Hi Reddit! I’m Steven Strogatz and I’m the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, New York. I have broad interests in applied mathematics. At the beginning of my career, I was fascinated by mathematical biology and worked on a variety of problems, including the geometry of supercoiled DNA, the dynamics of the human sleep-wake cycle, and the collective behavior of biological oscillators, such as swarms of synchronously flashing fireflies. In the 1990s, my work focused on nonlinear dynamics and chaos applied to physics, engineering, and biology. Several of these projects dealt with coupled oscillators, such as lasers, superconducting Josephson junctions, and crickets that chirp in unison. In each case, the research involved close collaborations with experimentalists.
I enjoy branching out into new areas, often with students taking the lead. In the past, this has led us into such topics as mathematical explorations of the small-world phenomenon in social networks (popularly known as “six degrees of separation”), and its generalization to other complex networks in nature and technology; the role of crowd synchronization in the wobbling of London’s Millennium Bridge on its opening day; and the dynamics of structural balance in social systems.
I’m here to answer questions about a recent paper my group published in the journal eLife (https://doi.org/10.7554/eLife.30212; plain-language summary: https://doi.org/10.7554/eLife.30212.002) – where we used a simple mathematical model to discover why diverse infectious diseases and cancers show similar distributions in their incubation periods – or queries related to anything about mathematics and its applications more broadly. I’ll start answering questions at 1pm EST. AMA!
How much did Ed Lorenz’ seminal work on chaos theory influence you personally?
How quickly did applied mathematicians in other fields pick up on his ideas? Were the consequences recognized immediately?
P.s. On behalf of graduate students everywhere, thank you for your textbook on chaos and nonlinear Dynamics.
It certainly got me very interested in the subject of chaos. When I was just beginning as a graduate student in 1981, I remember hearing a lecture about Lorenz’s work on strange attractors given by Prof. Keith Moffatt at Cambridge University. It was fascinating, and I couldn't wait to learn more about it. Soon afterward, I started reading the books by Abraham and Shaw, which were sort of like picture books or comic books that explain the visual aspect of chaos, along with the basic concepts. Then in 1983, a book by Guckenheimer and Holmes came out which explained the math in a pretty understandable way. After that I was hooked.
It was also a fantastic thrill for me when I got to meet Prof. Lorenz. My first job was as an assistant professor at MIT, where I taught a graduate-level nonlinear dynamics and chaos course that eventually grew into the material for my textbook. Every year I would ask Prof. Lorenz to come and give a guest lecture for the course. He'd say, what would you like me to talk about? And I'd say, how about the Lorenz equations? And every year, his answer was always the same – that little model? And then he wouldn't talk about them. Instead he would tell the class about whatever he was working on at that point. He was still doing interesting research well into his 70s and 80s. To my chaos students (and to me too), it was like hearing about relativity theory from Einstein himself. Wow! We were all awestruck.
Prof. Lorenz was also a very nice, modest man. I often saw him in the Walker Memorial cafeteria at MIT when I stayed late to eat dinner after work, and he would come walking in with his wife, holding hands, one of them hobbling a little bit with a cane.
What do you think is the biggest challenge in the math/biology field?
That's a tough one! There are so many challenges. Just to name a few, how about these: Modeling cardiac arrhythmias, and understanding them well enough to help prevent them. Same thing with the dynamics of cancer. Understanding the behavior and occasional overzealous reactions of the immune system, and what's going on with chronic inflammatory diseases. Then, of course, there's consciousness. Good luck with that one. And how about origin of life? Personally, I find that one of the most fascinating mysteries – to try to understand how life could've evolved from chemistry.
A different answer would be to say that the biggest challenge is the difficulty of bridging the cultural divide between mathematics and biology. It's often the case that people who like one of these subjects are less inclined toward the other. Yet to make real progress in mathematical biology (and in its more modern form, computational biology) we need people who are comfortable and knowledgeable and respectful of the ideas, techniques, and styles of thinking used in mathematics and biology, both. But fortunately it seems like enormous progress is being made in that direction. When I was getting started, mathematical biology was a fringe subject. Today, just about every serious research institution has exciting efforts going on in which physicists, mathematicians, engineers, and computational scientists are teaming up with doctors and biologists – and it even looks like computational social sciences are starting to be hot now too. So many fascinating things to work on!
