Archive for the ‘Women and Math’ Category.

## 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

When 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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## A Woman in Numbers

I am used to thinking that a “woman in numbers” means a female number theorist. But not anymore. I just discovered drawings by Svetlana Bogatyr. From now on the expression a “woman in numbers” will convey an additional meaning to me.

I am grateful to Svetlana for permitting me to post several of her drawings. The “Mature Woman” is on the left. “Eurydice”, “Girl in Scarf” and “Holland Woman ” are below.

Enjoy.

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## Women in Numbers

This year I am again on the organizing committee of the Women and Mathematics program at Princeton’s Institute for Advanced Study. Our subject is “p-adic Langlands Program.” It is a fashionable, advanced and very influential program connecting number theory and representation theory.

We invite undergraduate students, graduate students and post-docs to apply. In 2009-2010 the Institute has been running a special year in Analytic Number Theory. That has brought many number theorists to the institute already, so there will be a lot of people to talk to.

Last year I promised to hold a math party during the program. But I had to cancel it due to a scheduling conflict with George Hart’s ZomeTool Workshop. I am planning a party this year. Either way, we’ll have fun.

If you want to learn about the Langlands program, to spent time on the beautiful grounds of the Institute, to eat in one of the best cafeterias around, and to make new friends with other women interested in number theory, then please apply. The application deadline is February 20.

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## Gelfand’s Memorial

Israel Gelfand’s memorial is being held at Rutgers on December 6, 2009. I was invited as Gelfand’s student.

My relationship with Gelfand was complicated: sometimes it was very painful and sometimes it was very rewarding. I was planning to attend the memorial to help me forget the pain and to acknowledge the good parts.

I believe that my relationship with Gelfand was utterly unique. You see, I was married three times, and all three times to students of Gelfand.

Now that I know that I can’t make it to the memorial, I can’t stop wondering how many single male students of Gelfand will be there.

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Israel Gelfand was my scientific adviser from the time I was 15. This is the story of how Gelfand helped me, when at 20 I was an undergrad at Moscow State University. At that time, I was married to Sasha (Alexander) Goncharov, who was also Gelfand’s student.

Sasha was more driven by mathematics than I. I had a lot of different interests: I wanted to hang out with friends, go to movies and read books. Sasha only wanted to do mathematics. His only other obsession was with what our colleagues (including me) were doing mathematically. So he was constantly asking me about the math problems I was thinking about.

For example, I was sitting at my side of the desk working, and he asked me to tell him about my problem. A few minutes later, I was forced to interrupt my work to go grocery shopping, because the household chores fell to me. As soon as I returned with bread and milk, Sasha excitedly told me the solution to my problem. It made me feel stupid, as if I should have solved it while I was waiting in the line for bread and milk. That feeling blocked out all the other feelings I should have been noticing, such as frustration and annoyance with Sasha.

Without his interference, I would have happily solved the problem myself. I was about to start my serious research, but I worried that I’d end up as a supplier of new problems for his papers.

You might wonder why I didn’t stop sharing my math with Sasha. But at that time, I wasn’t very in touch with my feelings and I prided myself on being a logical person. The idea that a husband and wife would discuss their work together seemed logical. Besides, even though I wasn’t particularly interested, Sasha was always ready to tell me about his math problems. It seemed important for me to be fair and to reciprocate. So I was stuck in a situation I didn’t know how to resolve.

I never confided this issue to any math colleagues. I never mentioned it to Gelfand — mostly because I was too scared of him to initiate any conversation. Besides, Gelfand delegated most of his responsibilities to others, because he was quite famous and busy. For example, all official paperwork related to his adviser role was done by Alexandre Kirillov. With me avoiding Gelfand and Gelfand being busy, we almost never spoke one-on-one.

You can understand my surprise when one day Gelfand approached Sasha and me to have a chat. He told us that we were about to start our own research, and while he permitted me to ask Sasha about what he was doing, he would not allow Sasha to interfere with my research.

Gelfand was a great judge of character. Without anyone telling him, he perceived what was going on in our marriage and gave me an excuse to stop Sasha’s prying. It was an appreciated gift.

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