Archive for the ‘Women and Math’ Category.

Should You Date a Mathematician?

The book How to Drive Your Man Wild in Bed by Graham Masterton has a chapter on how to choose a lover. It highlights red flags for men who need to be approached with caution. There is a whole list of potentially bad signs, including neglecting to shower in the previous week and talking only about himself.

The list of bad features also includes professions to avoid. Can you guess the first profession on the list? OK, I think you should be able to meta-guess given the fact that I am writing about it. Indeed, the list on page 64 starts:

Avoid, on the whole, mathematicians…

I am an expert on NOT avoiding mathematicians: in fact, I’ve married three of them and dated x number of them. That isn’t necessarily because I like mathematicians so much; I just do not meet anyone else.

When I was a student I had a theory that mathematicians are different from physicists. My theory was based on two conferences on mathematical physics I attended in a row. The first one was targeted for mathematicians and the second for physicists. The first one was very quiet, and the second one was all boozing and partying. So I decided that mathematicians are introverts and physicists are extroverts. I was sure then that my second husband chose a wrong field, because he liked booze and parties.

By now, years later, I’ve met many more mathematicians, and I have to tell you that they are varied. It is impossible and unfair to describe mathematicians as a type. One mathematician even became the star of an erotic movie. I write this essay for girls who are interested in dating mathematicians. I am not talking about math majors here, I am talking about mathematicians who do serious research. Do I have a word of advice?

I do have several words of caution. While they don’t apply to all mathematicians, it’s worth keeping them in mind.

First, there are many mathematicians who, like my first husband, are very devoted to mathematics. I admire that devotion, but it means that they plan to do mathematics on Saturday nights and prefer to spend vacation at their desks. If they can only fit in one music concert per year, it is not enough for me. Of course, this applies to anyone who is obsessed by his work.

Second, there are mathematicians who believe that they are very smart. Smarter than many other people. They expand their credibility in math to other fields. They start going into biology, politics and relationships with the charisma of an expert, when in fact they do not have a clue what they are talking about.

Third, there are mathematicians who enjoy their math world so much that they do not see much else around them. The jokes are made about this type of mathematician:

What is the difference between an extroverted mathematician and an introverted one? The extroverted one looks at your shoes, rather than at his own shoes.

Yes, I have met a lot of mathematicians like that. Do you think that their wives complain that their husbands do not notice their new haircuts? No. Such triviality is not worth mentioning. Their wives complain that their husbands didn’t notice that the furniture was repossessed or that their old cat died and was replaced by a dog. My third husband was like that. At some point in my marriage I discovered that he didn’t know the color of my eyes. He didn’t know the color of his eyes either. He wasn’t color-blind: he was just indifferent. I asked him as a personal favor to learn the color of my eyes by heart and he did. My friend Irene even suggested creating a support group for the wives of such mathematicians.

While you need to watch out for those traits, there are also things I like about mathematicians. Many mathematicians are indeed very smart. That means it is interesting to talk to them. Also, I like when people are driven by something, for it shows a capacity for passion.

Mathematicians are often open and direct. Many mathematicians, like me, have trouble making false statements. I stopped playing —Mafia— because of that. I prefer people who say what they think and do not hold back.

There is a certain innocence among some mathematicians, and that reminds me of the words of the Mozart character in Pushkin’s poetic drama, Mozart and Salieri: —And genius and villainy are two things incompatible, aren’t they?— I feel this relates to mathematicians as well. Many mathematicians are so busy understanding mathematics, they are not interested in plotting and playing games.

Would I ever date a mathematician again? Yes, I would.

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The Second IMO Gold Girl

Me in 1975Janet Mertz encouraged me to find IMO girls and compare their careers to that of their teammates. I had always wanted to learn more about the legendary Lida Goncharova — who in 1962 was the first girl to win an IMO gold medal. So I located her, and after an interview, wrote about her. Only 14 years later, in 1976, did the next girl get a gold medal. That was me. I was ranked overall second and had 39 points out of 40.

As I did in the article about Lida, I would like to compare my math career to that of my teammates.

I got my PhD in 1988 and moved to the US in 1990. My postdoc at MIT in 1993 was followed by a postdoc at Bar-Ilan University. In 1996 I got a non-paying visiting position at Princeton University. In 1998 I gave up academia and moved to industry, accepting an offer from Bellcore. There were many reasons for that change: family, financial, geographical, medical and so on.

