Author Archives: lsloman

AMS panel on math and motherhood

To top off JMM, I went to the AMS panel discussion on mathematics and motherhood hosted by Carrie Diaz Eaton of Bates College. Panelists Karoline Pershell, executive director of the Association for Women in Mathematics, Karen Saxe, the associate executive director of the AMS, and Talithia Williams, an associate dean and professor at Harvey Mudd, discussed everything from how having children affected their careers, family leave policies and informal support networks, role models, and accepting outside help. Here are some highlights from the panel, edited for clarity:
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James Tanton’s proofs that 1 = 2

This afternoon at Mathemati-Con, James Tanton proved to us 12 different ways that 1 = 2. Here are just a few of his arguments:

1. Since $i^2 = -1$,
\[ 1 = {i^2 + 3 \over 2} = {(\sqrt{-1})(\sqrt{-1}) + 3 \over 2} = {\sqrt{(-1)(-1)} + 3 \over 2} = 2 \]
2. Take a unit square. The diagonal has length $\sqrt{2}$ by the Pythagorean theorem. But we can also approximate the diagonal using a staircase. The lengths of the horizontal segments of the staircase must add to 1, as must the lengths of the vertical segments. So the staircase, no matter how closely it approximates the diagonal, always has length 2. As we make the approximation better, this means that $\sqrt{2} = 2$ and therefore $2 = 4$ implies $1 = 2$.
3. Write $ 0 = 0 + 0 + 0 + \dots$. Since $0 = 1 – 1$, we have
\[ 0 = (1-1) + (1-1) + (1-1) + \dots \]
But if we rearrange the parentheses, we get
\[ 0 = 1 + (-1+1) + (-1+1) + (-1 +1) + \dots = 1 + 0 + 0+ \dots = 1. \]
Thus $0 = 1$; adding 1 to both sides gives $1 = 2$.

See if you can spot the mistakes in each of the arguments. Tanton ended with the following irrefutable “proof by shopping”: At a store holding a 2-for-1 sale, he asked the sales associate how much it would cost to buy 1 item. She responded that it was the regular price–that is, the same price as buying 2 items. Hence it must follow that $1 = 2$.

World Premiere of “Secrets of the Surface”

This evening I went to the premiere of “Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani”, a film chronicling the life and mathematical achievements of the first female and first Iranian Fields medalist. The film walked through Mirzakhani’s life, from childhood in Iran up to her work on the moduli spaces of Riemann surfaces, her family, and her death from cancer in 2017.

Following the screening was a Q&A moderated by Hélène Barcelo of MSRI, with panelists Ingrid Daubechies, Amie Wilkinson, Jayadev Athreya, Tatiana Toro, all mathematicians who knew Mirzakhani; also on the panel were Erica Klarreich, a math journalist who narrated the film, and George Csicsery, the director and producer. Here’s an incomplete and slightly edited transcript of the panel.

CSICSERY: This film could not have been made without MSRI. One person who I know would love to be here is David Eisenbud, who initiated this project a little over two years ago and could not come tonight. He was extremely obsessed with the accuracy of the mathematical narration and animation.

Q: Can you talk about raising a family as a woman mathematician (or otherwise)?

DAUBECHIES: It’s a question I thought about when I was young, because the famous mathematicians I knew of (Emmy Noether) didn’t have families. But then I met Cathleen Morawetz and she had 4. And you only need one counterexample to know it’s not a theorem.

CSICSERY: I interviewed Cathleen Morawetz a few years ago and she said it was a piece of cake.

DAUBECHIES: You just need to keep track of what is important to you.

TORO: I think they complement each other very well. Having kids allows you to keep your sanity in mathematics.

WILKINSON: I would say the same is true for me, but I do come from quite a position of privilege, so I don’t want to answer this on behalf of all the women who are facing a choice between having children and having a career. I think it can be a very serious obstacle. I’ve been told that it’s been helpful to say that there was a moment in my life as a mathematician when I wondered if I would ever prove a theorem again, when I had just had a baby.

