Mathematics Awareness Month 2009

With April on its way out, it behooves me to take a moment and mention the focus of this year’s Mathematics Awareness Month. April has been bestowed with this glorious title every year since 1986 – last year the topic was Mathematics and Voting, which I discussed at some length in three earlier posts (see here, here, and here).

This year’s focus is on Mathematics and Climate. On the homepage you can find links to a variety of articles, most of which focus on the difficulty in coming up with mathematical models that can accurately reflect the complexity of the interconnected world in which we live. This is perhaps best summarized by Professor Pat Kenschaft, who writes the following in her essay, “Climate Change: A Research Opportunity for Mathematics?”:

How do we analyze the dynamics of the atmosphere, the oceans, the solid earth (especially volcanic emissions) and the biosphere (the system of plants, animals, and other living things)? Scientists have studied pieces of these systems, cutting them both conceptually and geographically, but even the pieces are not tractable by current mathematics, and the challenges as we try to understand the interplay of all phenomena involved are far beyond current conceptual and computational capabilities.

This is a theme that comes up in quite a few of the articles related to this year’s focus on the intersection of math and climate. As we begin to demand more from our models, those models will necessarily need to become more sophisticated. This requires mathematicians to create models that not only reflect reality, but are also optimized so that we can obtain results within a reasonable time frame.

There are a host of other articles discussing the interplay between climate and mathematics. Some of the articles cover related topics as well – for example, Professor Margot Garritsen’s article “Mathematics in Energy Production” provides a good example of the essential role mathematics plays in our current methods for procuring gas and oil, and briefly discusses the relationship between math and alternative energies.

With city-sized blocks of ice crumbling off of the Antarctic, there can be little doubt that climate change is happening, even if we don’t understand everything that underlies it. Will mathematics come to our rescue? Don’t worry – if it doesn’t, I’m hopeful that Captain Planet will.

Captain Planet: Math Spokesman for the 21st century?
April 30, 2009 | No Comments »

How Low Can We Go?

I’m not sure, but this seems like a good candidate for a new bar. According to a recent study out of the University of Washington, as many as half of the population may fail to understand simple probability statements, in the context of weather forecasts.

Here’s the summary:

If, for example, a forecast calls for a 20 percent chance of rain, many people think it means that it will rain over 20 percent of the area covered by the forecast. Others think it will rain for 20 percent of the time, said Susan Joslyn, a cognitive psychologist at the University of Washington who conducted the study.

Coming out of Washington, one would think that the participants would have a better than average understanding of rain forecasts, but now I certainly hope that’s not the case.

That’s American math education for you. Maybe everyone should just move to LA – at least here, the forecast is the same every day.

April 24, 2009 | 1 Comment »

Ballpark Mathematics

Like the dawn of a new day, the start of the baseball season carries with it tremendous promise. These first few weeks provide a reprieve from the breakneck pace of March Madness, where every team is burdened with the knowledge that one loss is all it takes to prevent it from total victory. Instead, the major leagues are a product of the season in which they begin, and just as the warming weather invites us to spend weekend afternoons on grassy knolls looking for shapes in the clouds, so too do the opening games of the baseball season encourage us to let our hair down and reacquaint ourselves with this traditional American pastime.

The American Dream personified?

However, eventually Spring must give way to Summer, and Summer must give way to Fall. As the days grow shorter, so does the window of opportunity for a team to make it into the playoffs, making every game in that final stretch increasingly important for teams that may be on the cusp of attaining a trip to the post-season. As with many things in life, the factors that determine which teams on the cusp will make it through to the playoffs are not entirely within that team’s control – of course they must play to the best of their ability, but they must also hope that those teams in the race with them will falter.

For many, this is what makes the rush to the postseason so exciting. However, if you can mine the treasure trove of data that the MLB generates every year, perhaps you can take some of the mystery out of which teams will ultimately prevail.

With this goal in mind, Professor Bruce Bukiet of the New Jersey Institute of Technology developed a mathematical model 9 years ago to help predict how well each baseball team will perform throughout the season. His model has beaten the odds for several of the past years, and has garnered enough recognition for this article on Yahoo News that was posted towards the beginning of the month.

Predictions versus actuals for the National League Central. The rest of the prediction data can be found here.

What parameters does Prof. Bukiet use to model the season? The model itself seems to be somewhat shrouded in secrecy, but according to the article, “his model computes the probability of a team winning a game against another team with given hitters, bench, starting pitchers, relievers and home field advantage.”

So, if you’re a Yankees fan or a Red Sox fan, this mathematical model has good news for you this year. The Cubs are poised for a good season as well. Of course, there’s no guarantee that the model will give accurate predictions (and for you Giants and A’s fans, you should hope that it won’t), but based on historical performance, the evidence suggests that these predictions will go beyond other expert predictions.

Whether your team fares well under the model or not, the fact that this model can consistently beat the odds speaks to the power of mathematical modeling, when the correct parameters are used. Of course, you may be skeptical about the robustness of this model, given that it has stumbled, and the fact that information about the model is kept somewhat secret.

So, if you think your team is undervalued, I’d encourage you to make a model that you think can best this one. Especially if your model predicts that the Giants will get their act together, already.

April 21, 2009 | No Comments »

Numb3rs in Real Life

For those who don’t believe we can actually use math to fight crime, the story of Harry Markopolos, the man who blew the whistle on Bernie Madoff, shows that a dream of using math to catch criminals need not be untenable. In a recent interview for 60 Minutes, Mr. Markopolos describes how he harnessed the power of mathematics to discover that whatever Mr. Madoff was doing, it had to be illegal.

