When I got my new MacBook Air, I had to switch from my old email client, Eudora, to MacMail, which seemed like the best alternative. I often work offline, especially when traveling, because the internet can be expensive and often unavailable, so I need some such client. MacMail has the following inferiorities:

1. When you draft or edit draft messages, it forgets the new subject line.

2. You can't edit a message you've received, despite its misleading subject line or your own need to annotate. Of course you can take it out of MacMail, edit it, and put it back, but that is much too much trouble.

3. You can't reply to multiple messages at once. If I email 1000 students about a tutoring opportunity and get 25 responses, I want to reply to all at once, not one at a time or build a reply list one at a time. I installed a script for this by Aral Balkan using Script Editor, although I had to remove the line that reads "set the was replied to of (item i of the themessage) to true from the script."

4. If you're filing a message and find the desired mailbox doesn't yet exist, Eudora let you choose "New Mailbox." MacMail makes you leave the task and go create a new mailbox.

5. Otherwise filing messages works as in Eudora, but if you want to open a mailbox, you have to use the sidebar, very awkward if you have thousands of nested mailboxes as I do.

6. You can't queue a message for sending at a specified time, perhaps as a meeting reminder or solutions to a take-home exam.

7. In Eudora you could take a list of 100 email addresses and put them in a group. In MacMail, you have to add them (one by one?) to Contacts, and then move them (highlighting them one by one from the list of perhaps thousands of contacts) to the group. In practice, I don't bother, or I create an alias under SystemPreferences:Keyboard:Text.

8. You can't easily embed animated gifs in MacMail as in Eudora.

9. If using MacMail with gmail, each message is limited to 99 recipients, so when I email our 900 math/stats students, I either have to break the message up into 10 messages or go to gmail online to send it. Apple should have its own unrestricted smpt server without such a limit and figure out the requisite security.

Why in the world wasn't Eudora maintained? And after all these years, why hasn't MacMail corrected these inferiorities?

Tell me if you know how to do or work around these obstructed tasks.]]>

When I ask my kids what's 4x7, they answer "28."

7x4? "28."

4+7? "11."

7+4? "You already asked us that."

Commutativity of multiplication is true, but commutativity of addition is obvious. Later on those youngsters will find that for matrices, addition remains commutative but multiplication does not.

"Now if you could only make fractions interesting," the math faculty responded. The thing about adding fractions such as 1/3 + 1/4 is that nothing sensible works. If you add the numerators, what do you do with the denominators? Add them? Doesn't work. Multiply them? Doesn't work. Of course if the denominators are the same, it's easy: 1/5 + 2/5 = 3/5 just as 1 apple + 2 apples = 3 apples. But if the denominators are different, you apparently can't add the fractions any more than you can add 1 apple + 2 oranges. There's no such difficulty with multiplication; you just multiply the numerators and multiply the denominators: 2/3 x 2/5 = 4/15. OK, there is a way to add fractions by finding a common denominator, but this is not the time to tell the students that. Here's how their test should go:

Question: 1/5 + 2/5. Correct answer: 3/5.

Question: 1/3 + 1/4. Correct answer: Can't be done.

After all, this is how we teach square roots. √4 = 2, √-4 can't be done. Only much later do we thrill them with √-4 = 2

Then, later, when you note that 1/3 = 4/12 and 1/4 = 3/12, a bright rebel student will say, "So we can add them!" At that moment they are ready to delight in doing the impossible.]]>

This study suggests that at least under certain circumstances it may be more effective as well as less expensive to have the best teachers teach large lectures than to have many, less effective teachers teach smaller sections.

The paper actually discusses two studies at two different universities. The first compared small (30) to large (90), and neither large nor small classes had recitation sections. The second study compared small (20-35) to large (150-240) and both large and small had additional small recitation sections, so in one sense even the large sections had some small class time. In both cases teacher effect dwarfed all else, and the class size effect was not significant. Only at the second university did the study look at dropout rates.