DAUBECHIES: I had that same experience. When you’ve just had a child, your body has been taken over as a factory to make a baby. After the delivery, I couldn’t concentrate, and I was thinking, “I make my living with my brain, and it’s gone?” and I was really afraid. I didn’t even tell my husband—and he’s my best friend. Then it came back, and I was relieved.

WILKINSON: I told a colleague of mine this story and he passed it on to someone else, and she came back to me and told me it was tremendously helpful.

TORO: Ingrid talked about her husband as her best friend. A supportive partner is the key to success.

WILKINSON: That’s something that Maryam had in Jan.

Q: Can you say something about shooting the film in Iran?

CSICSERY: I visited Iran alone and worked with an Iranian film company, as talented filmmakers as I’ve ever met. We shot for ten days in Isfahan and Tehran and there were no difficulties. This happened in March and I doubt if it could be reproduced today. I learned a lot about the reverence that the people at those schools I visited have for Maryam. Really a kind of stature that I don’t think American mathematicians could aspire to in a thousand years. Maryam is a national hero. I don’t know what kind of employment you’d have to have here to achieve that. It was kind of impressive to see how she was thought of. The respect for her and reverence is trying to become a foundation of higher education and educational reform.

Q: Maybe you can say a little more about the education system and the specifics of Mirzakhani’s trajectory? How did she get to the middle school and high school?

CSICSERY: I think that the students who show talent are caught up in the system immediately, with teachers pulling them forward and shepherding them into the next level, and making the contacts with university and the IMO and competition programs. There’s a community that passes people on so they don’t fall away. We have that in this country but not in the same way, it happens more haphazardly. It’s quite possible—likely—that it’s a tiny elite that I was seeing. They have produced a talented pool all over the world.

Q: I’m curious if there are practical applications of this mathematics that Mirzakhani developed.

DAUBECHIES: As the applied mathematician at the table, I don’t know of any concrete applications of her work. But I can name so many examples of pure mathematics that nobody thought would ever have an application that I think it’s highly likely that her work will have an application. One that I cannot foretell—if I knew, I would do it. I strongly believe that as an applied mathematician you have to learn as much pure mathematics as you have an appetite for. So no, I don’t know of an application and no, it doesn’t bother me in the slightest.

ATHREYA: Even though the trajectory of billiards is a simple problem in Newtonian mechanics, surprisingly enough we don’t know that much about it. So I think there could be applications in physics, and in things like supersymmetric quantum field theory, as long as you ask me no further questions.

Q: Has Maryam’s work changed the math you do or the way in which you do it?

ATHREYA: Maryam was a year ahead of me so was someone who I looked up to, but was a near peer. Seeing what she did encouraged me to take bigger swings, at harder problems.

Q: I have a question for the journalist on stage. I’ve been thinking about all the appreciation of Mirzakhani’s work in Iran, but what is your impression of the appreciation of her work here in the US?

KLARREICH: I don’t really know, because in general when I write something it goes out into the world and then I don’t know that much about how it’s received. But when Mirzakhani won the Fields medal, I wrote a profile of her and I believe that it’s one of the most popular articles that I’ve written for them. I don’t’ know how many people read it but I believe it’s still read quite a bit. As George was talking about how respected she is in Iran, I was thinking about how many of my non-mathematician friends could even name a mathematician and I don’t think it’s that many.

Q: What are the challenges of writing mathematics for a popular audience?

KLARREICH: It is difficult. In this case, it helped a lot that my very ancient Ph.D. was in this area of mathematics. I think, though, that for any mathematics out there, there is some little nugget you can convey to a general audience and you just have to keep digging until you find that.

CSICSERY: Some of the most exciting moments in the production process were the discussions between Jayadev and Erica about the scripting of the explanations and the animation.

Q: What advice do you think Maryam would have had or do any of you have for young women who aspire to be mathematicians?

DAUBECHIES: If you or your friends like mathematics and have fun doing it, then you should do it. And you can do it. And there will be ways in which you can use that mathematics in your life. There are jobs for mathematicians in academia and elsewhere. Mathematicians are a kind of universal donor. The skills you learn becoming a mathematician you can apply in so many different directions.