Hi there! I'm a physicist with a similar career, on a much smaller scale, to yours. I work with medical imaging only.
One thing that puzzles me a lot is how the intersection of super advanced difficult to grasps algorithms with easily impressionable life scientists who believe in "teh powerz of computah" result in some incredible bullshit, that nonetheless is believed and sometimes survives as a hype for 5 years or so. We're even reviving phrenology under a different name!
Did things like that also happen to you? What's the best way to prevent things like this from happening?
Sometimes there are goofy ideas that survive longer than they should. I don't have any good suggestion for what to do about this, beyond hoping that science will ultimately be self-correcting. There are always hungry and skeptical young people coming up, who want to make a name for themselves by straightening out errors in the earlier literature and getting down to the truth. As long as there are ambitious and talented people coming up, and as long as science and medicine stay exciting enough – as I think they always will – then ultimately I think nonsense will be beaten back and the truth will out. As I type those words, I realize what I just said may sound ridiculously optimistic, naïve, and idealistic and I do realize that sometimes it does takes a frustratingly long time for the truth to prevail. But at least in science and math our outlook is better in this respect than in just about any other branch of human endeavor.
Hey. I'm a math major undergraduate student and I'm thinking of applying for a PhD next year. What advice can you give me regarding this?
I guess it depends whether you are going for pure math or applied math. For pure math, you want to make sure that you’re strong in analysis, algebra, and topology. For applied math, it's important to be good at numerical work – make sure to practice your computational skills, and to take some good computer science courses, as well as courses in numerical analysis, statistics, and machine learning. Plus you will also need the usual fare for applied math: differential equations, both ordinary and partial, and comfort with some application area like physics, biology, or social sciences. And since applied math is part of mathematics, make sure that you also bulk up on the fundamentals of algebra and analysis. Differential geometry would help too. There's a lot to learn! Good luck!
thank you for doing this. I'd like to ask some questions related to your recent paper.
If I understood correctly, the variability of incubation periods can be described by the "Dispersion Factor", which - in your model - depends on the "Relative Fitness" r of the mutant/pathogen irrespective of heterogeneity. I was wondering if
- 1. is there any way of deriving an optimal (from the point of view of the pathogen) relative fitness r that maximizes the chances of survival, i.e., an r that minimizes the chances of either dying or triggering an immune response?
- 2. is there any way of relating dispersion factors with the dynamics of epidemics? I would (intuitively) think that a larger dispersion factor probably increases the chances of survival on an epidemic level.
- 3. (perhaps a bit unrelated) are there other areas of applied math where the same modelling is used?
And would you have any ideas on how the understanding of dynamics of incubation periods could help us improve public health?
(if I may ask yet another question: is the code for your simulations available online for students to play around?!)
Thank you again!
Off the top of my head, I'm not sure what to say about those first two questions. We would need to do research to answer either of them. They are fascinating directions to consider – trying to link our work more with public health and epidemiology. As far as whether our code is available, yes. You should be able to download it from eLife, where the paper has been published. Here's the open access link to the paper:
Another question: Do you plan on making another book in a similar vein to The Joy of x (with interesting stories about how math and the real world overlap) in the near future?
Yes! I'm glad you asked. For the past year I've been writing a new book, tentatively titled "The language of the universe". It's the story of calculus – how it was discovered, who discovered it and why, what they were hoping to achieve, and how calculus has helped make the world modern. I know that lots of students take calculus and learn how to do derivatives and integrals, but many have trouble seeing the point of it all. Unless a student goes on in science or engineering, the usual advanced placement course in calculus could seem like a pointless exercise. That course is usually taught in such a rush (because there is so much to cram in for the exam) that there isn't much time to learn about why the subject was created or what amazing things it has done for the world. In this new book, I'm trying to tell the stories and big ideas behind calculus. I want people to see that calculus is truly one of the greatest inventions and works of art that humanity has ever come up with. The book should be out sometime in 2019. Stay tuned!