David Bressoud, former president of the Mathematical Association of America, called section size

"a question I'm often asked and for which I know of no direct and meaningful evidence, at least with regard to undergraduate mathematics. ... Let me add that if a university chooses to go with small classes taught by graduate students and adjunct faculty, then coordinating and supervising the teaching of those classes is extremely important. I've seen disasters with small classes. Our study has also seen successes with large classes, but there are certain things that are critical to making them work: careful preparation and supervision of the recitation instructors, a highly supportive mathematics tutoring center with mechanisms for ensuring that students who need it use it, and the provision of feedback loops to the lecturers that foster active participation by the students in the class and give them a sense that the instructor is responsive to their questions and confusions. Clickers are one way of accomplishing this. One-minute quizzes, 'pair and share,' on-line feedback, or just long pauses built into the class to give students time to formulate questions are other techniques that can be employed successfully."

My own best personal experiences in teaching calculus have been with larger lectures: untypical special sections close to 100 at Williams, typical lectures of about 250 at MIT, and the very best a lecture of 100 at Rice. On the other hand, at least for more advanced courses, very small classes from about 10 to 20 seem much better. In general, the most popular courses at Williams are our signature Tutorials, in which students meet with faculty in groups of two once a week and present the material themselves.

______________

*"Class Size and Teacher Effects on Student Achievement and Dropout Rates in University-Level Calculus" by Tyler J. Jarvis of Brigham Young University.]]>

A student in my Investment Mathematics class asked me how often you need to rebalance. The most common guideline we found was "at least once a year," but we knew that the answer should depend on the value

For a portfolio of valueV, letabe the cube root of 15000/V, and rebalance everyayears.

So if you have $15,000, rebalance every year. If you have $120,000, rebalance every

six months. If you have $1 million, rebalance every three months. If you have $25 million, rebalance every month. If you have $1 billion, rebalance every nine days. If you have $1 trillion, rebalance every day.]]>

The only bad effect is in the summer, if it's hot, and I'm home, and I have lights on. I don't often have them on during the daylight hours, which last late in the summer. And even if it's dark and I'm home, I don't have them on if I'm watching TV. But once in a while I'm home in the evening and I'm reading or working and I need to have a light on. This is a relatively small consideration. And even this I can address, by having a few lamps around with energy-efficient, fluorescent light bulbs.

Here's a much better way to save energy. When you take a bath or a shower in the wintertime, don't let the hot water go down the drain. Keep it in the tub for a couple of hours to warm up the house.]]>

You start out heading about 25 degrees right of the base line and run with acceleration of constant maximum magnitude. You slow down a bit coming into first, hit a local maximum speed as you cross second, start the final acceleration home a bit before crossing third base, and reach maximum velocity as you sprint across home plate. The total time around the bases is about 16.7 seconds assuming maximum acceleration 10 feet per second per second, about 25 percent faster than following the baseline for 22.2 seconds (coming to a full stop at first, second, and third base), and about 6 percent faster than following a circular path for 17.8 seconds. The record time according to Guiness is 13.3 seconds, set by Evar Swanson in Columbus, Ohio, in 1932. His average speed around the bases was about 27 feet per second.

Is it legal to run so far outside the base path? The relevant official rule of Baseball says:

7.08 Any runner is out when -- (a) (1) He runs more than three feet away from his baseline to avoid being tagged unless his action is to avoid interference with a fielder fielding a batted ball. A runner's baseline is established when the tag attempt occurs and is a straight line from the runner to the base he is attempting to reach safely.The rule just says that after a tag attempt the runner cannot deviate more than three feet from a straight line from that point. The rule doesn't apply until the slugger is almost home, when our fastest path is nearly straight. So our path is legal.

The model's only assumption is an upper bound on the runner's acceleration vector, which we take to be 10 feet per second per second for the times given here. For a better runner the acceleration might be greater and the times correspondingly faster, but the optimal path is the same. The markings around our optimal paths show the acceleration, forward at the beginning and end, leftward for most of the path.

The fastest path to second base is more symmetric, because you start from rest at home and you must come to rest on second base. So the path veers outward just as much from home to first as from first to second, starting at a larger 28 degrees to the right of the baseline. It takes 10.4 seconds (assuming acceleration of 10 feet per second per second), as compared to 12 seconds along the baseline, coming to a full stop to make the sharp turn at first and second base.

The fastest path to third base is also symmetric, with the big bulge between first and second and a smaller initial angle of 23 degrees right of the baseline. It takes 14.5 seconds, as compared to 18 seconds along the baseline, coming to a full stop at first, second, and third base.

This work began with a colloquium talk I advised at Williams College by undergraduate Davide Carozza:

My colleague Stewart Johnson was in the audience. He realized that he could compute the fastest path accurately, as pictured above.

