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Thanks to An Explainer


Conrad explains all

Keith Conrad is a professor in the mathematics department at UCONN—the University of Connecticut. My dear wife Kathryn Farley and I are about to move to join him—not as faculty but as another resident of the “Constitution State.”

Today we thank him for his work on explaining mathematics.

Conrad is a prolific writer of articles on mathematics. He makes hard concepts clear, he makes easy concepts interesting. He has a sense of humor; his website is filled with fun of all kinds.

He has interesting license plates, photos of streets that bear his first name, a Russian update to Tom Lehrer’s “Elements” song, and more links to others’ items.

If you’d like a video example of fun see this for a short video illusion. Too bad it is an illusion and it does not work in reality. As a chocolate lover I wish it worked—free chocolate forever. A similar illusion with a triangle was once featured by Martin Gardner, leading Ken as a teenager to make a cardboard cutout version overlaid on a map of the Bermuda Triangle as an “explanation” of disappearances there.

Articles

I have not yet met Conrad in person, but have enjoyed reading his articles on math of all kinds. He has a giant {63 \times 4} grid of clickable titles, grouped by subject area.

For an example, he has a title “Roots on a Circle.” The file name says “numbers on a circle” and the essay begins disarmingly enough with a picture of the 7th roots of unity. The next page shows a simple polynomial where most but not all roots lie on the circle:

This is enough to draw you in and stay attached as things become more complicated beginning on page 3. It helps that Conrad does not stint on algebraic details. This essay supplements a cautionary tale in mathematics: a link to a MathOverflow list whose top item is that the factors of {x^n - 1} over {\mathbb{Q}} have no coefficient of absolute value greater than {1} for {n = 1,\dots,104}. Before you try to prove this by induction check out {n = 105}.

Finite Groups

One of my favorite articles is titled, “Orders Of Elements In A Group.” Conrad starts with the humble concept of the order of an element in a group. Then he builds up a theory that explains various properties of order.

I especially like that he supplies examples to help you with your intuition. For me, and Ken, finite groups are just counter-intuitive. Groups have magical properties but my naive conjectures about them usually fail. Conrad’s article ends with a discussion of primality testing which is dear to us in complexity theory.

Finite Groups that Encode Information

I recently needed a finite group with a certain structure. In complexity theory we sometimes use matrices to encode information in a way that makes an algorithm more efficient. Algorithms like matrices since they can be stored and multiplied efficiently. As an example, suupose that we have two matrices {A} and {B} so that

\displaystyle  AB = -BA.

That is the matrices anti-commute. Then we can use such matrices to encode information about a string {S} of {A,B}‘s. Every string {S} can be written as:

\displaystyle  \pm A^{k}B^{l}.

The {\pm} encodes the number of inversions in the string {S}. That is the number of times {B} is followed by an {A}:

\displaystyle  \dots B \dots A \dots

So {ABBA} has {2} inversions, and {AAABA} has {1}. Here are matrices that anti-commute.

\displaystyle  A = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}

\displaystyle  B = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}

We can do even better. There are matrices so that

\displaystyle  AB = \lambda BA,

where {\lambda} is a root of unity. They exist as the example below shows for fourth roots of unity. But finding such matrices was curiously hard, at least for me. Lots of web searching.

\displaystyle  A = \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0 \end{pmatrix}

\displaystyle  B = \begin{pmatrix} 1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i \end{pmatrix}

Open Problems

Conrad’s articles are helpful. For a wide variety of topics he presents both theorems and history of math concepts. What I find most attractive is the examples and additional comments that pepper his writing.

Check him out.