Bait and switch

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Katherine Socha

Katherine Socha’s talk, Sea battles, Benjamin Franklin’s lamp, and jellybellies, began with an apology. Socha said the was actually replacing three fluid dynamics and fluid modeling problems with the three terms in the tile, because they were more fun and evocative. The bait and switch, if you will.

Chapter I, on Sea battles, was really about surface waves. Why is it that sometimes (for example, when a bomb is dropped in the ocean) the waves with larger wavelengths travel faster, and other times (like when raindrops fall on water) they travel slower? This has to do with

Samedi Gras

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Vi Hart

Saturday, to me, was about the magic and beauty of mathematics: how it connects to art, how do we learn it, how do we make fun of it, and the big questions ahead.

It was a great pleasure to wake up and go see internet celebrity Vi Hart talk about her approach to hyperbolic space through art. Hart said she didn’t just want to know what it was abstractly, she said that to really understand it, she needed to “feel hyperbolic space”. And so she set out to find a good way of representing it, starting with the usual Poincare disk representation. She tried knitting (a very popular hobby among mathematicians these days), strings and beads, twisting balloons, and even giant models using water bottles as really big beads. It was fun to hear her relate her train of thought from one representation to another. This is where you see mathematicians and artists really think the same way, and in Hart’s case, she is constantly thinking about both mathematics and art simultaneously, in a way that most of us can’t replicate. If you haven’t seen her videos or website, do yourself a favor and click here http://vihart.com/. (A bit of trivia: her dad, George Hart, has recently become Chief of Content at the Museum of Mathematics, which I will talk a bit more about below.)

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Kannan Soundararajan and Brian Conrey

Later in the day I got to see a different kind of genius, Kannan Soundararajan talking about The Riemann zeta-function and related L-functions: A progress report. What I loved about this talk (besides the eloquent delivery by this great speaker) was that it really was a progress report on the zeta-function: from the moment it was conceived by Riemann (1859) to present day. The historical details and even images of the original documents really gave a whole new dimension to a problem that most of us are familiar with, the Generalized Riemann Hypothesis. He also mentioned the applications of proving GRH, from the most immediate, in a way that would tells us more about primes (like finding a polynomial time algorithm to determine primality), to the more “exotic”, like Arithmetic Quantum Unique Ergodicity. He then presented partial progress towards proving GRH and connections with L-functions (which are a generalization of zeta-functions and thus exhibit a lot of the same properties). It was also a real treat to see how Sound’s (as he is called by his friends) work fit into this big picture. There is something really beautiful (there’s that word again) about seeing mathematics in a historical context, and to see a true leader in the field give credit to all the mathematicians that have worked or are still working on a problem.

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The French Quarter

On the way to lunch I was inspired by the light and this strange cloud-line in the sky (are the clouds organizing themselves in an homage to mathematics?), so I took this (overexposed) photo.

After lunch I walked around the exhibits, sacrificing the Current Events Bulletin series of talks. I was surprised to find that Vi Hart was doing a demonstration/group project to build a representation of the hyperbolic plane using balloons, as described in her morning talk. Hart, her father (George Hart) and Glen Whitney (President of the Museum of Mathematics, or the MOMATH, http://momath.org/), encouraged people to blow up their balloons and explained how they should be twisted. Then Hart helped the participants put the pieces together. I’m not sure everyone was understanding what they were making (this is the picture we were trying to replicate http://vihart.com/balloons/hyperbolic/hyperbolictilingsnub2_medium.png), but I’m sure the non-mathematicians went home to google more about the subject, and the rest of us tried to think of how we could do something like this in our classes. All in all, it was really fun to participate in this group effors (my piece of the puzzle wasn’t very symmetric, but I think I got some cool photos, which I have posted on here).

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Hyperbolic fun!