On the practical level, I had had two children and raising them was my first priority. But there was also a psychological element to this change: my low self-esteem. I believed that I wasn’t good enough and wouldn’t stand a chance of finding a job in academia. Looking back, I have no regrets about putting my kids first, but I do regret that I wasn’t confident enough in my abilities to persist.

I continued working in industry until I resigned in January 2008, due to my feeling that I wasn’t doing what I was meant to do: mathematics. Besides, my children were grown, giving me the freedom to leave a job I did not like and return to the work I love. Now I am a struggling freelance mathematician affiliated with MIT. Although my math blog is quite popular and I have been publishing research papers, I am not sure that I will ever be able to find an academic job because of my non-traditional curriculum vitae.

The year 1976 was very successful for the Soviet team. Out of nine gold medals our team took four. My result was the best for our team with 39 points followed by Sergey Finashin and Alexander Goncharov with 37 points and by Nikita Netsvetaev with 34 points.

Alexander Goncharov became a full professor at Brown University in 1999 and now is a full professor at Yale University. His research is in Arithmetic Algebraic Geometry, Teichmuller Theory and Integral Geometry. He has received multiple awards including the 1992 European Math Society prize. Sergey Finashin is very active in the fields of Low Dimensional Topology and Topology of Real Algebraic Varieties. He became a full professor at Middle East Technical University in Ankara, Turkey in 1998. Nikita Netsvetaev is an expert in Differential Topology. He is a professor at Saint Petersburg State University and the Head of the High Geometry Department.

Comparing my story to that of Lida, I already see a pattern emerging. Now I’m curious to hear the stories of other gold-winning women. I believe that the next gold girl, in 1984, was Karin Gröger from the German Democratic Republic. I haven’t yet managed to find her, so can my readers help?

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Lida Goncharova — The First Gold IMO Girl

LidaJanet Mertz wrote several papers about the gender gap in mathematics. One of her research ideas was to find girls who went to the International Math Olympiad (IMO) and compare their fate to that of their teammates with a similar score. She asked me to find Soviet and Russian IMO girls. All my life I had heard about Lida Goncharova, the first girl on a Soviet team, and the first girl in the world who took a gold medal, but I had never dared to reach out to her. A little push by Janet Mertz was enough for me to find Lida’s phone number in Moscow and call her.

My conversation with Lida Goncharova

Lida

Lida got interested in mathematics when she was five years old. Luckily, many of her relatives were mathematicians and she started bugging them for math puzzles.

Her involvement with math was interrupted by the death of her parents — her mother when she was seven, and her father when she was nine. She ended up living with her sister, but felt very lonely.

After several years of personal turmoil, she renewed her pursuit of mathematics. Lida started discussing math with her mother’s first husband. She joined a math circle which was run at Moscow State University. When she was 13 she went to a summer camp and found a mentor there to study trigonometry. Eventually she ended up at School Number 425, one of the first schools in Moscow that opened for children gifted in math.

At the end of high school she went to the IMO as part of the Soviet team and won a gold medal there. After that she enrolled in the most prestigious Soviet institute for the study of math — Moscow State University (MSU).

Half of her high school classmates went to MSU, including her high school sweetheart Alexander Geronimus. Lida married Alexander when she was a sophomore and they had their first son in her fourth year of undergraduate school.

Meanwhile, she wasn’t doing as well in her studies as she had hoped. Lida was very fast to pick up math ideas during conversations, but she had difficulty reading books. As ideas were becoming more complicated and involved, this became a problem. She started feeling that she was falling behind her friends. When her friends gathered together to discuss mathematics she couldn’t understand everything. She wanted to ask questions, but was too shy. Plus, she didn’t want to impose on them. She made a decision to be silent. As a result she started ignoring the conversations of others and became discouraged as she fell behind.

She had her second child at the beginning of graduate school, where she studied under the supervision of Dmitry Fuchs. Lida was already losing her self-esteem and so she chose a self-contained problem that didn’t require a lot of outside knowledge. The solution involved some combinatorial methods, but Lida didn’t quite understand the big picture and the problem’s goal.

I contacted Dmitry Fuchs and asked him about Lida’s thesis. He told me that Lida’s main result is extremely important and widely cited. It is called Goncharova’s theorem.