WILKINSON: I think that is essentially the kind of advice Maryam would give. She was very positive. She also acknowledged when things were difficult. She never had any problem saying when things were difficult or minimized her challenges. But she was always positive and was a very encouraging person. I think what she would say would have been very practical and encouraging.

TORO: I would say persevere, and look at her life. That’s proof you can do it. When we think of heroes, we usually think of people much older than us. But she was my hero. She accomplished many things, mathematically but also otherwise. The way she is respected in Iran is proof of that.

Q: It sounds like she directly changed life for people in Iran, changing the image of women and becoming a national hero. Do you have more examples of the impact she had?

CSICSERY: The impact visible in the film is simply the inspirational effect, primarily on high school and college students who see her as a role model and feel that it’s acceptable to do what she did. I think that’s a very important step. I think that’s a concrete accomplishment.

Q: How aware was Maryam’s daughter of her stature and does she show any interest in mathematics?

CSICSERY: Maryam’s husband, Jan, has tried to shield Anahita from all of the attention. I was able to persuade him to allow us one day of filming the two of them. I didn’t ask for an interview with Anahita, she’s eight years old. She’s a strong young girl who is very attached to her father and that’s all I wanted to say about it.

BARCELO: Maryam was very private and Jan is definitely respecting that.

Q: Are there any writings of how she faced death and disease? Did she become philosophical toward the end of her life?

CSICSERY: She stayed away from questions about religion. People I interviewed who knew her beliefs would not tell me. I think this was on her part, an act of diplomacy which has cemented her status in Iran because she did not make commitments of that type, and it also protected her family.

Q: Does anyone know what happened to her notes that were seen in the film?

CSICSERY: They’re in the possession of her husband.

Q: Maryam broke the mold in so many ways. It seems like she was constantly supported. Was there any pushback?

WILKINSON: I don’t know very much at all about resistance she got from her colleagues or peers. I do know that when she got the Fields medal, people on the Internet wrote nasty things about how she got it because she was a woman and so on.

CSICSERY: I think she got a lot of support from her teachers and there’s a whole generation of Iranian mathematicians who owe a lot to those very teachers. That was an opportunity that’s not open to everyone.

WILKINSON: The few times we talked about it, she never said anything about any resistance she got in her career or in Iran. The only things she talked about were personal.

BARCELO: Any last words?

CSICSERY: I want to thank the panelists, and also there are a few DVDs left outside.

Skip Garibaldi on “Uncovering lottery shenanigans”

When Lawrence Mower, an investigative reporter in Florida, examined a list of over a million prizes worth \$600+ that had been claimed in the Florida lottery, he noticed something interesting—there were a few outliers in the data. Not a couple of people with somewhat high winnings, either: there were several people who had won more than 100 of these prizes. One man from Pompano Beach, Florida had 252 wins worth \$600 or more; another Hollywood, FL resident had claimed 578.

What was suspicious about these numbers? Garibaldi mentioned a few possibilities. Clerks verify that a ticket is a winner by scanning a barcode. If a gambler approaches a clerk with a winning ticket, the clerk can claim it’s a loser or worth less than it actually is, and keep the winnings for themselves. If a ticket is worth a lot of money—over \$600 in Florida—then gamblers need to travel to the state lottery office and register themselves or their earnings for tax purposes. If a gambler wants money right away or is wary of registering themselves with the state government, they might be willing to sell their tickets for less than they’re worth, letting the buyer claim the winnings. These buyers are called ticket aggregators, and in some cases they even use this system for money laundering.

