A matter of preference

From Incomplete Preferences to Ranking via Optimization
P. Chebotarev, E. Shamis

This fascinating problem deals with how to determine a full ranking of a finite list of candidates given only partial preference data. For instance, if I prefer cola to coffee and green tea to black tea, and you prefer coffee to black tea, the following are compatible rankings for these preference data:

1. cola
2. coffee
3. green tea
4. black tea

and

1. cola
2. coffee
3. black tea
4. green tea

In this simple example there are two possible rankings; but in real-world situations, there are often none. How can one use a ranking to best approximate sets of incomplete prefence data, given that there will probably be no single ranking consistent with all preferences? I (and many others) would really like to know.

Wrapped in a riddle

Slashdot has reported on a project to decode three original Enigma messages intercepted in 1942. The method employed is, of course, brute force distributed computing, combined with statistical methods and some clever optimization tricks.

I had thought that the Enigma had been completely unraveled; as it turns out, even with today's computers, very sophisticated computational tools are required to answer some of its questions.

Revisionism

The first draft of the first paragraph of the paper I've been writing needed a little more depth, so I added a few clauses and then cleaned up the grammar. Once it was printed out and in black and white, though, I saw that the new version was horribly written; repeated words, confusing syntax, and who knows what other barriers to readability. Less can be more, and still not be enough.

Asymptotically approaching truth

Over the past month, I've been writing a paper based on some results that I completed last October. At least, I thought I had completed them. As I type, there are more and more subtle details that I realize I need to consider. What I thought would have been a ten-page note, including an introduction and technical padding, has grown to be twice that length, and I haven't even started adding examples. I really am close to putting it on the arXiv, I swear...

Return to roots

A curious identity and the volume of the root spherical simplex
C. De Concini, C. Procesi

Take a maximal parabolic subgroup of an affine Weyl group. Take the product of its exponents, and divide by the product of its degrees. Now, sum that over all maximal parabolic subgroups; you get 1. Now that's an elegant equation.

And further, acconding to E. Vinberg, this also tells us about the ratio of the volumes of the Weyl chamber and its intersection with the unit sphere. I'm excited to see where this goes.

Fusion coefficients

Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas
E. Ardonne, R. Kedem

The second author is my "academic aunt", having been advised by B. McCoy.

I love combinatorics in mathematical physics.

Fossil fuel lovers

Diesel Sweeties by R. Stevens has featured math humor on two occasions. I hope Stevens includes such gags again.

In 2006 we use the technology of 1958 to recreate the technology of 1849

A. Carol has built a difference engine No. 2 from LEGO.* Not only does this evoke fond childhood memories of building trucks and castles, but it also calls me to consider the role of mechanically assisted computation in mathematics.

The difference engine sits at a fascinating point in the history of the relationship between mathematicians and machines. Because the mechanism by which its atomic components produce its output can be seen, and even touched, the user can maintain intimate contact with the algorithm as it progresses. Compare this to the simplest of modern calculators, or the first vacuum tube computers; their processes are invisible, necessarily rendering them "black boxes", even to their creators. A tremendous leap of faith is required to accept their output as valid. Because the difference engine is what an educator might call a "manipulative", no such leap is needed.

However, I would never trade my computational tools for C. Babbage's technology. The importance of computational mathematics both with an eye to applications and as a means of generating examples, as well as methods of computer-assisted proof (as used for the four-color theorem or the work of S. B. Echad), cannot be understated. Of course, this doesn't even include the immeasurable societal benefit of computational technology that has proceeded in parallel with these advances in the mathematical arts and sciences.

But still, there's that irreplaceable satisfaction that comes from seeing an algorithm happen. As it is, I can only achieve this from examples worked out by hand.

*The plural of LEGO is in fact LEGO.

On the lighter side

A New Upper Bound on Rubik's Cube Group
Silviu Radu

The Rubik Cube has had more success inhabiting pop consciousness than any other big discrete object. I never got really good at it myself; I could solve one side perfectly well, but that's no big achievement. I suppose my ability to hold parallel data in my head has improved in the past 15 years, but I doubt that would be the best use of my time and mental energy. It truly is a mathematical gift that keeps on giving.

From the arXiv

Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection
A. Kuniba, M. Okado, R. Sakamoto, T. Takagi, Y. Yamada

This paper elucidates the relationship between rigged configurations and Young tableaux. Being able to pass back and forth between them is essential to the application of my first result.


Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux
A. Ram

Includes an introduction to crystals and some of the latest exciting generalizations and applications of them.

Just apply yourself!

I'm very fortunate to be in a math department where little to no distinction is made between pure and applied mathematics. In D. Zeilberger's opinion, one who sees such a distinction as impacting the value of their work deserves a rather harsh qualitative evaluation as a mathematician.

Consider the areas of computer science, theoretical physics, and theoretical ecology. Most of the questions addressed by researchers in these fields would be right at home in "pure" math journals; they simply are viewed within the context of other intellectual concerns, and (in the best cases) the research is guided by insights arising from the greater context.

For example; computer science research consists (in part) of questions in probability, combinatorics, logic, complexity, statistics. Mathematical physics involves itself with differential geometry, differential equations, representation theory, and homological algebra. Theoretical ecology is essentially the study of complex systems; historically the only mathematical method for studying such objects was differential equations, but more recently cellular automata, both probabalistic and deterministic, have been used to describe such systems in potentially mathematically tractable terms.*

In all of these cases, the application only enhances the value of the mathematics. Does it really hurt algebraic geometry (or algebraic geometers) to use varieties in the pursuit of genetic alignment? If it does, I'd love to hear precisely how.

If you do "pure" math, get off your high horse. It doesn't make you better than anyone else, in any sense of the word.

*Omissions of mathematical topics are due to my own ignorance; please bring others that deserve mention to my attention.

It's a pun, not a spelling error

So, the natural (albeit somewhat narcissistic) thing to do once you start a blog is to do a blog search for the name, right? And what do I find, but a whole lot of individuals who don't know the difference between "discrete" and "discreet".

This leads to two unpleasant situations:

• A reader who doesn't know the difference sees this blog, reinforcing their error (and not seeing the humor in the name);
  
• A reader who does know the difference sees this blog and assumes that I don't know the difference.

Seriously, learn the language and don't rely on spell-check to clean up your act.

Let $n\in\mathbb{Z}_{\geq 0}$.

This is a repository for my thoughts, observations, and reactions to goings-on in the world of mathematics. If you notice a slant towards discrete math/combinatorics, it's because that's the kind of math I do.

If you can't understand the title of this post, you really should learn TeX.