Meanwhile, her husband finished his PhD in math and secured a great job in an academic institution. They had started as peers, but her work was interrupted by having their children. Lida finished her PhD a couple of years after her husband and got a very boring job as an algorithm designer. She even wrote some papers at the job, but she was not much interested. She continued her attempts to do mathematics and continued asking everyone for problems, but it didn’t go anywhere. Her friends were not very interested in her calculations and after the birth of her third child she began to lose hope in her research.

When Lida and her husband entered graduate school they became religious. Ten years later, Alexander decided to pursue the Russian Orthodox religion as a career and got a parish in 600 km from Moscow. They didn’t want to move their children away from Moscow, with its educational and cultural opportunities. So they started living in two places with long commutes. This didn’t help her math either.

Eight years after the third son, the fourth son was born. Although Lida sporadically continued her calculations, she still didn’t talk about them to anyone.

When the older children went to high school, Lida enjoyed solving their math problems tremendously. In 1990 perestroika started and Lida lost her job. She got an offer to create a private school and teach there. By this time she had had two more children, a son and a daughter. Lida continued working for the private school until her six children grew out of it. Lida enjoyed teaching and inventing methods to teach mathematics. The school ended in 2004. But she continues working with kids sharing with them her joy of mathematics.

Lida believes that she has had an extremely lucky life in many ways. The only exception was her unsuccessful math career. She can’t live without math, and will continue working with kids, solving fun problems and doing her private research.

When I first called her and said I wanted to talk about her and math, she told me: —There is nothing to talk about. I stopped doing math after my PhD. Almost.— That —almost— kept me asking questions.

Lida’s teammates

Lida

Janet Mertz was considering a serious research project comparing the fates of IMO medal girls with the fates of their teammates, to see whether gender plays a role later. However, due to the language and cultural differences and the fact that most of the girls changed their last name, it was difficult to locate them. So Mertz put this research project on hold.

She had asked me to find and contact the Russian women and I was so fascinated with Lida’s story that I decided to write it up in this article. And because the research is on hold, I decided to include the fates of Lida’s teammates.

Lida Goncharova got her gold medal in the 1962 IMO with 42 points and was ranked third. The teammate with the closest score was Joseph Bernstein with another gold medal and 46 points. I don’t even have to check Wikipedia to tell you about Joseph, as I was once married to him. He used to be a professor at Harvard University and is now a professor at Tel-Aviv University. He is a member of the Israel Academy of Sciences and Humanities and the United States National Academy of Sciences. He achieved a lot and is greatly respected by his peers.

Joseph Bernstein might not be the best person to compare Lida to as he had a perfect score. Some might argue that a perfect score indicates that he might have done better if the problems had been more difficult.

The two Soviet teammates whose scores were the closest to that of Lida, but below her, were Alexey Potepun with 37 points and Grigory Margulis with 36 points.

Alexei Potepun got a PhD in mathematics and is now a professor at Saint-Petersburg University. He has published eleven papers.

Next to Alexey Potepun is Grigory Margulis, who is a professor at Yale and was awarded the Fields Medal and the Wolf Prize. He is a member of the U.S. National Academy of Sciences.

You might notice that the two people who moved to the US are much more famous than those who stayed in Russia. You might say that moving to the US is a better predictor of success than gender. Sure, living in a free country helps, but Margulis got his Fields medal while he was in the USSR. And Bernstein invented his famous D-Modules while in Russia also.

My conversation with Lida was personally inspiring. I loved the tone of her voice when she talked about mathematics. There were many elements that prevented her from having the mathematical career she might have had: the untimely death of her parents, her shyness, raising six children, many years of long commutes. When we look at the achievements of her closest teammates, we can’t help but wonder what kind of mathematics we lost.

This conversation was very encouraging for me. I felt there were similarities between Lida and myself in more ways than I expected. What we share most of all is a love for mathematics. I could hear that in her voice.

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Sparsity and Computation

Once again I am one of the organizers of the Women and Math Program at the Institute for Advanced Study in Princeton, May 16-27, 2011. It will be devoted to an exciting modern subject: Sparsity and Computation.

In case you are wondering about the meaning of the picture on the program’s poster (which I reproduce below), let us explain.

WaM 2011 Poster Picture

The left image is the original picture of Fuld Hall, the main building on the IAS campus. The middle image is a corrupted version, in which you barely see anything. The right image is a striking example of how much of the image can be reconstructed from the corrupted image using clever algorithms.

