I think it is great when graduate students come up with problems to work on by themselves. There is probably no better preparation for a research career. Unfortunately, grad students in math only have about 3-4 years to learn about a field, produce new results, publish their research and write a thesis. This does not leave a lot of time for exploration. The role of the advisor is to try help select the most fruitful directions. Here are some of the questions that I found helpful:
1) Is the problem new, or has it been answered in some form before?
This is essential since you don’t want to have a student writing a thesis or a paper on something that is already known – and if it is known, it is good to find how others did it. This can then naturally lead to further questions.
2) Is the question interesting?
This is harder. One way is to have students think about how they could convince other graduate students that this problem is worth studying. After they can think about how to convince non-scientists.
3) Is it the question related to what we are doing in the lab/research group? Can it be answered in a reasonable time?
This is a more practical point. More senior graduate students that already have the majority of the thesis written, can have more leeway.
I am sure there are a number of other criteria here. In his TED talk, Uri Alon says that he approaches research like improv theater. This is great once a question is clearly articulated, and the group is looking for a way forward. Picking the right problem to work on is tricky. What other questions can help guide graduate students to a good problem, while encouraging them to take a role in designing their own research?
In his short story “On Exactitude in Science” Jorge Luis Borges describes a map so detailed that it is the exact size of the empire it represents. Every bridge, road and house on the map is the size of their real counterpart. Of course, such a map is absurd. Maps are useful because they show the most important features of a location – they give us only the information we need. Even a map that is more than a few feet across is useless for navigation. The job of a mapmaker is then to discard enormous amounts of detail, and show us only what we need to know to get from one place to another.
This is not only true for maps. Our megapixel cameras are similar to the cartographers in Borges’ story. Each photo we take contains enormous amounts of data. However, the information that we need to reconstruct a good approximation of each image is much smaller. This is why photos are almost always compressed when stored. Indeed, most compression works by discarding some of the unnecessary information in the original pictures.
But if much of the information in a photo is thrown away afterwards, then why do we need megapixel cameras? Why not just record the essential information about the scene in front of us to start with. If we did so, we would not have to compress the files later.
How to do this has long puzzled scientists and mathematicians. The problem is that one approach may work for some pictures, but fail for others. How can we make a camera that will always only make measurements we need to reconstruct a picture?
The mathematician Emmanuel Candés stumbled upon the answer almost by chance while trying to remove noise from an image. He knew that an exact algorithm that would identify only relevant information would require far too much time to run. He therefore tried an approach that at first sight should not have worked well. To his surprise he recovered almost all the information in the original image – the process worked like magic. Candés said later: “It was as if you gave me the first three digits of a 10-digit bank account number — and then I was able to guess the next seven.”
This was the first step in the development of what is now called “compressive sensing”: A way to take good pictures by taking very few measurements. Surprisingly, rather than carefully planning which measurements to take, it is best to take measurements at random. When this was first proposed, many thought it must be wrong. How could taking measurements at random work better than a sophisticated algorithm? Later mathematical analysis answered these doubts, however. As a result, compressive sensing is used in different areas of medical imaging, and its applications are growing.
We are awash in information. But the main problem is not how to acquire more. Like cartographers, we need to find what matters, and discard the rest. And here, like in so many other matters, it is mathematics that shows the way.
Here is an interview with one of the discoverers of compressive sensing, Emmanuel Candés. Follow the link to a more technical review article.
Here is a good article in Wired that gives a more detailed overview of the origins of compressive sensing Here is another overview with a bit more math, but also very understandable. Here is also an understandable lecture.
Here is the text of another episode for Engines. For a great illustration of this idea, see this excellent web page created by undergrads at UC, Berkeley. I will try to write a second on compressive sensing, and record them together. Comments are welcome, of course:
In 1973 the University of California at Berkeley was sued for sex discrimination in graduate student admissions. The case seemed clear cut: only 35% of female compared to 44% of male applicants were admitted to graduate programs across the university. However, when statisticians looked at the data in more detail they found a surprise. When looking at the admission rates of individual departments, the apparent bias disappeared. Across departments, women were either more likely to be admitted or about equally likely to be admitted as men. Indeed, at the level of individual departments, women seemed to fare slightly better.
This is an example of Simpon’s paradox – a paradox that can affect averages whenever we combine, or pool, data. Here is another example involving two New York Yankees players, Derek Jeter and David Justice. In both 1995 and 1996 David Justice had a higher batting average than Derek Jeter. However, when we compute the batting average over both seasons, then Derek Jeter is ahead of David Justice. Again, pooling the data gives a different picture than when looking at smaller chunks.