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Had to get that shot

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It's all about the teamwork

After the hands-on fun, I went to a panel on hands-on teaching (and learning) titled Inquiry-proof intrusctional techniques, featuring Dev Sinha, Glenn Stevens, Michael Starbird, David Pengelley, and Margaret Robinson. The five panelists brought different experiences and shed light on different aspects of inquiry-based classrooms. The moral of the story was that every person has their own way of approaching active learning, but there are some key principles that seem to be common to all these styles. Stevens, from Boston University, has worked with the Ross and PROMYS programs, which are geared towards pre-service teachers and talented high school students, and is a problem-based approach to dicovery learning. Stevens said that in these programs the emphasis is on experiencing mathematics, and on using mathematical language as a tool for understanding the experience. Starbird, from the University of Texas at Austin, talked about his experiences using modified Moore method and listed the main principles (which all the other panelists seemed to echo in their own words). The goals for the students, according to Starbird, are: to understand simple things deeply, to ask questions, to learn to learn from failure (not a typo). Robinson, from Mount Holyoke, talked about her writing intensive lab courses. Basically, students learn to understan mathematics through a series of modules and the approach is more geared towards their being able to read and write mathematics, as a bridge from the entry-level courses like Calculus to the upper-level courses like Real Analysis and Abstract Algebra. Pengelley, from New Mexico State University, talked about using historical sources instead of textbooks. Pengelley claims the value of this is it gives students an approach to mathematics that is similar to the approach of the mathematicians who discovered it, more so than the digested and packaged content from a textbook. Sinha, from the University of Oregon, talked about pre-lecture owrk, as a way to get students to think about the material before it is covered in class. This panel was definitely a great source of ideas for people who want to teach classes in this way, and the panelists were very diplomatic in the sense that they did not claim this is the best way to teach period, but rather it’s the way that’s worked best for them.

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Yours truly posing with the debaters and moderator (from left to right, Derivative, Integral, and moderator)

After this I gave my 10-minute talk, which turned out better than I thought it would. Again, I think it’s impossible to give any information in a talk of this lenght other than “I did this.”

In the evening (after waiting for the silly Sheraton elevators for about 15 minutes… I think they should hire one of the many mathematicians in the Sheraton to come up with a better elevator algorithm) I went to the dramatic presentation Derivative vs. Integral: The Final Showdown, by Williams College professors Tom Garrity and Colin Adams (“Integral” and “Derivative”, respectively, and pictured on the left). I will not do this debate justice by relating it to you, but mostly, Derivative’s platform was “derivatives are useful for applications and are easy to compute” and Integral claimed to be more “beautiful and poetic”, on top of going further than the derivative. Our moderator asked for a vote, and Integral won by a small margin. I hope that doesn’t mean I have to teach my Calculus course without using derivatives, because I have no idea how I would go about doing that (which is why I, personally, voted for derivative).

Then I closed the day by eating some more beignets at Cafe du Monde.

Day 4 highlights

Unfortunately, I will be on an airplane going back to real life at Bates College. But here are the talks/events I would go to if I could:

All the AWM Workshop-related activities, like the presentations by recent Ph.D.s and the graduate student poster session.

8:50-9:50a.m. Expander graphs in pure and applied mathematics, III, Alexander Lubotzky, Sheraton Ballroom.

10:05-10:55a.m. Lessons from the Netflix Prize, Robert M. Bell, Sheraton Ballroom.

3-4p.m. From flapping birds to space telescopes: The mathematics of origami, Robert Lang, Sheraton Ballroom.

4:30-5:30p.m. Between the Folds: watch ten artists and theoretical scientists fuse mathematics and sculpture in the medium of origami. AMS-MAA special film presentation in the Sheraton Ballroom. 

Day 3 plan

8:20-8:40a.m. Hyperbolic Planes Take Off!, Vi Hart. Rhythms I, 2nd floor, Sheraton

9-11a.m. WRITING

11:10a.m.-noon The Riemann zeta-function and related L-functions: A progress report, Kannan Soundararajan, Sheraton Ballroom.