Female undergraduates, graduates and postdocs are welcome to apply to the program. You will learn exactly how the corrupted image was recovered and much more. The application deadline is February 20, 2011.

Eugene Brevdo generated the pictures for our poster and agreed to write a piece for my blog explaining how it works. I am glad that he draws parallels to food, as the IAS cafeteria is one of the best around.

by Eugene Brevdo

The three images you are looking at are composed of pixels. Each pixel is represented by three integers corresponding to red, green, and blue. The values of each integer range between 0 and 255.

The image of Fuld hall has been corrupted: some pixels have been replaced with all 0s, and are therefore black; this means the pixel was not “observed”. In this corrupted version, 85% of the pixel values were not observed. Other pixels have been modified to various degrees by stationary Gaussian noise (i.e. independent random noise). For the 15% observed pixel values, the PSNR is 6.5 db. As you can see, this is a badly corrupted image!

The really interesting image is the one on the right. It is a “denoised” and “inpainted” version of the center image. That means the pixels that were missing were filled in and the observed pixel integer values were re-estimated. The algorithm that performed this task, with the longwinded name “Nonparametric Bayesian Dictionary Learning,” had no prior knowledge about what “images should look like”. In that sense, it’s similar to popular wavelet-based denoising techniques: it does not need a prior database of images to correct a new one. It “learns” what parts of the image should look like from the original image, and fills them in.

Here’s a rough sketch of how it works. The idea is to use a new technique in probability theory — the idea that a a patch, e.g. a contiguous subset of pixels, of an image is composed of a sparse set of basic texture atoms (from the “Dictionary”). Unfortunately for us, the number of atoms and the atoms themselves are unknowns and need to be estimated (the “Nonparametric Learning” part). In a way, the main idea here is very similar to Wavelet-based estimation, because while Wavelets form a fixed dictionary, a patch from most natural images is composed of only a few Wavelet atoms; and Wavelet denoising is based on this idea.

We make two assumptions that allow us to simplify and solve this problem, which is unwieldy-sounding and vague when the texture atoms have to be estimated. First, there may be many atoms, but a single patch is a combination of only a sparse subset of them. Second, because each atom appears in part in many patches, even if we observe some noisily, once we know which atoms appear in which patches, we can invert and average together all of the patches associated with an atom to estimate it.

To explain and programmatically implement the full algorithm that solves this problem, probability theorists like to explain things in terms of going to a buffet. Here’s a very rough idea. There’s a buffet with a (possibly infinite) number of dishes. Each dish represents a texture atom. An image patch will come up to the buffet and, starting from the first dish, begins to flip a biased coin. If the coin lands on heads, the patch takes a random amount of food from the dish in front of it (the atom-patch weight), and then walks to the next dish. If the coin lands on tails, the patch skips that dish and instead just walks to the next. There it flips its coin with a different bias and repeats the process. The coins are biased so the patch only eats a few dishes (there are so many!). When all is said and done, however, the patch has eaten a random amount from a few dishes. Rephrased: the image patch is made from a weighted linear combination of basic atoms.

At the end of the day, all the patches eat their own home-cooked dessert that didn’t come from the buffet (noise), and some pass out from eating too much (missing pixels).

If we know how much of each dish (texture atom) each of the patches ate and the biases of the coins, we can estimate the dishes themselves — because we can see the noisy patches. Vice versa, if we know what the dishes (textures) are, and what the patches look like, we can estimate the biases of the coins and how much of a dish each patch ate.

At first we take completely random guesses about what the dishes look like and what the coins are, as well as how much each patch ate. But soon we start alternating guesses between what the dishes are, the coin biases, and the amounts that each patch ate. And each time we only update our estimate of one of these unknowns, on the assumption that our previous estimates for the others is the truth. This is called Gibbs sampling. By iterating our estimates, we can build up a pretty good estimate of all of the unknowns: the texture atoms, coin biases, and the atom-patch weights.

The image on the right is our best final guess, after iterating this game, as to what the patches look like after eating their dishes, but before eating dessert and/or passing out.

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Modern Coin-Weighing Puzzles

I usually give a lot of lectures and I never used to announce them in my blog. This time I will give a very accessible lecture at the MIT “Women in Mathematics” series. It will be on Wednesday October 6th at 5:30-6:30 PM in room 2-135. If you are in Boston, feel free to join. Here is the abstract.

