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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## AMS panel on math and motherhood

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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 $1 = 2$.

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## Advice from Haimo Award Winners

What do award-winning mathematicians Federico Ardila-Mantilla, Mark Tomforde, and Suzanne Weeks have to say about teaching? Each of these mathematicians received the MAA Haimo Award at JMM 2020 and gave 20 minute talks. Here is a summary.

**Federico Ardila-Mantilla** used his time to ask us to reflect by thinking about three “open questions.” First, a story about Federico. A mathematician during the day, DJ at night, it is not surprising that Federico is interested in music. This interest led him to enroll in a *timbales* class at his local community center. His teacher? One of the top timbaleros of the 1970’s. After just two weeks of classes, this timbalero invited Federico to play with him at an event. It was a moving experience for Federico. **What does a similar experience look like in a math environment? When do we see top mathematicians inviting newcomers into their work?**

Undisturbed beech forest behave in a weird but spectacular way. “Apparently, the trees synchronize their performance so that they are all equally successful. And that is not what one would expect. Each beech tree grows in a unique location, and conditions can vary greatly in just a few yards. The soil can be stony or loose. It can retain a great deal of water or almost no water. It can be full of nutrients or extremely barren. Accordingly, each tree experiences different growing conditions; therefore, each tree grows more quickly or more slowly and produces more or less sugar or wood, and thus you would expect every tree to be photosynthesizing at a different rate… [However] The rate of photosynthesis is the same for all the trees. The trees, it seems, are equalizing differences between the strong and the weak.” – The Hidden Life of Trees. **What does this synchrony look like in a mathematics classroom?**

Born in 1952, bell hooks experiences the initial racial integration after segregation was deemed illegal. “Bussed to white schools, we soon learned that obedience, and not a zealous will to learn, was expected of us. That shift taught me the difference between education as the practice of freedom and education that merely strives to reinforce domination.” – bell hooks. **What does a liberating math classroom look like?**

**Mark Tomforde **had four original beliefs that have changed throughout his teaching experiences. They are:

– Old Belief 1: My main objective is to teach math content. The more content the better. Updated belief: **My main objective is to teach my students to think.** In 10 years what will students remember from this course? What do I want them to remember? I want to prepare my students for life, to develop good habits when engaging with facts, to work on improving their skills such as problem solving, attention to details, ability to communicate technical ideas, and to think deeply about a few topics.

– Old Belief 2: The more students I can reach the better. Updated belief: **Quality is more important than quantity.** I am a fixed quantity and the more projects I am involved with, the less of myself I can devote to them. For me, it is better to devote my time and energy to one or two projects and do it well rather than spending little time in a lot of projects.

– Old Belief 3: The best students are our future, teach to them. Updated belief: **Direct teaching to all students paying particular attention to the ones that are struggling.** Are our best students more important? Sometimes we think we are just training the future mathematicians and try and identify the “best students.” But, how many students are we not able to reach? Our goal should not be to find talent but rather to help students develop their talent.

– Old Belief 4: Good students are the ones who can prove theorems, solve problems and do well. Updated belief: **There are many ways for students to be successful and contribute.** When I teach math majors, I regard all of my students as the future of math. This is because math requires a much larger ecosystem than a group of researchers. In addition to researchers, we need teachers, advocates, PRs, writers, and many others are needed. In my classes, I ask students to find a need and contribute to fill that need. Some of the projects students conduct are: leading math circles, tutoring, creating guides for incoming students in the department, starting newsletters, and organizing department picnics.

**Suzanne Weeks **is a co-director of MSRI-UP and PIC Math, and has worked in many industrial projects. Her experiences have taught her it’s important to:

– **Show up** – Take advantage of the opportunities that arise for you.

–** Sit in front** – Be engaged, be involved, be fully in the room.

– **Raise your hand** – Be an active participant, raise your hand, raise your voice, ask for stuff when you need it.

– **Make friends** – You need to have a support group.

– **Be yourself** – Your path is your own. Do what is best for you.

– K**now your worth because you are worthy** – Celebrate your successes.

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## 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.

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## UC Berkeley attendees on JMM 2020

Past AMS president, Ken Ribet, and five University of California, Berkeley undergrads share their thoughts on JMM 2020.

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## A conversation with Shyam Naraynan

Former *Who Wants to Be a Mathematician* contestant, Shyam Naraynan, talks about what he’s been up to since he last competed in the game.

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## 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.

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## Mariah Birgen on JMM 2020

Mariah Birgen, Wartburg College, discusses her time at JMM.

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## Biomedical Data Sharing

AMS Invited Address speaker Bonnie Berger gave a captivating talk on Biomedical Data Sharing. There is a recurring issue in Computational Biology: **genomic data is growing exponentially faster than computing power and data storage. **If we want researchers to have access to the data, compressing data, sharing it, and decompressing it is not a viable solution due to the fact that decompressing is time consuming and requires a lot of storage. Bonnie’s solution: Compress the data and operate on the compress data. How?

Bonnie’s research team provided an algorithm that plots data points in a high-dimensional Euclidean space and covers the data points with spheres.

Then, by only using one point per sphere, the algorithm searches through the spheres, decides which are of interest to the particular problem of study and then goes back to these spheres and does a more thorough search in a region slightly larger than these spheres. The result? An algorithm that is much faster than the available methods and that recovers more than 99% of the results that slower algorithms recover. This work has been cited thousands of times.

Bonnie then went to describe other problems for which they provided similar algorithms to reduce run time while providing great accuracy. One that seemed really interesting was to combine and cluster cell data that was obtained via multiple experiments by different research groups.

For more information, visit Bonnie Berger’s website: http://people.csail.mit.edu/bab/computing.html

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## A conversation with Preeti Dubey

Preeti Dubey shares her goals for JMM 2020.

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