I remember well the beginning of the first project. Yakov Isaevich and I went to see a young urologist Alexei Velikanov, the son of one of Moscow’s top physicians. Yakov Isaevich was a longtime friend (and patient) of the older Velikanov, who asked Yakov Isaevich to help his son. Alexei showed us a huge sheet of paper with various data collected from about a hundred patients who had had a prostate adenoma removed (this is a benign tumor of the prostate frequently found in older men). The data included various characteristics, such as blood pressure and other test results, before and after the surgery. He was hoping to use these data to come up with some conclusions about when surgery was more likely to be successful, thereby enabling him to make a set of recommendations as to when to remove the tumor.
He needed help analyzing the data and hoped we could do it. As I learned later, this was a typical situation. Doctors, engineers, and others would often expect that mathematicians have some magic wand that enables them to quickly derive conclusions from whatever data they had collected. Of course, this is wishful thinking. We do know some powerful methods of statistical analysis, but very often these methods cannot be applied because the data are not precise or because there are different types of data: some objective and some subjective (describing how the patients “feel,” for example); or some quantitative, such as blood pressure and heartbeat rate, and some qualitative, such as “yes” or “no” answers to some specific questions. It is very difficult, if not outright impossible, to feed such inhomogeneous data into a statistical formula.
On the other hand, sometimes asking the right questions may allow one to realize that some of the data is irrelevant and should be simply thrown away. My experience is that only about 10–15 percent of the information that the doctors collected was ever used when they made the diagnosis or treatment recommendations. But if you asked them, they would never tell you this directly. They would insist that all of it is useful and would even try to come up with some scenario or other in which they would take this information into account. It would take a while to convince them that actually in all of those cases they ignored most of the data and made the decision based on very few essential criteria.
Of course, sometimes there were questions that could be answered by simply feeding the data into some statistical program. But working on these projects, I gradually came to realize that we mathematicians are most useful for doctors not so much because of our knowledge of these statistical programs (after all, this isn’t by itself very difficult, and anyone can learn it), but because of our ability to formulate the right questions and then to go through a cold and unbiased analysis to get the answers. It is really this “mathematical mindset” that seems to be most useful to those who are not trained to think as mathematicians.
In my first project, this approach helped us to weed out irrelevant data and then find some non-trivial connections, or correlations, between the remaining parameters. This wasn’t easy and took us a few months, but we were happy with the results. We wrote a joint paper about our findings, and Alexei used them in his doctoral thesis. Yakov Isaevich and I were invited to the thesis defense, along with another Kerosinka student, Alexander Lifshitz, my good friend, who also worked on this project.
I remember how at the thesis defense one of the doctors asked for the name of the computer program used to derive these results, and Yakov Isaevich answered that the names were “Edward and Alexander.” This was true: we did not use a computer, instead doing all computations by hand or with a simple calculator. The main point was not to calculate (that was the easy part) – it was to ask the right questions. An eminent surgeon, present at the defense, then made a comment to the effect that it was very impressive that mathematics turns out to be so useful in medicine, and perhaps will become even more useful in the years to come. Our work was received well by the medical community, and Yakov Isaevich was pleased.
Soon afterward, he asked me to work on another project in urology that had to do with kidney tumors (for another doctoral thesis), which I was also able to complete successfully.
The third, and last, medical project I worked on was the most interesting one for me. A young doctor, Sergei Arutyunyan – who also needed help to analyze his data for a thesis – and I had a great rapport. He was working with patients whose immune systems were rejecting transplanted kidneys. In such situation the doctor has to make a quick decision whether to fight for the kidney or remove it, with far-reaching consequences: if they kept the kidney, the patient could die, but if they removed it, the patient would need another one, which would be very difficult to find.
Sergei wanted to find a way to tell which recommendation was statistically most viable, based on quantitative ultrasound diagnostics. He had much experience in this area and collected a lot of data. He hoped that I could help him to analyze this data and come up with meaningful objective criteria for decision-making that could be useful to other doctors. He told me that no one had yet been able to do this; most doctors thought this was impossible and preferred to rely on their own ad hoc approaches.
I looked at the data. Like in our previous projects, there were about forty different parameters measured for each patient. During our regular meetings, I would ask Sergei pointed questions, trying to figure out which of these data were relevant and which weren’t. But this was hard. Like other doctors, he would give his answers based on specific cases, which was not very helpful.
I decided to use a different approach. I thought, “This man makes these kinds of decisions every day, and obviously he is very good at it. What if I manage to learn to ‘be him’? Even if I don’t know much about the medical aspects of the problem, I could try to learn his methodology following his decision-making process, and then I could use this knowledge to come up with a set of rules.”
I suggested that we play a kind of a game.2 Sergei had collected data on approximately 270 patients. I chose, randomly, the data for thirty of them and put aside the rest. I would take the history of each of these randomly chosen patients and have Sergei, who was sitting at the opposite corner of the office, ask me questions about the patient, which I would answer by consulting the file. My goal in all this was to try to understand the pattern of his questions (even if I could not possibly know the meaning of these questions as well as he did). For example, sometimes he would ask different questions, or the same questions, but in a different order. In such a case, I would interrupt him: “Last time you did not ask this. Why are you asking it now?”