If you consider the different voter turnout rates in different states, you could win with still fewer votes by winning states with low turnouts. In 1992, voter turnout varied from 42 percent of the voting-age population in Hawaii to over 70 percent in Maine, Minnesota, and Montana. A recent article in

For such reasons many recommend that the Electoral College voting system be replaced by election by popular vote. Popular vote not only would be fairer, but also would encourage the candidates to campaign in the whole country, instead of focusing on a few battleground states.

Karen King, Director of Research for the National Council of Teachers of Mathematics and MathFest's 17th Falconer Lecturer for the Association for Women in Mathematics, explained that high school teachers and students need to focus on meaning rather than just procedures. It's one thing to know you cannot divide by 0; it is more to know why. (The official reason is that x = 1/0 means that 0x = 1, which is impossible. The intuition is that 1/a, as a gets small, grows without limit. And what if a is negative? Actually mathematicians have a special number system in which infinity ∞ is a number, -∞ = ∞, and 1/0 = ∞.) King emphasized that searching for understanding is a good habit that has to be learned and relearned for real education. A high purpose of education is to develop such mental habits. We need to explain that to future teachers, students, parents, client disciplines, and government.

Karen King, Director of Research for the National Council of Teachers of Mathematics

My old friend Sylvia Bozeman of Spelman College and 14th Leitzel Lecturer spoke on the importance of mentoring, especially for minority women. One needs good mentors and a culture to support them. Historically black colleges, which provide such a culture, enroll just 10% of African-American students but award 30% of all African-American baccalaureate degrees in mathematics and statistics.

Sylvia Bozeman of Spelman College

John Ewing, President of Mathematics for America, told me of the importance of a community of excellent mathematics teachers in our high schools:

John Ewing, President of Mathematics for America

Mathematics itself is always growing. Robert Ghrist of the University of Pennsylvania, in his invited address, presented algebraic topology as the new mathematics which will revolutionize education and society. To a mathematician, all interconnections and relationships can be viewed geometrically and then understood algebraically.

Robert Ghrist of the University of Pennsylvania

Student talks are a highlight of mathematics meetings. A student team from the University of Colorado, Boulder -- Christopher Corey, Stephen Kissler, and Sean Wiese -- talked about their winning entry in the Mathematical Contest in Modeling. They designed a plan to get the maximum number of rafting trips through the Grand Canyon.

Stephen Kissler, Christopher Corey, and Sean Wiese

Another winning student team from Cornell, Dennis Chua and Alvin Adrian Wijaya, explained and predicted the shapes of trees and leaves.

Alvin Adrian Wijaya and Dennis Chua

Everyone deserves to feel the joy of discovering and understanding some new bit of mathematics.]]>

Or maybe you copy something you'd like to keep for a while to share with everyone you write to. But if in the meantime you use copy and paste for something else, it's gone.

It wouldn't have to be like that. From the beginning one of the most fundamental computer operations,

If you'd just been to say bridge nationals, you could write a few pertinent sentences,

Incidentally this feature is available as aliases in my beloved mail application Eudora, which unfortunately doesn't run under the latest Mac operating system.

A laptop's long-term memory is huge, and it never forgets as long as it lives. But a laptop's short-term memory, in the form of copy-paste, can remember only one thing at a time. That's got to change.]]>

I was on my way to a math conference in Granada, Spain. I have lots of friends in Granada, including my collaborators on the Double Bubble Theorem, my only paper to appear in

Fortunately they made it back to Granada in one piece, although Sean had his wallet and passport stolen in the train station.

But most of the time they did math in their rooms, and they proved some nice results. Alex summed it up like this:

So it was fun to see familiar places and friends on my return this week. The conference boasted talks by leading theorists and experimentalists. The three local organizers were my old friends Rafa López and Paco Martín and Antonio Ros.