I will discuss several coin-weighing puzzles and related research. Here are two examples of such puzzles:

1. Among 10 given coins, some may be real and some may be fake. All real coins weigh the same. All fake coins weigh the same, but have a different weight than real coins. Can you prove or disprove that all ten coins weigh the same in three weighings on a balance scale?

2. Among 100 given coins, four are fake. All real coins weigh the same. All fake coins weigh the same, but they are lighter than real coins. Can you find at least one real coin in two weighings on a balance scale?

You are not expected to come to my talk with the solutions to the above puzzles, but you are expected to know how to find the only fake coin among many real coins in the minimum number of weighings.

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Marriage Proposals, Or How I Learned to Say No

In the name of privacy, I have changed the names of the men I did not marry. But there is no point in changing the names of my ex-husbands, as my readers probably know their names anyway.

I received my first marriage proposal when I was 16. As a person who was unable to say “no” to anything, I accepted it. Luckily, we were not allowed to get married until I was 18, the legal marriage age in the USSR, and by that time we broke up.

To my next proposal, from Sasha, I still couldn’t say “no”, and ended up marrying him. The fact that I was hoping to divorce him before I got married at 19 shows that I should have devoted more effort in learning to say “no”. I decided to divorce him within the first year.

My next proposal came from Andrey, I said yes, with every intention of living with Andrey forever. We married when I was 22 and he divorced me when I was 29.

After I recovered from my second divorce, I had a fling with an old friend, Sam, who was visiting Moscow on his way to immigrate to Israel.

Sam proposed to me in a letter that was sent from the train he took from the USSR to Israel. At that point I realized I had a problem with saying “no”. The idea of marrying Sam seemed premature and very risky. I didn’t want to say yes. I should have said no, but Sam didn’t have a return address, so I didn’t say anything.

That same year I received a phone call from Joseph. Joseph was an old friend who lived in the US, and I hadn’t seen or heard from him for ten years. He invited me to visit him in the US and then proposed to me the day after my arrival. The idea of marrying Joseph seemed premature and very risky, but in my heart it felt absolutely right. I said yes, and I wanted to say yes.

I was very glad that I hadn’t promised anything to Sam. But I felt uncomfortable. So even before I called my mother to notify her of my marriage plans, I located Sam in Israel and called him to tell him that I had accepted a marriage proposal from Joseph. I needed to consent to marry someone else as a way of saying “no” to Sam.

After I married Joseph, I came back to Russia to do all the paperwork and pick up my son, Alexey for our move to the US. There I met Victor. I wasn’t flirting with Victor and was completely disinterested. So his proposal came as a total surprise. That was the time I realized that I had a monumental problem with saying “no”. I had to say “no” to Victor, but I couldn’t force myself to pronounce the word. Here is our dialogue as I remember it:

  • Me: I can’t marry you, I am already married.
  • Victor: I am sure it’s a fictitious marriage; you just want to move to the USA.
  • Me: That’s not true. It’s a real marriage.
  • Victor: If it were a fictitious marriage, you wouldn’t admit it. So, it’s a fictitious marriage. My proposal stands.

My sincere attempt at saying “no” didn’t work. I moved to the US to live with Joseph and I soon got pregnant. Victor was the first person on my list to notify — another rather roundabout way to reject a proposal.

The marriage lasted eight years. Sometime after I divorced Joseph, I met Evan who invited me on a couple of dates. I wasn’t sure I wanted to get involved with him. But he proposed and got my attention. I was single and available, though I had my doubts about him.

Evan mentioned that he had royal blood. So I decided to act like a princess. I gave him a puzzle:

I have two coins that together make 15 cents. One of them is not a nickel. What are my coins?

He didn’t solve it. In and of itself, that wouldn’t be a reason to reject a guy. But Evan didn’t even understand my explanation, despite the fact that he was a systems administrator. A systems administrator who doesn’t get logic is a definite turn-off.

So I said “no”! That was my first “no” and I have mathematics to thank.

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My Sister

Ingrid and Me 2010When the Women and Math program at IAS was coming to an end, Ingrid Daubechies invited me to a picnic at her place for PACM

(The Program in Applied and Computational Mathematics at Princeton University). I accepted with great enthusiasm for three reasons: I was awfully tired and needed a rest; PACM was my former workplace, so I was hoping to meet old acquaintances; and most of all, I loved the chance to hang out with Ingrid.