How is this possible? Let’s look at the case of graduate school applicants to the University of California at Berkley. It turns out that more women applied to departments in the humanities, while men tended to apply in higher numbers to engineering and science departments. Humanities departments had fewer available slots, and rejected more applicants. Thus female applicants applied mostly to departments which admitted fewer students, whether male or female. As a result, the overall fraction of women admitted was lower than that of men. A bias may have existed, but it was not a bias in the rate of admissions. Rather, it was a bias in the number of women who chose to apply graduate studies in technical fields.
Simpson’s paradox can have important consequences. For example medical researchers compared a less invasive treatment for kidney stones to the established surgical methods, and found the new treatment to be better overall. However, the less invasive treatment was more frequently applied to small kidney stones. Since smaller kidney stones are easier to treat, this gave an advantage to the new, less invasive method. When the treatments were compared separately on small kidney stones and large kidney stones, the traditional treatment proved to be more successful. Taking into account kidney stone size completely changed the conclusion about which treatment is better.
The outcomes of lawsuits, promotions, and our choice of medical treatments are frequently based on numerical evidence. Yet our intuition can easily mislead us when we think about numbers. Mathematics and statistics can help – they can give us answers to the question we are asking. But it is up to us to make sure that we are asking the right questions.
The Wikipedia article on Simpson’s Paradox has a number of other good examples.
The mathematician John Tukey is credited with saying that “Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.”
I have recently revisited the following old problem in a new paper with Alan Veliz-Cuba and Zack Kilpatrick: Suppose that you are trying to decide between different alternatives. Maybe you are trying to find the best place to fish, behind which bush to look for your dinner, or between different products in a super market. Humans, animals, and even animal collectives decide between alternatives all the time. Such decisions are frequently not based on a single observation. Indeed, animals accumulate evidence, and combine bits of information to arrive at a better choice.
There has been a lot theoretical work to determine the best way to integrate information that arrives over time. There are also fascinating experiments that suggest that certain animals make decisions in a way that is very similar to the theoretical models. Indeed, neuroscientists have even capture the neural signature of such processes.
This figure shows the activity of neurons in area VIP of monkeys that are deciding whether a cloud of dots moves predominantly to the right or to the left. As the animal accumulates information for one or alternative, the activity of a cell increases. Figure from Gold and Newsome, Annual Review of Neuroscience, vol. 30, p. 535 (2007).
However, in most classical studies of such decision making the correct (or better) choice is fixed during a trial (For a recent study where this is not the case, see here). Unless there is a some pressure to make a decision quickly, in such cases it is best to accumulate as much evidence as possible, that is, wait indefinitely to make sure that you are making correct choice. However, the natural world constantly changes, and what is a correct choice or better option at one instant, may no longer be so in the next. Our goal was to extend classical models to the case where the truth is not constant.
The model that we derived has some interesting features: An ideal evidence accumulator will discount prior evidence in a way that is determined by the volatility of the environment. In other words, to perform well in a changing environment it is best to forget evidence that is too old, as it is no longer pertinent. Indeed, our model shows that you want to keep evidence that has arrived over about one average environmental epoch. As a consequence, even if for some reason the best option does not change for an extended time, there is a limit to the certainty you can attain about your choice. Your certainty is limited by the amount of information you keep in memory.
The nice thing about differential equation models is that they suggest plausible neural implementations. We proposed such a model with the activity in different neural populations representing the evidence for different choices. Interestingly, unlike in classical models, the different populations here are coupled through excitation.
The amazing variety of life on this planet is a product of evolution. However, it took billions of years for sharks, chimps and magnolia to evolve from their common ancestor. Given that evolution operates on such an enormous timescale, how could we possibly study it in a laboratory? Human life just seems too short.
But not all evolution is slow. Within our lifetime bacteria have evolved defenses against the most powerful antibiotics. Indeed, many antibiotics are themselves the result of evolution fueled warfare between different bacteria and between bacteria and the fungi they attack. We simply borrow some of their weapons. But, bacteria that are not killed by antibiotics can prosper. They give rise to new resistant generations, rendering our weapons useless. This type of evolution can occur within days, and if we don’t discover new drugs, the resulting antibiotic resistance may end up costing millions of lives by the middle of this century.
Can we control how organisms evolve? Our ancestors have done this to suit their own ends: dogs and wheat are in their current form a result of evolution that humans have been steering.
Scientists have tried to do this more deliberately. Perhaps the first was the Reverend William Dallinger. Just over 20 years after Darwin published his theory of evolution, Reverend Dallinger examined whether single celled organisms could adapt to slow changes in their environment. He started with an incubator filled with microbes that could initially only survive at room temperature. Over six years, the Reverend slowly increased the temperature inside the incubator to 158 degrees F to see whether the microbes would adapt.