Lunch and exhibits

2-3p.m. Counting special points: logic, Diophantine geometry and transcendence theory, Thomas Warren Scanlon, Mardi Gras D, 3rd floor, Marriott.

or

2:15-3:05p.m. Binary quadratic forms: From Gauss to algebraic geometry, Melanie Matchett Wood, Sheraton Ballroom.

or

2:35-3:55p.m. Inquiry-proof instructional techniques panel, La Galerie 6, 2nd floor, Marriott.

3-4p.m. Spaces of graphs and surfaces – On the work of Soren Galatius, Ulrike Tillmann, Mardi Gras D, 3rd floor, Marriott.

4:30p.m. Talk!

6-7p.m. Derivative vs. Integral: The final showdown, dramatic presentation by Colin Adams and Tom Garrity. Armstrong Ballroom, 8th floor, Sheraton.

7:30-8:15p.m. Increasing the pool of underrepresented mathematicians, Robert Bozeman, Waterbury Ballroom, 2nd floor, Sheraton.

8:30-10:30p.m. MAA/Project NExT Reception, Gallery Ballroom, 1st floor, Sheraton.

Thank Gauss it’s Friday

Day two was mostly about applied mathematics and running around to get to places on time for me.

I have noticed that it’s impossible to walk in a straight line, or at least at constant speed, at these meetings. Every few paces you will recognize someone, or they will recognize you, and you will start chatting. Usually it’s someone you haven’t seen in a long time, and chatting to catch up takes a while. Suddenly, you’re late to everything. I have even been avoiding the gaze of some people I recognize and I put my best “I’m writing a blog, giving a talk, and trying to eat and sleep a reasonable amount, so sorry I can’t talk” face on. Of course, you can also embrace this. I was talking to someone yesterday who said that even when he wasn’t at a meeting, he would run into people at the exhibits and suddenly they were having a serious conversation, and then, a meeting. Andrew Bernoff joked, as he introduced Andrea Bertozzi’s SIAM invited address, that he would like to model (with Bertozzi) the random walks of drunk mathematicians on Bourbon street. I think they should extend their research to the modeling of Joint Math Meetings walks, which are not random, but more pinball-like.

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Katherine Socha

The day started with Katherine Socha’s lovely talk entitled Sea battles, Benjamin Franklin’s oil lamp, and jellybellies. I will post a more complete summary later, but I will share with you now the main reason Socha’s talk was so compelling to me.  She focused on making the modeling of fluid flow (which at its most formal requires a deep understanding of PDEs, or Partial Differential Equations) accessible for teching undergraduates how Multivariable Calculus and Ordinary Differential Equations relate to some real-life examples. She also managed to put these ideas into a historical and biological context, even quoting poetry and reading us limericks.

Later, I went to the end of the Beauty and Power of Number Theory session to catch Harold Stark’s talk, entitled Landau’s Class Number Theorem: A Gem That Wasn’t. That was a nice, albeit rushed, number theory talk. I wish he’d had an hour rather than 20 minutes to give it.

I attempted to get to the second Lubotzky Colloquium on time, but was about 10 minutes late. I was able to appreciate that expanders have been a useful tool in understanding the Hardy-Littlewood conjecture, which is basically a more general form of the twin primes conjecture. Expanders are so essential, Lubotzky says, “if they hadn’t been discovered by computer scientists they would have been discovered by number theorists.” He said that this was an example of when tools have existed before the theorems they are most useful for (“the tools develop faster than the dreams”), which seemed pretty poetic to me. In the end, there is a proof for the twin primes conjecture but with “almost primes” using expander graphs and the equivalent statements on property (tau).

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Andrea Bertozzi

Soon after, I watched Andrea Bertozzi give her invited address on Self-organization in human, biological, and artificial systems. In this talk, Bertozzi explained how she models crime and swarms in terms of their self-organization, and how this could be applied to create self-organizing artificial intelligence. I will write more about this soon.

At the end of the day, with a belly full of crawfish etouffee and beignets, drinking a Purple Haze at the Marriott bar, I found myself unable to upload anything to the blog, because the internet is too slow. I believe we should have a new phrase, more appropriate for our times, that substitutes “It’s like watching paint dry” and “It’s like watching the grass grow.” My proposal is “It’s like watching the progress bar fill up.”