Ingrid is a great cook, so she prepared some amazing deserts for the picnic. While I was helping myself to a second serving of her superb lemon mousse, a man asked me if I was Ingrid’s sister. Ingrid overheard this and laughingly told him that we are soul-sisters.

I admire Ingrid, so at first I took this as a compliment and felt all warm and fuzzy. But when my critical reasoning returned, I had to ask myself: Why would someone think I am Ingrid’s sister with my Eastern European round face, my Russian name and my Russian accent?

I started talking to the man. He asked me what I do. I told him that I am a mathematician. He was stunned. What is so surprising in meeting a mathematician at a math department picnic?

Now I think I understand what happened. It never occurred to him that I was a mathematician. I was clearly unattached, so he couldn’t place me as someone’s wife. As the picnic was at Ingrid’s house, he must have concluded that I had to be Ingrid’s relative. Very logical, but very gender biased.

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Math Careers and Choices

More and more I stumble upon the claim that the difference in individual personal choices between men and women is one of the main contributors to the gender gap in mathematical careers. Let me tell you some stories that I’ve heard that illustrate some choices that women made. The names have been changed.

Ann got her PhD in math at the same time as her husband. They both got job offers at places very far from each other. As Ann was pregnant, she decided not to accept her offer and to follow her husband. In two years she was ready to go back to research and she started to search for some kind of position at the University where her husband worked. At that time there was a nuptial rule that prevented two spouses from working at the same place. There was no other college nearby, and Ann’s husband didn’t yet have tenure and freaked out at the thought of changing his job. They decided to have a second child. After two more years of staying home, she felt completely disqualified and dropped the idea of research.

Olga was very passionate about her children’s education. She felt that public education was insufficient and that parents needed to devote a lot of time to reading and playing with their kids. Her husband insisted that the children would be fine on their own and he refused outright to read to them or to participate in time-consuming activities. Olga took upon herself the burden she was hoping to share. At that same time, she started a tenure track position. In addition to all of this, she took her teaching very seriously. Her students loved her, but her paper-writing speed declined. She didn’t get tenure and quit academia. Later, she realized that in fact her husband had shared her sense of the importance of their children’s education, but he had played a power-game which left her doing all the work.

Maria told her husband Alex that she wanted a babysitter for their two children now that she was ready to get back to mathematics. Alex sat down with her to look at their finances. Before looking for a job Maria needed to finish her two papers. That meant they had to pay a babysitter even though Maria wasn’t bringing in any money. Maria agreed to postpone her comeback until the kids went to kindergarten. She somehow finished her papers and found her first part-time job as an adjunct lecturer. Alex sat down with her again to discuss their finances. They calculated how much time she would need to prepare to teach after such a long break. The babysitter would cost more than her income. In addition, they would have to buy a second car and some professional clothes for Maria. They summed everything up: it appeared that they couldn’t afford for her to take this job. Maria rejected the offer.

Two years later, Alex was offered an additional job as a part-time mathematical consultant. Instead of accepting it Maria’s husband suggested that the company interview Maria, who was longing to return to work. Maria got an offer, but at half the fee her husband was offered. The manager explained this difference by pointing to her husband’s superior work record. Maria and Alex sat down together again and calculated that it would more profitable for them as a couple if he took the job and dropped all his household responsibilities. Maria couldn’t find a way to argue.

I recently wrote about my own decision to quit academia twelve years ago. Although what I really wanted to do was to work in academia, my family responsibilities took precedence.

Yes, personal choices are a great contributor to the gender gap in mathematical careers. I just do not like when people assume that women chose freely. Some choices we were cornered into making.

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Women, Science and The Right Tail of a Bell Curve

by Rebecca Frankel

The article Daring to Discuss Women in Science by John Tierney in the New York Times on June 7, 2010 purports to present a dispassionate scientific defense of Larry Summers’s claims, in particular by reviewing and expanding his argument that observed differences in the length of the extreme right tail of the bell curves of men’s and women’s test scores indicate real differences in their innate ability. But in fact any argument like this has to acknowledge a serious difficulty: it is problematic to assume without comment that the abilities of a group can be inferred from the tail of a bell curve. We are so used to invoking bell curves to talk about group abilities, we don’t notice that such arguments usually use only the mean of the curve. Using the tail is a totally different story.