Following you on Twitter has been such a great decision!
What's your favorite article in the past few years that asked a question people haven't thought of before?
I love this article about how the genetic code may have evolved:
Collective evolution and the genetic code, by
Kalin Vetsigian, Carl Woese, and Nigel Goldenfeld
In biology class I was taught – as many of us were – that the genetic code was a frozen accident. That there was no particular sense to it, that it was just an accident of history. This article makes a strong case that there is something special about the genetic code. I thought it was a really fascinating creative application of quantitative ideas – math, basically – to answer a deep question about a milestone in the history of life on earth.
What are your favorite popular math/science books? What do you think are the key ingredients in making such books appeal to a broad audience?
Here are a few that immediately come to mind, because I read or reread them recently:
Black Hole Blues
I wouldn't necessarily want to elevate them as being my favorites, because there are so many books that I love on math and science, but these are five great ones. A key ingredient in all of them is great clarity in their exposition of the science. But it's more than that – all of them have a voice. I love it when I can hear the author through the pages.
Hi Steven, loved your collaboration with 3blue1brown.
Question: Where do you think mathematical education should be focused to bridge the divide between seemingly complicated concepts and intuitive learning?
Eg. Trigonometry was a seemingly complicated topic for me, until I learnt it's duality with the coordinate points on a unit circle, which kind of shifted the focus of my reasoning from triangles (somewhat counter intuitive) to circles (wayyy more intuitive)
I hear you about trigonometry. I learned it on circles first, and triangles later. That did seem to help. But I'm not sure that would be true for everyone. What makes your question so difficult is that different people find certain things easier and other people find them harder. For me, and perhaps for you, the more visually the subject is presented, the easier more it makes sense to me. But I know others who prefer everything to be symbolic. They have trouble picturing things, but as soon as the variables and letters start flying across the page, they are happy. So I think the only thing that a teacher can do is try everything. Teach applications, history, visual thinking, symbolic thinking, intuition, rigorous argument, all of it. The hope is that you'll push somebody's button some of the time and keep everyone so stimulated that they will go on to teach themselves – that's for the best learning of all occurs.
Your mathematical applications to the social sciences sound amazing. Have you published a document in plain English (my highest level math was calculus) that people with a more basic understanding of math could read and understand? Are there any other publications in this field you would recommend?
You could start with this NY Times article:
It mentions an application to social balance and the run up to WWI.
I'd recommend the book Nexus, by Mark Buchanan, which gives a fun, readable treatment of how network theory relates to the social sciences (some of it deals with my work with Duncan Watts). Also try Watts's book Six Degrees.
Within the scope of network sciences, how do you think availability of large scale web-based datasets help us gain better and fundamental understanding of human networks? and what kinds of questions do you think will be answered using intersection of Data Science/Machine Learning and available theoretical knowledge about networks?
The advances in machine learning are staggering. Maybe I'm falling for the hype, but I was overwhelmed by the way that AlphaZero played chess against Stockfish in the recent match. Such genuine creativity (or at least the appearance of it) and the other successes of machine learning in various domains (e.g. language translation) do make it seem likely that machine learning will have a revolutionary impact on various parts of science, including network science. So yes, I suppose I do agree with the thrust of your question. We now have about 1000 well curated network data sets. Perhaps machine learning will tell us something fundamental about them that we didn't know before. Which raises the big question – how will the machines teach us what they are figuring out? I read a recent article in the New York Times Sunday Magazine, if I recall correctly, that discussed recent research on the question of how to get machines to explain to us what they are "thinking". If and when that problem is solved, it could be a new era for science.
If you could change one thing about math education, what would it be?
Hi Steven, thank you for your time. A couple of questions.
First, why did you decide to publish in eLife rather than a more established journal?
Second: I only skimmed the paper; the models and analysis looked interesting, but I didn't see any validation against real data. Do you have any plans of validating this model? If so, how do you plan to do so? It seems like it would be difficult to use these models to make predictions.