It’s just as well, since I needed to finish writing my talk for Saturday. There is something extremely challenging about summarizing your own research into a ten-minute talk. I hope it makes sense.

Indeed, Thank Gauss it’s Friday.

More Photos

JMM 2011

Thursday in Pictures

I decided to finish my accounts of Day One comic-book style. After all, this is the city of laissez faire, so I will let myself do as I do.

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Denis Auroux

The first talk I attended at the meetings was Denis Auroux’s, entitled The symplectic geometry of symmetric products and invariants of 3-manifolds with boundary, in the Sheraton Ballroom at 10a.m. Auroux started by apologizing for how technical-sounding his title was, and said that it was in fact a very accessible talk. He said he preferred this to talks with simple-sounding titles that turn out to be very technical. As someone who got lost quite quickly (Heegard-Floer TQFT -Topological Quantum Field Theory – was mentioned in the first five minutes) I think the title was just right…  (To the left, Denis Auroux answers some questions from an audience member.)

After this I had lunch at the Cafe Fleur de Lis, had a decent spinach salad, but my lunch buddy ordered the BLT which turned out to be a triple decker with a ton of bacon in it. Let’s just say it’s not for the faint of heart.

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Alexander Lubotzky

From 1-2pm I attended Lubotzky’s first colloquium lecture, which was so good it got its own blog entry. Search January 6 posts for more on that.

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Yuval Peres

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Yuval Peres

Immediately after I attended Yuval Peres’s intriguing talk Laplacian growth and the mystery of the abelian sandpile: A visual tour. Peres explained a few computer simulations that were very closely related and which exhibited interesting common patterns. They all have the same initial setup: a lattice on the plane, and a particle at the origin. In Diaconis-Fulton addition, one adds another particle at the origin, and this particle takes a random walk on the lattice until it finds an unoccupied spot. If this experiment is repeated many many times on a computer, the limiting shape seems to be a disk. Interestingly enough, in a slightly different example, the so-called Rotor-Router model, each spot on the lattice has a direction, and the particle moves according to the direction the “origin” is pointing in. This arrow rotates clockwise (or counterclockwise? I forget) every time the particle moves. The particle again stops moving when it encounters an open spot, and the limiting shape is still a disk. The third model is the sandpile model. In this case, the lattice points all have mass 1, and when we add mass at any point, the site divides the excess equally among its four neighbors. If one starts with only mass 1 at the origin and adding mass 1 repeatedly, the limiting shape also seems to be a circle, with mass 1 in the interior, fractional mass on its boundary, and zero mass on the exterior. The big result is that one can use known results in free boundary problems in PDEs to understand and prove these patterns, namely that regardless of the model the limiting region is a disk. Multiple starting points are more complicated, but seem to still be bounded by algebraic curves. Finally, Peres presented the “mysterious” abelian sandpile problem. I found it interesting that it seems like the simulations are ahead of the math. In my experience, simulations are used to understand the math, so it is nice to see these roles reversed.

I then went to a couple of receptions. The first was a reception for FOCUS magazine affiliates (I just became a member of the editorial board), and to honor outgoing editor Fernando Gouvea for his many years working for the magazine. This was a fun event and a great way to meet other math-writing enthusiasts.

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Harry Lucas

The next reception was organized by Project NExT (I was a Project NExT Fellow last year) and the Educational Advancement Foundation. The EAF is focused on promoting and supporting Inquiry Based Learning, and was founded by Harry Lucas (pictured on the left). Lucas took a mathematics course taught by R.L.Moore in the University of Texas at Austin, and from then on he was hooked on Inquiry Based Learning and helping other people learn about this style of teaching. I myself have taught a few courses this way, and have found it a very rewarding (albeit challenging!) experience. It was great to see a lot of the people who have mentored me on these teaching issues, as well as other Project NExTers.