Here Ros, an illustrious theoretician, describes in five words why he likes such conferences:

One excellent talk was by a University of Delaware graduate student, Nicholas Brubaker. In 2007, he had worked on a research project I suggested along with other undergraduate students of Ron Umble at Millersville University. They studied soap bubbles in a cube. Previously a graduate student team at a summer school at the Mathematical Sciences Research Institute in Berkeley, California, had computed candidate double bubbles:

Nick and his collaborators managed to reproduce the candidates physically, thus demonstrating their physical stability. Here are their "center bubble" and "slab cylinder":

Here, Nick tells about creating these soap bubbles:

The best single bubble (

It is hard to see a pattern. Yes, the individual bubbles are becoming hexagonal, the optimal shape for filling the whole plane, as described in our recent post. Most of the interior bubbles have six sides, and most of the boundary bubbles have five sides, but some (colored black) have more sides, and others (colored gray) have fewer sides. Especially focusing on such cases as

We made progress at the conference. After further computations, the next month Cox sent me the following email message:

Date: Wed, 25 Apr 2012

From: Simon Cox

To: Frank Morgan

Subject: large circular clusters

Congratulations Frank!

You win. ...

Simon

For example, for

We posted a joint paper with these results on Tuesday. It is not yet clear that the clusters will keep getting rounder, but I think that they will.

Most of these results are just numerical. The only cases mathematically proved optimal are the single, double, and triple bubble (

The problem is that inserting a leap year every four years makes for an average calendar year of 365.2500 days, whereas the actual "tropical" or seasonal year is 365.2422 days long. After 100 years, summer would arrive 100(.2500-.2422) ~ .78 days earlier. So we should eliminate about 3/4 of a leap day every century, or about three leap days every 400 years. And we do! Only one in four century years is a leap year: 2000 was a leap year, so 2100, 2200, and 2300 will not be leap years. By 2092, summer will come very early: at 1:14 a.m. on June 20, but the omission of a leap year in 2100 will mean that 100 years from now, in 2112, summer will be back at 9:14 p.m. June 20. And then in another hundred years, after the omission of a leap year in 2200, it will make it back to June 21. (I'm comparing just leap years, since summer comes a bit later on non-leap years.)

These century adjustments give our current calendar 365.2425 days per year, still a tiny bit longer than the actual 365.2422 days. This suggests that we should skip an additional leap day in about 1/.0003 years, or about 3000 years.

But there is much more to the story. The length of the year is changing! First of all, the earth's revolution is gradually slowing down, about one day every ten million years. (Then will the year become 364 days?) Amazingly enough, there is a much bigger effect in the opposite direction. Like a spinning top running down, the wobbling or precession of the earth's axis speeds up. Since our seasons are caused by whether the axis tilts towards or away from the sun, this increasing precession causes the year to speed up, currently at about one day every 167,000 years. This would cause an accumulated error in the calendar of one day after about 600 years, around the year 2600. Note that this effect is much greater than that due to the current mismatch between the calendar and the year.

There is another big effect, often overlooked: The day is getting longer, and longer days mean fewer days per year. The lengthening day requires the occasional addition of a "leap second" and the readjustment of all accurate clocks worldwide. Why is the day getting longer? Friction with the tides for example is causing the earth's rotation to slow, although there seem to be other complicating influences and this effect seems irregular and hard to measure. Ancient rocks that recorded daily lunar tides indicate that about a billion years ago the day lasted only about eighteen hours. If in a billion years the day lengthens by a factor of 4/3, the number of days in a year goes down by 3/4. In a billion years from now the year would lose about 91 days, or one day in about 11 million years. Actually, current controversial measurements suggest we are now losing about one day per 150,000 years, causing an accumulated error in the calendar of one day after about 550 years. One possible cause is the increased inertia of the earth due to huge water reservoirs behind dams. Combined with the previous effect of increasing precession, this yields a one-day error in about 400 years.

Meanwhile a controversy has raged over a peculiar definition usually used by astronomers. They mark the year by the beginning of spring, while they probably should average over all the seasons. It makes a difference because the earth's orbit is not perfectly round, but a bit elliptical. Precession has a larger effect on the summer solstice, when we are farther from the sun, and a smaller effect on the other seasons. As far as I can tell, the effects of this peculiar definition currently may be canceling out the other effects and keeping our calendar in almost perfect agreement with the tropical year, at least for the coming millennia. So nothing to worry about.

Wednesday morning I visited the spectacular new home of the new National Science Foundation mathematics institute at Brown University, the Institute for Computational and Experimental Research in Mathematics (ICERM). Their 11th floor lecture hall has three floor-to-ceiling glass walls overlooking Providence and the waterfront:

In the lobby of the lecture hall you can technologically eavesdrop on everything occurring inside. From this impressive lobby, Director Jill Pipher describes their recent "Day of Data":

The institute has all kinds of activities for experienced mathematicians, post-docs, graduate students, and undergraduates.