And he would explain, “Because for the last patient the volume of the kidney was so and so, and this ruled out this scenario. But for the current patient it is so and so, and so this scenario is quite possible.”
I would make notes of all this and try to internalize this information as much as possible. Even so many years later, I can picture it well: Sergei sitting in a chair in the corner of his office, deep in thought, puffing on a cigarette (he was a chain-smoker). It was fascinating to me to try deconstructing the way he thought – it was kind of like trying to undo a jigsaw puzzle to find out what the essential pieces were.
Sergei’s answers gave me extremely valuable information. He would always arrive at the diagnosis after no more than three or four questions. I would then compare it with what actually happened to each patient. He was always spot on.
After a couple dozen cases, I could already make the diagnosis myself, following the simple set of rules that I learned while interrogating him. After half a dozen more, I was practically as good as he was in predicting the outcome. There was in fact a simple algorithm at play that Sergei was following in most cases.
Of course, there were always a handful of cases in which the algorithm would not be useful. But even if one could derive effectively and quickly the diagnosis for 90 to 95 percent of the patients, this would already be quite an achievement. Sergei told me that in the existing literature on the subject of ultrasound diagnostics, there was nothing of this sort.
After completing our “game,” I derive
d an explicit algorithm that I’ve drawn as a decision tree below. From each node of the tree there are two edges down to other nodes; the answer to a specific question at the first node dictates which of the next two possible nodes the user should go to. For example, the first question is about the index of peripheral resistance (PR) of the blood vessel inside the transplant. This was a parameter Sergei himself had come up with in his research. If its value was greater than 0.79, then it was highly likely that the kidney was being rejected, and the patient required immediate surgery. In this case, we move to the black node on the right. Otherwise, we move to the node on the left and ask the next question: what is the volume (V) of the kidney? And so on. Each patient’s data therefore gives rise to a particular path on this tree. The tree terminates after four or fewer steps (it is not important to us at the moment what the remaining two parameters, TP and MPI, were). The terminal node contains the verdict, as shown on this picture: the black node means “operate” and the white node means “do not operate.”
I ran the data of the remaining 240 or so patients, whose files I had put aside, through the algorithm. The agreement was remarkable. In about 95 percent of the cases, it led to an accurate diagnosis.
The algorithm described in simple terms essential points of the thought process of a doctor making the decision, and it showed which parameters describing the patient’s condition were most relevant to the diagnosis. There were only four of them, narrowing down the initial slate of forty or so. For example, the algorithm showed the importance of the index of peripheral resistance that Sergei had developed, measuring the flow of blood through the kidney. That this parameter played such an important role in the decision-making was, by itself, an important discovery. All of this could be used in further research in this area. Other doctors could apply the algorithm to their patients, test, and perhaps fine-tune it to help make it more efficient.
We wrote a paper about this, which became the basis for Sergei’s doctoral thesis, and applied for a patent that was approved a year later.
I was proud of my work with Yakov Isaevich, and he of me. Despite our good relationship, however, I kept my “other” mathematical life – my work with Fuchs and Feigin and all of that – secret from him as I did from most other people. It was as though applied mathematics was my spouse, and pure mathematics was my secret lover.
Still, when the time came that I had to look for a job, Yakov Isaevich told me that he would try to hire me as an assistant at his lab at Kerosinka. That in turn would allow me to become a Ph.D. student there a year later, which would open a clear path to my employment in the foreseeable future. This sounded like an excellent plan, but there were many obstacles, not least that, as my dad had been warned when he went to Kerosinka before I applied there, I would again have to confront anti-Semitism.
Of course, Yakov Isaevich was well aware of this. He had been at Kerosinka for several decades and knew how everything worked. He had actually been hired by Rector Vinogradov himself, whom Yakov Isaevich held in high regard.
Questions of my appointment would be handled by mid-level bureaucrats, not by Vinogradov, and those guys would be sure to close all doors to anyone whose last name sounded Jewish, but Yakov Isaevich knew how to work the system. At the beginning of the spring semester of my last year at Kerosinka, when the question of my employment became urgent, he had typed a letter appointing me to his laboratory. He carried the letter with him in his briefcase, so that, should he get a chance to talk to Vinogradov about me personally, he would be well-prepared.
The opportunity soon presented itself. One day he bumped into Vinogradov as he was entering Kerosinka. Vinogradov was pleased to see him and asked, “How are you doing, Yakov Isaevich?”
“Terribly,” replied Yakov Isaevich grimly (he could be a good actor).
“What happened?”
“We have done wonderful things at my lab in the past, but we can’t do this anymore. I can’t get new talent. I have this great student who is graduating this year, but I am unable to hire him.”
I suppose Vinogradov wanted to show Yakov Isaevich who was the boss – which is what Yakov Isaevich’s goal was – so he said, “Don’t worry, I will take care of this.”