Dinner was at a Bennachin Restaurant in the French Quarter, which was quite good (they serve African food).

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Marriott Lobby

Afterwards, I worked for a while at the Marriott Lobby with a friend. It was quite funny to see so many mathematicians gathered at the bar working, talking, and checking email. I don’t think I’ll ever get used to how many of us there are everywhere. On the left, you can get an idea of what it was like.

As expected, I didn’t go to everything I wanted to go to and I didn’t get to write as much as I had hoped, but I ran into a ton of people and I had a blast.

2011 Who Wants to Be a Mathematician

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Expanders are not just for dentists

Today, I had the pleasure of watching Alexander Lubotzky, in a packed ballroom, give the first part of his colloquium lectures entitled Expander Graphs in pure and applied mathematics. Lubotzky managed to pack a lot of information into a one-hour talk and to do so in a delightful and understandable manner. I will attempt, in these next few days, to write about all three colloquium lectures and summarize the main ideas.

In this first installment, Lubotzky focused on introducing the main concepts and motivating the importance of these expanders in computer science. I will not write the formal definition here, but I will say that a graph X is called an expander (rather epsilon-expander) if it is somehow highly connected: any subgraph with less than half as many vertices as the total graph has lots of “neighboring” vertices. So in a sense expanders are “fat and round”. It makes sense then that they are the building blocks of many communication networks (since one would like these netwroks to be very connected).

Expanders were first studied in 1967 by Barzdin and Kolmogorov, in terms of networks of nerve cells in the brain, but they were not formalized until 1973, when Pinsker used them to study communication networks. In general, computer scientists have become interested in studying families of expanders, with a number of vertices that approaches infinity, k-regular for a fixed k (as small as possible), and a fixed epsilon (as large as possible). The graph with the largest epsilon would be a complete graph in n vertices, but this is not computationally feasible (would require millions and millions of connections), which is why we restrict the degree of the vertices.

But what are the applications of expanders? Lubotzky decided to google the term “expanders”, and got 4 million hits. He realized, however, that many of these hits were related to dentistry, not mathematics. So he refined his search by googling “expander graphs” instead, for which he got 400,000 hits,

including communication networks, psudorandomness and Monte Carlo algorithms, derandomization, and error-correcting codes. He also found many great images, but quickly cautioned that “the worst way to learn about graphs is to look at them. You learn nothing from looking at a graph.”

To really apply anything about expanders one needs to be able to construct explicit graphs with these desirable properties, and herein lies the challenge. This is where Kazhdan’s property (T) and property (tau) from representation theory lend a hand. I will now skip some details, but the upshot is that given a finitely generated group G and a family of finite index normal subgroups of G (denoted L), G has property (T) implies that the Cayley graphs of G/N_i (N_i in L) are expanders, and G has property (tau) with respect to L if and only if the Cayley graphs of G/N_i (N_i in L) are expanders. So this gives us a way of constructing explicit expander graphs: take a group with property (T) (like SL_n(Z) for n>2) and the Cayley graphs of G/N_i are a family of expander graphs. There are also equivalent statements relating expander graphs to random walks, measure theory, and geometric and analytic aspects of Riemannian manifolds.

The big, final theorem of the day (conjectured bu Lubotzky, Babai, and Kantor in 1989 and proved recently byt the work of many mathematicians) is the all non-abelian finite simple groups are expanders in a uniform wya (that is, same k and same epsilon). Lubotzky ended the talk by teasing the next lecture, the fact that the generalizations that went into proving this last theorem have far reaching number theoretic applications.

On a final note, here is my favorite moment from the talk (not related to the mathematics, but rather the delivery): Lubotzky stops mid-talk and says to the audience “In Israel, we are very informal. But I was told that you need to wear a suit and tie to show respect for your audience. OK, now you know I respect you…” and he took off his jacket and tie.

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Photo by E. David Luria

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Photo by E. David Luria

Day 1 Photos

Some more photos taken on the First Day.

2011 Joint Math Meetings Day 1