More recent versions of this experiment are being carried out by many scientists, including Tim Cooper at the University of Houston, and Richard Lenski at Michigan State University. In 1988, Lenski started growing bacteria giving them just enough food to survive from day to day. He has been observing 12 different lines of bacteria ever since. This amounts to over 60,000 bacterial generations, equivalent to about 1,500,000 human years – longer than our species has been around.
The results of this experiment are giving extraordinary insights into how life changes and adapts. The 12 different lines of bacteria have all evolved to thrive on their meager diets. Looking at their genes reveals that they have often used the same tricks – the same mutations – to achieve this. But in one line something unexpected happened. The bacteria started feeding in a completely new way, a change similar to us evolving the ability to eat wood.
So the churn of mutations, and transfer of genes, keeps creating variants of organisms that have never before existed. Most quickly disappear. A few succeed and create offspring that inherit their parents’ characteristics. And so – as Darwin wrote at the end of his Origin of Species – “from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.”
We have become so used to antibiotics that it is hard to imagine what medicine would look like without them. Antibiotics are not only used to treat infections, but also make many other medical treatments possible by preventing infections in the first place. Here evolution works against us – medicine without antibiotics will look very different indeed.
Some of Richard Lenski’s reflections on his long running experiment.
The last two issues of SIAM News both have an article that describes some collaborative efforts with different colleagues. Eric Shea-Brown writes about our joint effort to relate the patterns of connections between the neurons in the brain to their joint dynamics, and ultimately function. Dana Mackenzie gives a nice overview of interactions between mathematicians and synthetic biologists. This includes a description of some joint work with Will Ott (UH) and Matt Bennett (Rice). Unfortunately, the second article does not mention the two postdocs who did most of the work: Faiza Hussain (Rice) who did the majority of the experimental work, and Chinmaya Gupta (UH) who did most of the modeling. The press release from UH is here.
When asked why he no longer frequented Ruggeri’s restaurant in St. Louis, Yogi Bera famously replied: “Nobody goes there any more. It’s too crowded.” As other Yogi Bera quotes, this one is silly, yet insightful. And it is related to an interesting problem at the interface of mathematics and economics.
The problem was inspired by the El Farol bar in Santa Fe, near the famous Santa Fe Institute for the study of complex systems. Scientists at the institute like to frequent El Farol on Thursdays to listen to live Irish music. However, the bar is too small for all of them. Let’s assume that if more than half of them decide to go, it is so packed that nobody has a good time. However, when the bar is not overcrowded, an evening of music is more fun than staying at home. Importantly, in our example everybody needs to make up their mind at the same time about whether to go or not. Nobody can call ahead to see how many people are at the bar, or coordinate with others.
How can a patron decide whether to go or not? If everybody acts the same, then everybody either stays at home, or everybody goes to the bar. Neither of these outcomes is optimal, since the bar is either completely empty or completely packed.
But these are scientists. They observe the bar and track how crowded it is from one week to the next. They look for patterns in attendance to help them decide whether to go or not. Maybe the bar was nearly empty the last couple of weeks, indicating that it might be empty again this week. Or maybe there is a cycle with the bar overcrowded one week, and nearly empty the next. Thus each scientist develops a strategy to translate these observations into a decision.
But here is the catch – there is no one strategy that guaran-tees success. If there was, everybody would be using it. But with the same strategy everybody would again be making the same decision each week, and the bar would be either overcrowded or empty. We would be where we started. The best hope is that the scientists choose different strategies. Some stay home, and each Thursday the bar is filled exactly to capacity.
This may seem like a frivolous problem. However, it has been studied extensively by mathematicians and economists as a simple model of a market. Indeed, the scientists compete for a resource – music at a bar. They are rational, as they monitor attendance and use this information strategically. But, as in a real market, they have limited information – they do not know the strategies of others, only how many show up each week. To do well, everybody needs to keep learning and adapting.
But the El Farol problem applies more widely. Suppose that instead of a bar and patrons we are think of the ocean and fisher-man. If all go out to fish, the stock will collapse. But if none do, many will go hungry. Or think of how we make use of our environment. And consider, that unlike the patrons of the El Farol bar as residents of planet Earth, we will not get a chance to try again, if we overcrowd it and overtax its resources.
The original El Farol problem was proposed in a very readable paper by Brian Arthur. In the original paper, the assumption is that if more than 60% of potential patrons visit the bar, then it will be overcrowded. I have changed this to make the exposition a bit simpler, and kind of combined the El Farol problem and the Minority Game. These are similar, but not the same. You can find the origi-nal paper here.
As noted, the El Farol problem has an even simpler version – the Minority Game. The idea is that each agent plays in successive rounds of a game where there are only two choices A and B. Any agent in the minority wins a round. Everybody in the majority loses. For instance if during one round most agents chose A, then all those who chose B win a set amount. Each agent learns a strategy based on their previous choices. Here is an introduction, and an accessible overview which discusses implications for economics can be found here.