At which point, Yakov Isaevich produced my appointment letter. Vinogradov had no choice but to sign it.
Normally, this letter had to be signed by a dozen people before landing on Vinogradov’s desk: the heads of the local Komsomol organization and Communist Party, and all kinds of other bureaucrats. They would surely find a way to stall the process so that this would never happen. But now it already had Vinogradov’s signature! So what could they do? He was the boss, and they couldn’t possibly disobey his wishes. They would grind their teeth and stall for a while, but eventually they would all give up and sign. You should have seen their faces when they saw Vinogradov’s signature at the bottom! Yakov Isaevich had played the system brilliantly.
Chapter 13
Harvard Calling
In the midst of all that stress and uncertainty, in March of 1989, a letter arrived from the United States, on the letterhead of Harvard University.
Dear Dr. Frenkel,
Upon the recommendation of the Department of Mathematics, I would like to invite you to visit Harvard University in the fall of 1989 as the recipient of the Harvard Prize Fellowship.
Sincerely yours,
Derek Bok
President of Harvard University
I had heard about Harvard University before, though I must admit I did not quite realize at the time its significance in the academic world. Still, I was very pleased. Being invited to America as a winner of a fellowship sounded like a big honor. The President of the University personally wrote to me! And he addressed me as a “Doctor,” even though I had not yet received my bachelor’s degree. (I was then in the last semester of my studies at Kerosinka.)
How did this happen? The word about my work with Borya was spreading. Our first short paper had already been published, and we were finishing three other, longer, papers (all in English). The physicist Lars Brink, visiting Moscow from Sweden, solicited one of them for a volume he was editing. We gave him our paper for the volume and asked him to make twenty or so copies and send them to mathematicians and physicists abroad we thought would be interested in our work. I had found their addresses in published papers that were available at the Moscow Science Library and gave the list to Lars. He kindly agreed to help us because he knew how difficult it would be for us to send copies ourselves. That paper became widely known, in part because of its applications to quantum physics.
This was several years before the usage of the Internet became widespread, but the system of dissemination of scientific literature was quite efficient: authors would circulate typed manuscripts of their articles before publication (they were called preprints). The recipients would then copy and forward them further to their colleagues as well as university libraries. The twenty or so people to whom Lars Brink sent our paper must have done the same.
In the meantime, tremendous changes were happening in the Soviet Union: it was the time of perestroika, launched by Mikhail Gorbachev. One of the results of this was that people were allowed to travel abroad with greater freedom. Before then, mathematicians like Feigin and Fuchs received many invitations to attend conferences and visit universities in the West, but foreign travel was closely regulated by the government. Before getting the usual entry visa to another country, one had to get an exit visa, which enabled the person to leave the Soviet Union. Very few of those visas were given out, out of fear that people would not return (and indeed, many of those who were granted exit visas didn’t come back). Almost all requests were denied, usually on bogus grounds, and Fuchs once told me that it had been years since he had last tried to get one.
But suddenly, in the fall of 1988, several people were allowed to travel abroad, one of whom was Gelfand. Another, a talented young mathematician and a friend of Borya named Sasha Beilinson, also traveled to the United S
tates to visit his former co-author Joseph Bernstein, who had emigrated a few years earlier and was a professor at Harvard.
In the meantime, some scientists in the West also recognized that changes were coming and tried to use this opportunity to invite scholars from the Soviet Union. One of them was Arthur Jaffe, a renowned mathematical physicist who was then the chairman of the Mathematics Department at Harvard. He decided to create a new visiting position for talented young Russian mathematicians. When Gelfand, who had an honorary degree from Harvard, came to visit in the fall of 1988, Jaffe enlisted his help in order to convince the president, Derek Bok, whom Gelfand knew personally, to provide funding and support for this program (some funding was also provided by Landon Clay, who later founded the Clay Mathematics Institute). Jaffe called it the Harvard Prize Fellowship.
Once the program was in place, the question was whom to invite, and Jaffe polled various mathematicians for suggestions. Apparently, my name was mentioned by several people (including Beilinson), and this was the reason that I was selected among the first four recipients of the award.
The letter from President Bok was quickly followed by a longer letter from Jaffe himself, which described the terms of the appointment in more detail. I could come for a period between three and five months; I would be a Visiting Professor but wouldn’t have any formal obligations except for giving occasional lectures about my work; Harvard would pay for my travel, housing, and living expenses. Practically the only thing Harvard wasn’t providing was a Soviet exit visa. Fortunately, and to my great surprise, I got it in a month.
Arthur Jaffe wrote in his letter that I could come as early as the end of August and stay till the end of January, but I chose to stay for three months, which was the minimum period my stay specified in the letter. Why? Well, I had no intention of emigrating to the U.S., and I was planning to come back. Besides, I was feeling guilty that I would have to take a leave from the job at Kerosinka that Yakov Isaevich had won for me with such an effort.
Love and Math Page 16