Bernie’s luck was bound to run out sooner or later, as he must’ve known. His seeming success was really nothing more than a giant Ponzi scheme – in other words, he was able to pay his investors amazing returns by taking money from new investors, rather than by creating new wealth. It doesn’t take a mathematician to realize that such a plan is unsustainable, since the more successful your scheme becomes, the more new investors you require in order to keep the scheme successful. Eventually, the pool of new investors will become too small, and the scheme will collapse. Bernie Madoff must have known this, and this inevitability is perhaps part of the reason why he decided to confess.

If you’d like to put some numbers to such a wordy explanation, you’re more than welcome to. In fact, Professor Marc Artzrouni of the University of Pau in France has attempted to do just that, with a recent paper titled “The Mathematics of Ponzi Schemes.” In it, Professor Artzrouni models the amount in a fund taking into account a collection of 7 parameters, including the rate of return promised to investors (rp), as well as the actual interest rate at which the money is invested (rn). Notice that in the case where rp is larger than rn, investors are promised a rate of return that is less than the actual rate, as in the case of a Ponzi scheme.

Charles Ponzi, founder of the Ponzi scheme. The lesson here: never trust a man who uses that much Pomade.

Professor Artzrouni models the asymptotic behavior of the amount of money in a particular fund subject to different restrictions on the initial conditions. In particular, his models produce three types of funds: those that remain solvent (the amount of money in the fund is always positive), those that collapse, and those that collapse, but could regain solvency with a bailout.

Interestingly, he is able to produce examples of what would be considered Ponzi schemes that nevertheless remain solvent. This can happen in certain situations if the fund manager is able to supply enough outside capital to the initial investment – this initial input on behalf of the fund manager must be enough to offset the fact that the promised rate of return is larger than the actual rate of return. Professor Artzrouni discusses the existence of these so-called “philanthropic” Ponzi schemes in the context of Social Security (so named because the initial capital put in on behalf of the manager goes towards the solvency of the fund rather than being considered an investment on behalf of the manager), which has sometimes been criticized as being nothing more than a government sponsored Ponzi scheme.

Unfortunately, it doesn’t look like Madoff’s fund falls into such a category. What’s even worse, from the 60 minutes article it’s clear that Bernie Madoff was not the only one who knew that he was running a scam. Mr. Markopolos knew as well, and called shenanigans on Madoff to the SEC nearly 9 years ago. Sadly, it seems that the SECs own poor background in financial mathematics blindsided them to Madoff’s antics for nearly a decade, even as Markopolos continued to submit reports detailing Madoff’s fraudulent practices.

The conclusion here is that mathematics can only be used to fight crime when the people fighting it have a strong enough background in mathematics. Or, failing that, every investigative unit should have one go-to math guy, à la Charlie Eppes.

“A Ponzi scheme, you say? Quickly – to the faculty lounge!”

Either way, it’s unfortunate that nobody was able to put Madoff away before he screwed so many people out of so much money. One hopes that next time we will be able to act more quickly when the mathematical evidence so strongly suggests that something bogus is happening.

April 13, 2009 | No Comments »

Numerology Goes on Holiday (Again)

I made my reservations fairly clear regarding the double dose of math holidays last month. Despite my objections, I remained confident that the headlines they gathered would quickly fade away, and I wouldn’t have to worry about these faux math headlines for the next 12 months. In this way, I was able to sleep peacefully at night.

Unfortunately, it seems there are those who wish to disturb my slumber.

Dan Vergano over at USA Today recently wrote a brief article which highlighted the fact that this year there are a whopping 2 “square days,” one of which is today, 4/01/2009. The day is called a square day because if you read the date as a number, the number turns out to be square. In this case, 4,012,009 = 2003 * 2003.

The article attempts to be relevant by making a tenuous link between this sort of mathematical wizardry and the latest film excursion into numerology – Nicolas Cage’s most recent triumph, Knowing. Mr. Vergano was also kind enough to link to my article on Square Root Day, although based on the tone of his article, I’m not sure he appreciated the point I was trying to make. Perhaps he intended to address my concerns, but in the process of writing he got lost in Nic Cage’s eyes. Lord knows it can happen to the best of us.

What’s that, Mr. Cage? Sorry, I got a little distracted.

So, Mr. Vergano, if you’re reading this, I beg of you: use your powers for good. With a readership as large as I’m assuming yours must be, you have a venue to help dispel stereotypes about people who study mathematics. Of course, those stereotypes include, but are not limited to, the idea that mathematicians spend their days looking for significance in arbitrary dates.

To his credit, Mr. Vergano does point out the insignificance of these types of diversions. But if there isn’t any significance, what’s the point in writing about it? Does this happen in other fields besides mathematics?

I get that advances in math may not seem as sexy to the lay person as certain advances in the sciences, and sometimes the ideas can be difficult to communicate. But there are opportunities for those willing to look. Here’s one: why not write an article celebrating the contributions of Mikhail Gromov, recent recipient of the Abel prize? This, it seems to me, would be a much more worthy topic for a writer with such exposure.

I don’t think I’m alone in this, either. A look at the comments to Mr. Vergano’s article reveals many others who fail to see the importance of today as a square day.

Perhaps one day the USA Today blog will discuss some real mathematics. And on that day, I shall declare a legitimate math holiday. Until then, I will remain here and nervously await Pi Day 2010.

April 1, 2009 | No Comments »