... with the many VMI webcams (including a live feed of downtown Lexington) ...
- ... Southern Virginia University calendar, from over in Buena Vista.
- For community events, read The News-Gazette calendar or the local events calendar from the Lexington visitor's center.
- Here are the movie times at the Hull's Drive-In and the State and Valley cinemas.
- Finally, you can always check out the local weather...
  - the seven-day Weatherbug forecast,
  - the five-day Weather Underground forecast,
  - the daily Yahoo.com forecast (via the Weather Channel).

The Trail

As you no doubt know, the Appalachian Trail is only a few minutes from campus. I don't think I'll ever actually walk the whole thing, but I do like to get out and hike for a few hours on parts of the trail, accompanied occasionally by our faithful trail dog, Mr. Puppy. Here's a nice picture from up on the Blue Ridge Parkway (also a few minutes from campus), with the Peaks of Otter in the background.

This page was written in HTML by Prof. Dresden, using a template originally borrowed from UT-Austin. It's been up, in one form or another, since 1997. Yes, I should probably update the design. Someday.