When Mower asked the Florida lottery secretary about his findings, she dismissed them. But Mower wasn’t convinced that he hadn’t found evidence of an illicit scheme, so he contacted a few mathematicians to work out how likely it really was that someone could win so many huge lottery prizes. Among those mathematicians was Skip Garibaldi, who related all of this at the start of his invited address this morning, “Uncovering lottery shenanigans.” Garibaldi talked about he, along with Philip Stark of UC Berkeley and Richard Arratia of USC, worked out how much a gambler would need to spend in order to have a non-negligible probability of the kinds of winnings that Mower had discovered. You can probably guess how it turned out, but you might be impressed by how extreme the numbers really are—I certainly was.

First, they needed to decide what a “non-negligible probability” meant. There were a few options, but they went with their most conservative option of a one in a million chance. If a gambler is playing several different games, you can construct a vector of ticket costs and a vector of number of tickets bought at each game. In the simplest case, the games are all scratch-off tickets, meaning the probability of winning is described by a binomial distribution. By minimizing over the number of tickets bought under the condition that the probability of getting the desired number of wins is nonnegligible, the team found an estimate of how much a gambler with hundreds of wins should be spending.

What they found was that our Pompano Beach friend should have been spending roughly \$1,000 per day on lottery tickets. That’s a lot of lottery tickets—and an interview with Mower revealed that he was, in fact, a ticket aggregator. (The Hollywood gambler was innocent.) The results have prompted policy changes across the US to address the issues of ticket aggregation and money laundering.

MAA Workshop on Microaggressions

This morning I went to the MAA workshop “Identifying and Managing Microaggressions in the Academic Setting”. It was run by Lynn Garrioch and Semra Kilic of Colby-Sawyer College, Omayra Orteya of Sonoma State, and Rosalie Bélanger-Rioux of McGill University.

The workshop started out with a discussion of what the term microaggression means. Kilic explained that it refers to brief, commonplace assaults on marginalized people, intentional or not. That was something I’d heard before, but I’d never heard it broken down into subcategories—microassaults, microinsults, and microinvalidations. Microassaults refer to actions like displaying swastikas—intentionally provoking. Microinsults encompass any questions or comments specifically related to a marginalized identity, like asking a nonwhite person who was born in the US where they’re from. And microinvalidations are those little signs that you’re negating someone’s experience.

The rest of the workshop was devoted to discussing a hypothetical scenario that involved a microaggression. Here, try it for yourself: you enter an academic department and can’t help overhearing two TAs discussing one student’s recent exam. “This student asked me why they lost so many points for not writing down ‘trivial steps’. The student was very upset,” TA #1 explains to TA #2. TA #1 is not convinced this student knows the material because they are very quiet in class and never attend office hours.

TA #2, however, pulls out another student’s exam for comparison. “Oh, that one is a great student!” says TA #1. But TA #2 notes that this student did not show many steps either. However, TA #1 believes they know the material because they talk a lot in class.

If you overheard this conversation, what would you do? Would it depend on your identity and the identity of the TAs involved? How might the situation have been avoided, or what factors could have produced it? If you did step in, what would be the most productive way to do so?

After discussing this scenario, the organizers introduced a twist. Imagine that a student who appears disabled comes to your office a week later and complains that their TA is biased against students with disabilities. You suspect that this student is the “quiet student” you overheard the conversation about. Does this change your feelings about the conversation you overheard? How could you best be an ally to the student?

Try putting yourself in the shoes of different characters, as well: can you see yourself in TA #1 at all? How would you feel if you were the quiet student or the more outgoing student? Seeing the different assumptions people made about the characters and how that affected their interpretation of what was going on was a worthwhile experience.

Creating a math textbook accessible to the blind

I entered the press room at JMM this morning to find a bustle of excitement as four men prepared to introduce themselves on video. As they explained their roles to the camera, J. Brian Conrey of AIM pulled me aside to tell me about their project: translating math textbooks into Braille automatically, something that until now took six months and $10,000. I sat down and asked them a few questions.

Al Maneki, a retired mathematician who is blind, was an initial proponent of the project and tests the team’s results to make sure they actually work. He and Martha Siegel initially conceived of the idea, and recruited Alexei Kolesnikov of Towson University. When they hit a wall, they asked J. Brian Conrey and David Farmer (also of AIM) for help. A rough, edited—and incomplete despite my typing efforts—of my conversations follows.