Think about it: it is reasonable to question whether a single data point — the test score of an individual person — is a true indication of his/her ability. It might not be. Maybe a single test score represents a dunce with hyper-overachieving parents who push him to study all the time. So does that single false reading destroy the validity of the curve? No of course not: because some other kid might have been a super-genius who was drunk last night and can barely keep his eyes open during the test. One is testing above his “true ability” and the other is testing below his “true ability,” and the effect cancels out. Thus the means of curves are a good way to measure the ability of large groups, because all the random false readings average out.

But tails are not. On the tail this “canceling out” effect doesn’t work. Look at the extreme right tail. The relatively slow but hyper-motivated kids are not canceled out by the hoard of far-above-the-mean super geniuses who had drunken revels the night before. There just aren’t that many super-geniuses and they just don’t party that much.

Or let’s look at it another way: imagine that you had a large group which you divided in half totally at random. At this point their bell curve of test scores looks exactly the same. Lets call one of the group “boys” and the other group “girls”. But they are two utterly randomly selected groups. Now lets inject the “boys” with a chemical that gives the ones who are very good already a burning desire to dominate any contest they enter into. And let us inject the “girls” with a chemical that makes the ones who are already good nonetheless unwilling to make anyone feel bad by making themselves look too good. What will happen to the two bell curves? Of course the upper tail of the “boys'” curve will stretch out, while the “girls'” tail will shrink in. It will look like the “boys” whipped the “girls” on the right tail of ability hands down, no contest. But the tail has nothing to do with ability. Remember they started out with the same distribution of abilities, before they got their injections. It is only the effect of the chemicals on motivation that makes it look like the “boys” beat the “girls” at the tail.

So, when you see different tails, you can’t automatically conclude that this is caused by difference in underlying innate ability. It is possible that other factors are at play — especially since if we were looking to identify these hypothetical chemicals we might find obvious candidates like “testosterone” and “estrogen”.

The possibility of alternative explanations for these findings calls into question Tierney and Summers’ claims to superior dispassionate scientific objectivity. Moving from the mean to the tail of a bell curve makes systematic effects on averages irrelevant, true, but it is instead susceptible to systematic effects on deviations, which are irrelevant at the mean. An argument that uses this trick to dodge gender differences in averages cannot claim the mantle of scientific responsibility without accounting for gender differences in deviations. I am deeply disappointed that Tierney and Summers did not accompany their assertions with a suitable reminder of this fact.

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My Dr’s Orders: Hit on Men

I was terribly shy when I was a teenager. I worked on this problem and overcame it. But when I moved to the US my shyness returned in a strange form. I was fine around Russians but shy around Americans. At first I assumed that it was a language problem.

I became friends with a Russian sexologist and psychotherapist. He pointed out that I never initiated a conversation with Americans and so I realized that my shyness had returned. He prescribed an exercise for me: I had to invite a new American guy to lunch once a week.

Why guys? Maybe because he was a sexologist or maybe because my problems with self-esteem were more pronounced when I was around men. In any case, I decided to do the exercise.

To paint the full picture I need to add some relevant details. At that time I was married, although I didn’t wear a ring, and wasn’t especially interested in other men. The reason I didn’t wear a ring was that Joseph, my husband at the time, did not himself want to wear a ring. As I love symmetry in relationships more than I love rings, I didn’t wear one either.

The men I was about to invite to lunch were mere acquaintances, because I had not yet made any American friends. So although I didn’t intend to hide it, they may not have realized that I was married.

Two things surprised me in this exercise. First, it was very easy. Most people agreed to do lunch with me.

Second, every man I invited mentioned his girlfriend. This was unexpected. From my experience with Russians, I anticipated that every man would hide his involvement with someone else, even with a wife, at least for some time. At the very least, many Russian men would try to flirt.

The Americans were different. Unclear why I had invited them out, they wanted to be upfront with me from the start, just in case I was interested in them. Since that experience, I admire the way that American men come clean.

I never invited any of these guys out twice: I just needed a supply of new men for my exercise in overcoming my shyness. I wonder if they thought I was put off by their confessions. Perhaps my loss of interest in them after the first lunch confirmed their suspicions that I was attracted to them.

The sexologist’s exercise was a success. Today I have no trouble inviting someone to lunch.

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