We submitted to Nature and Nature Communications first and in both cases the paper got rejected without review. Then we heard about eLife. We had a very positive experience publishing with them, and I am so glad that we submitted the paper there. The openness of the review process, and the collaboration among the reviewers and the editors to come up with a consensus review, were two features distinctive to that journal that we found very constructive. Plus the Journal is extremely high quality in its production and in the depth and innovativeness of the papers that are published there. We just thought it was a really good journal! Plus all of us (meaning all three authors on the paper) are fans of the open access nature of the journal. All our code is freely available as are all the data sets. This is the modern way to go.
In terms of validation against real data, the model predicts the shapes of the distributions in terms of a quantity called dispersion factor, which can be tested against real data (and we do this in the paper). Furthermore, the model makes certain predictions. I don't know if this kind of experiment can be done, but the model says that if identical doses of identical pathogens could be used to infect two equally healthy and otherwise identical organisms, they should still show remarkably different time courses of disease. This would not be the prediction of the traditional model. So it seems that it it should be possible to discriminate between the two contending models in an experiment like this, though I'm not exactly sure how it would be implemented or whether it's even possible at the current time.
How interested are you in pure mathematics? (Is it something that interests you at all, or are you mainly focused on applications?) Thanks!
I love learning and teaching pure mathematics but it doesn't come easily to me and I've never done research in it. Analysis and differential geometry are my favorites. Complex analysis is especially gorgeous. Honestly, abstract algebra has always been a sticking point for me. The book Visual Group Theory gives me hope that I might be able to find my way in someday…
Have you ever discovered or helped a misguided janitor with a genius-level IQ who secretly solved a difficult graduate-level math problem? When they were inevitably arrested, did you make a deal to get leniency for them if they get treatment from a therapist? What’s it like to win a Fields Medal?
I recently had the chance to rewatch "Good Will Hunting” and enjoyed it very much. But you are certainly right that the scenario depicted in the movie is implausible. Math has just become too specialized for something like that to occur at the highest level. Still, there are occasional shockers coming from someone relatively unknown, as in the recent breakthrough by Yitang Zhang -- https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty Note however that Zhang did have a great deal of mathematical training, and was certainly not a beginner like the Matt Damon character in Good Will Hunting.
Who is your favorite author and what are your favorite works of fiction?
edit: also, do you have any interesting neuroses? I've had germophobe math teachers in the past and math teachers that somehow broke a bone every time they went skiing.
For nonfiction, I love the essays by Lewis Thomas. It's hard to pick favorites, but if I had one, I think it would be him. His writing is so beautiful, so sunny, and always so surprising. What a great stylist he was! I wish I could've met him.
I might be the opposite of a germophobe. My wife is always scolding me for not washing fruit before eating it. Does that count?
I'm currently studying A-Level Maths and Further Maths (In England), and in my spare time I enjoy looking into more complex Maths from outside of the curriculum. How would you recommend learning Maths autodidactically?
The books by William Dunham are wonderful. They show real mathematics but explain the proofs and calculations clearly and include nice history.
Journey through Genius: The Great Theorems of Mathematics
The Calculus Gallery: Masterpieces from Newton to Lebesgue
Euler: The Master of Us All
Have you worked on anything that would be of interest to "normal" (not obsessed with math) high school girls?
Either of my NY Times series might interest you:
For instance, this about the birthday problem: https://opinionator.blogs.nytimes.com/2012/10/01/its-my-birthday-too-yeah/
or this about why your friends probably have more friends than you do (that's true for all of us, not just you!):
And I once wrote a paper about how to use math to plot the course of a love affair.
here's a NY times column about it: https://opinionator.blogs.nytimes.com/2009/05/26/guest-column-loves-me-loves-me-not-do-the-math/
It was meant for fun, but it has turned out to be a pretty good way to teach a subject in math called differential equations. Students seem to get a kick out of it, at least as compared to the usual way that differential equations are introduced!
Pineapple on pizza or no?
Sometimes. But no thanks to anchovies. Blecch.
Do you ever feel like Andy from the office when you talk about going to Cornell??? I literally couldn't read anything after I read that Cornell bit 😂😂😂😂😂 will read when less drink
Check out the clip starting at 16:50 in this Cornell video for some vintage Ed Helms:
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