David: Martha and Al and Alexei were working on this and hit a wall. Then we had a meeting with the right people—Volker Sorge and Rob Beezer—at the last JMM. They had the key pieces that were missing.

How did you know to who to introduce Al, Martha, and Alexei to?

David: We had a workshop about screen readers. I knew the people involved in that would be interested in this. There’s really two parts: the narrative part of a book which we take for granted because we can see a title and we know that’s the chapter title. But in Braille everything is the same size. Rob Beezer had built this infrastructure that could handle that. Volker Sorge had been involved in MathJax which puts math on webpages. It has a visual display for sighted people and a structural “pronounce-it” for blind people, and now it also outputs Braille—it talks to the Braille reader. There’s a third part—the images—which Alexei is taking the lead on and that’s the hardest part. Say you want to show a graph that has two curves on it so that blind people can feel it. The resolution of your fingers isn’t big, so if you want it to be in Braille it needs to be much bigger. The images are still an art and we’re having a workshop in August, organized by the team, to try to turn that into a science.

How did the project get started?

Alexei: My colleague, Martha Siegel, a professor emerita at Towson University and a longtime secretary at MAA, was particularly moved by a student who had to delay taking a math course for six months because the textbook wasn’t available. People had to take the source textbook and type it out in Braille. I thought this was a little ridiculous. I thought surely, in 2018, a better way would exist. I started looking at what is available, and nothing worked. We could produce raised print, but we would give it to Al to test, and there would be a lot of problems. Eventually we realized the direct path of converting LaTeX to raised print is pretty difficult because of the styling and so forth. So PreTeXt offered a different path from mathematical text.

What is PreTeXt?

Rob: PreTeXt is a source format that separates content from presentation. We know exactly where the math pieces are and where the sections and chapters begin. It’s totally automated. We now have about 100 projects in PreTeXt, covering almost all the undergraduate curriculum besides topology.

PreTeXt separates the content of mathematical texts from its formatting, allowing for easier translation between visual text and Braille.

Is the problem of producing textbooks in Braille unique to math?

Al: It’s most difficult for math. Braille textbooks in English can be had if you have an electronic file. English can automatically be translated to Braille. When it comes to math with equations and graphics, tables and matrices, that’s the hard part.

Volker: It’s hard for STEM, generally.

Al: There’s a standardized way, Nemeth, for representing math, diagrams, tables, and charts in Braille.

Volker: That [Nemeth] used to be done by humans. Now you can do it mostly automatically.

The correspondence between Braille and visual notation.

Why has automating Nemeth not been done before?

Volker: It has been done before, but those systems don’t work with Latex as well. The system I’m using is that we actually restructure all the math into something that’s natural to speak. We put it in a more semantic representation.

Al: There are very complicated formulas too—subscripts inside exponents, etc. It’s clearer in Braille than when you listen to it spoken. That can be hard because you can’t go back to reference it. There’s an art to how you speak math, and if you don’t practice that art just so, then you get into ambiguities, whereas in Braille if something’s ambiguous you can reference it again.

Are the Braille textbooks being adopted anywhere yet?

Rob: We are past proof of concept, but it’s not being adopted anywhere yet.

Alexei: One challenge is that a 15-page pdf becomes a very long chapter in Braille (about 3 times longer). There is no ability to control the size of the font. And you have only forty characters to play with on a line. It’s also surprisingly difficult to make sure symbols are translated correctly, because some rules are rules, but some rules are only guidelines.

Rob: We will eventually be able to distribute an electronic version of this and people can read it from the web on their Braille reader.

Volker: And students will be able to emboss research papers for themselves in the future.

Nancy Reid Invited Address: “In Praise of Small Data”

Last night’s speaker, Nancy Reid of the University of Toronto, was introduced with so many accolades that I couldn’t keep track of them all. Her talk, on the ways “big data” has been overhyped in recent years, opened with a description of how statistics and data science have exploded over the last decade, thanks to Markov chain Monte Carlo algorithms that transformed the statistical sciences by allowing integrals to be replaced with easier-to-compute finite sums.

