…on the Importance of Computing for Mathematicians
“Computo, ergo sum—” Jeffrey Lagarias
David Speyer: “Three of my students solved major parts of their thesis problems after I insisted they create code capable of checking the first nontrivial examples of their claims.”
John Stembridge: “I think the most valuable thing that computation can provide for a pure mathematician is the confidence that can come from seeing how special cases play out. If you are confident that what you are trying to prove is true, that can make all the difference.”
Sarah Koch: “I am constantly generating data and drawing pictures with various computer programs to better understand the spaces I encounter in my research. The programs I use range from Maple and Mathematica, to dynamical systems software. Many of my new ideas and questions are inspired by fascinating geometric phenomena that I discover in these computer pictures.”
Michael Zieve: “I use Magma every day in doing my research, both for doing experiments to see what might be true and then sometimes for producing computer-aided proofs of various results. To first approximation, Magma has all known algorithms for algebra, number theory, algebraic geometry, linear algebra and combinatorics already built into it, so that if someone somewhere in the world has figured out how to compute something, then you too can compute it easily using Magma. For instance, you can easily compute the irreducible characters of the automorphism group of a curve, etc.”
Jeff Lagarias: “If you can compute, you can do mathematical experiments for yourself, gather data, formulate hypotheses, and test them, and make discoveries. Each skill you have widens the possible opportunities you can seize. The sooner you learn it, the sooner you can use it.”
“My advisor Harold Stark succeeded in part because he could compute. For starters, his PhD thesis. Later he formulated conjectures, which made startling predictions, only confirmed by computer experiments, which caught people’s attention. Some of these examples, verified computationally to hundreds of decimal places, are still not proved 40 years later.”
Kartik Prasanna: “In certain areas of number theory, computing can be extremely valuable. I work on special values of L-functions and performing computations can provide incredible insights. Many of the major conjectures in the subject (eg. the Birch and Swinnerton-Dyer conjecture) were suggested by computer calculations. Moreover, I find that trying to implement something on a computer is a good way to check if you really understand it … since the slightest error in understanding will typically cause your computer program to output garbage. And it can be fun to play to around with explicit numerical examples! Not just do they solidify your understanding, but they make things concrete and remind you that in the end, deep theorems in number theory are often reflected in rather simple, concrete statements about numbers. I highly recommend that all my students be familiar and comfortable with SAGE and MAGMA. ”
Victoria Booth: “I use Matlab all the time in my research.”
Wei Ho: “The Birch and Swinnerton-Dyer Conjecture originally arose from computations about elliptic curves!”
Jenny Wilson: “In an ongoing project, one of my co-authors used MAGMA and the lrs Vertex Enumeration/Convex Hull package to perform computations in H^2 (GL_2(O)) for certain rings O.”
Alex Wright: “At least in my case, if I know a graduate student can program, this opens up new avenues,
possibly allowing them to work [with me] on a more original thesis problem.”
“The best example I know of amazing use of computing in math research is in this paper of Kontsevich and Zorich, quote: We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry.“
Ralf Spatzier: “Without a doubt, computers have revolutionized pure mathematics as they allow us to study complicated examples in ways that escape mere mortals. This allowed us to see phenomena and formulate conjectures beyond any previous dreams. Acquiring at least rudimentary skills in some advanced programming language will allow mathematicians to engage in such experiments.”
Wei Ho: “In my field, it’s very useful to be able to code. Magma and sage (open source) are the main tools. I have a joint paper that’s 100% computational (creating a giant database and computing invariants for the entries).
There is a conference every two years called the “Algorithmic Number Theory Symposium (ANTS)” — almost all of the talks / papers accepted are based on computations.
One of the large Simons collaboration grants right now is in “Arithmetic geometry, number theory, and computation.” And they have hired postdocs who code instead of teach!”
Jeff Lagarias: “The computer is a more dependable assistant than any human.”
Harm Derksen: “I have used programming in research and so have many of the graduate students I have worked with. Languages that we have used for math include maple, matlab, magma, gap, python, C++, Macaulay2. We have used them for computations, testing conjectures and applications.”
Alexander Barvinok: “Numbers do not lie.”