That Wednesday afternoon I spoke at the final meeting of Connect, a successful collaborative project of six southeastern Massachusetts colleges and universities to modernize mathematics curriculum. My host, Mary Moynihan of Cape Cod Community College, says, "Imagine being given six years to discuss courses with your colleagues. It's been wicked fun." Moynihan goes on to describe what delights her the most:

Statistics has become a top national curricular priority; we have to deal with lots of data and understand what we're doing.

Friday through Sunday I attended the annual Geometry and Topology Conference at Lehigh University, one of the most open and welcoming conferences I know. Everyone is invited to come and give a talk, university dorm rooms are available for $35 a night, and the organizers treat everyone as important mathematicians.

David L. Johnson, chief organizer of the conference and a past mathematical collaborator and old friend of mine, describes what makes it all worthwhile for him:

One of the speakers, Christina Sormani of the City University of New York Graduate Center and Lehman College has developed with Stefan Wenger of the University of Illinois at Chicago a new way of deciding mathematically whether two universes closely resemble each other:

Their new way of measuring similarity is called the "intrinsic flat norm."

Don Davis at Lehigh coaches an amazing American Regions Math League (AMRL) high school team which has won the national championship for the past three years. He appears in the following video with one of his team members, Matthew Kilgore, whose disability does not seem to have slowed him down.

Matthew's mother Brita calls Davis and his assistant coach Ken Monks the "most devoted coaches ever." Monks recognizes social teamwork over years of training as the key ingredient:

Monks's daughter Maria won the 2011 national Morgan Prize for undergraduate mathematics research (endowed by my mom Brennie J. Morgan).

Paul Martino, a successful entrepreneur who sponsors the ARML Lehigh Valley team to the tune of $6,000 per year, describes why:

Why shouldn't math students have the same opportunities as sports teams?]]>

Still an open question is the optimal shape for a two-holed doughnut, perhaps something like this

Actually Robert Kusner of the University of Massachusetts at Amherst has a conjecture for the optimal doughnut of any given number of holes, once again based on lovely and famous twins in the parallel spherical universe, which were discovered in 1970 by Blaine Lawson of the State University of New York at Stony Brook.

Marques also announced another, newer, related result with Ian Agol and Neves on the optimal positioning of two linked curves. You might think it best to take two linked round circles, but mathematicians have a better idea: a horizontal circle around a straight vertical line -- which mathematicians think of as an infinite circle:

Another invited speaker, Simon Brendle of Stanford University has recently proved the Lawson conjecture that Willmore's optimal doughnut's lovely twin (the "Clifford torus") in the parallel spherical universe (the "three-dimensional sphere" or hypersphere) has a special feature: it has less area than any other equilibrium doughnut. Here "equilibrium" means that the curvatures of the doughnut surface in orthogonal directions are equal and opposite. In an everyday doughnut in the familiar "Euclidean" universe, this happens on the inside, which curves inward as you go around and outward as you go up and down, but it fails on the outside, which curves inward as you go around and inward also as you go up and down. On the other hand, the lovely twin in the parallel spherical universe has such delicate equilibrium everywhere. It's more like a soap film without bubbles on a wire frame, which always curves up in one direction and down in the other. Here's how Brendle describes his proof of the Lawson conjecture for you:

The Lawson conjecture is a classical question about the shape of soap films in a three-dimensional sphere. I have long been fascinated by this conjecture, and I have tried many different approaches to the problem. However, the Lawson conjecture remained a mystery until recently, when the different pieces of the puzzle finally started fitting together.

His proof, technical but short (just seven pages long), first shows that for the least area equilibrium doughnut, the much more complicated "Gauss" curvature must be constant. Then the celebrated Gauss-Bonnet Theorem, which relates this Gauss curvature to the number of holes, implies that the Gauss curvature must be 0, which identifies it as the lovely twin of Willmore's doughnut.

In addition to Marques' and Brendle's talks and six other invited, hour-long talks, the Geometry Festival has lots of time for informal discussions over meals, snacks, and a Saturday-night banquet.

Geometry Festival organizer Hubert Bray of Duke University, this year's host institution, told me why he loves these events:

I agree.]]>