 

This started in 1990, but in the past ten years, the field of statistics has grown more rapidly than ever and has also become more interdisciplinary, largely thanks to excitement about big data. This also led to the development of a new field: data science. Reid explained that data science includes concepts outside the realm of traditional statistics: acquisition and preservation of data, making data usable, reproducibility, and security and ethics considerations. These require expertise in math, statistics, computer science, and in the field the data comes from—making data science more “outward-looking” than statistics has traditionally been.

 

However, Reid told us, despite all the hype about “big data” many examples of real-world problems do not actually use huge amounts of data. (Hence the title of her talk: “In Praise of Small Data”.) One such example is the field of extreme event attribution, which tries to tease out the role of anthropogenic climate change in extreme weather events like wildfires. A recent paper on this topic, explained Reid, tried to figure out how much of British Columbia’s 2017 wildfires were caused by humans.

 

To do this, they started with a simulation of the global climate, then downscaled it to British Columbia. They modeled the relationship between the area burned by wildfires and climate variables–temperature and precipitation. (This actually required two statistical models.) The authors then simulated the climate 50 times for two time periods: the decade 1961-1970 and the decade 2011-2020. Using their model, they found a distribution for how much area was likely to have burned in wildfires during each of these decades—and found the amount of area burned in the BC wildfires in 2017 was far out in the right tail of the distribution for 1961-1970, and not so far for the 2011-2020 distribution.

 

This is certainly a powerful application of statistical inference techniques, but Reid pointed out that there’s actually not that much data involved in the study—it’s mostly simulations.

 

Reid discussed a better example of “big data”, in which 6710 people over 50 were followed and their mortality associated with “arts engagement” (attending museums, operas, etc.). She pointed out, though, that collecting complete data for such a project requires a huge amount of time and effort—even though Google and Facebook are collecting data on billions of people every day, it’s not high-quality data. The observations are not independent, and typically, she explained, scientists are looking for unusual events which limits the amount of data that is actually useful. Moreover, the more data, the more complicated it is to work with it. Reid summed it all by noting that when it comes to data, it’s quality over quantity.

MAA Invited Address by Scott Adamson: Mazes, riddles, zombies and unicorns

Scott Adamson, of Chandler-Gilbert Community College, kicked off his talk by asking the audience this: “How can we create a more joyful mathematical learning experience where the beauty of mathematics can be shared and be engaged in by our students?”

 

I’m reminded of the need to answer this question every time I introduce myself as a graduate student in math and am met with the classic response—“I was so bad at math in school.” But it can be difficult to try out new methods of teaching when instructors are pressed for time and resources. Adamson articulated the specific way our education system is failing students—by asking them to memorize formulas without any understanding of the underlying mathematics—through examples he’s seen in his work. Students, for instance, learn how to calculate the average of a set of numbers, but often they don’t understand what it means conceptually. Teachers, he explained, frequently try to make math “fun” by introducing gimmicks—following mathematical steps to reach the end of a maze or attacking zombies with a line of best fit—but in the process, send the message that math is too painful to endure without turning it into a game. At the college level, there is rarely an attempt to make math fun at all.

 

The result is students who implement formulas mechanically and cannot solve problems that differ from what they’ve seen in class. Adamson suggested an inexpensive way to getting students engaged in problem-solving: vertical non-permanent surfaces, or VNPSs (that is, whiteboards, or chalkboards, or some approximation). Research by Peter Liljedahl has shown that students who work on permanent surfaces—those where you can’t erase your work—participate less, persist less in their problem-solving, and take significantly longer to start writing—two minutes as opposed to 20 seconds. Any non-permanent surface is good, but vertical surfaces are slightly better than horizontal surfaces. They let students work together more easily, and also look around the classroom at one another’s work for ideas.

 

Although students may struggle at first with a teaching method that requires them to solve new kinds of problems, Adamson suggested students are empowered when they are able to “discover” math for themselves.