# Research

'...we must clear our minds from all causes which are liable to make people blind against the truth, e.g., inveterate custom, party-spirit, rivalry, being addicted to one's passions, the desire to gain influence, etc....it is our duty to proceed from what is near to the more distant, from what is known to that which is less known, to gather the traditions from those who have reported them, to correct them as much as possible, and to leave the rest as it is...'

From The Chronology of Ancient Nations by Al-Biruni

Research is something mathematicians will often say they would like to do all the time. I'm not so sure about this, but maybe it's fair to say most of them feel like they never have enough time for research. Finding a reasonable equilibrium balancing creative energy and the other duties of professional life, namely, teaching as well as contributing to the good management of a workplace and the wider community, is a critical skill most aspiring mathematicians have to learn. Creative work is eventually supposed to result in some kind of output, which for a mathematician usually refers to research papers.

This is a painful process. Coming up with ideas and discussing them with friends is great fun. Having the discipline to present the argument systematically with precision is much harder. Many mathematicians secretly would just like to chat and throw out inspiration in the manner of a religious leader, such as Pythagoras might have been.

Pythagoreans celebrate the sunrise (1869) by Fyodor Bronnikov

The mathematics storytelling club (2022) Artwork by SUNYOO

Nonetheless, the process of writing is important. It's related to my favoured answer to a question from a student 'Why do we need rigorous mathematical proofs? Isn't an intuitive idea the really important thing?' Well, it's true that the essential idea is important. But writing it down in clear and rigorous form seems to allow intuition to move to the next level.

I should admit right away that I've never been as productive as I should be. This is mostly laziness and an inclination to get distracted easily. At the bare minimum, I should have written roughly twice as many papers as I have to qualify as a respectable professional. If you are still young, I recommend the habit of writing frequently, even regularly. It doesn't always or even primarily have to be what's called original work. It can just be writing in your own words what you learn together with whatever personal insights come to your mind. I think this is an advice common among authors, even of fiction. If you wait to write until inspiration strikes, you never get anything done.

There is a common myth that a great genius only writes a few important papers. There are a few examples of this nature, but they are quite rare. (There is the related statement that even among all the papers written by a great genius, only a few are important, which is closer to correct.) Einstein, for example, wrote over 300 papers in his lifetime, including 12 in the Annals of Mathematics, which is sometimes thought of as the most important academic journal in pure mathematics. Moreover, in the single year 1905, the Annus Mirabilis that saw the publication of his 4 epoch-making papers on the photoelectric effect, Brownian motion, special relativity, and the equivalence of mass and energy, Einstein wrote 22 papers in total. For the most part, creativity and productivity go together.

Books are often regarded as less important in mathematics, in contrast to practice in the humanities. Nonetheless, there must be great satisfaction in getting out a complete opus in which one's ideas and insights can be presented in coherent and systematic form. It is definitely an excellent service to the community, regardless of how much it contributes to one's status. I am the editor-in-chief, together with my wonderful colleague Katrin Wendland, of a book series Springer Monographs in Mathematics. Do contact me with any ideas for your own writing.

Even though I'm not very productive, I have spent an inordinate amount of time preparing presentation slides, essentially each time I have had to give a research lecture. In the link, you will see a haphazard selection of these. Mathematicians have mixed feelings about this mode of presentation. There is some advantage to the preparation that goes into it, as the speaker is required to think through the important points with a degree of care. However, because it also tempts the speaker to cram in material that goes by in a rush during the actual delivery, some people vastly prefer to attend a talk using a chalkboard. (People in the humanities might find it strange that mathematicians and theoretical physicists are unable to give or attend a lecture without substantial use of visual aids.) When I'm in the audience, I somewhat sympathise with this view. When I'm lecturing, I prefer slides mostly because I can't stand my own handwriting.

On the topic of visual aids, maybe it's acceptable to include one that came out of my own research.

The cursed curve plotted using Sage Math by Sachi Hashimoto

Solved completely by Jennifer Balakrishnan, Netan Dogra, Stefan Mueller, Jan Tuitman, and Jan Vonk.

What they proved is that the only solutions in integers are

(1, 1, 1) (1, 1, 2) (0, 0,1) (−3, 3, 2) (1, 1, 0) (0, 2, 1) (−1, 1, 0)

except for the obvious possibility of scaling each triple. Here, I am allowing myself this appropriation in part because Netan and Jan (Vonk) were my DPhil students at Oxford from 2011 to 2015, while Jennifer was a postdoctoral fellow there during roughly the same period. Furthermore, their work grew out of a theory developed by me as described in another article in Quanta, written from far too generous a standpoint by Kevin Hartnett, exaggerating greatly the importance of my ideas. Nonetheless, this result and the article do convey a sense of the rather concrete goal of the whole theory, which is to devise a method for solving completely all such equations. I am still working on this, albeit slowly amid all my distractions.

What we have here is a prototypical example of a theory and a problem that belong to arithmetic geometry, an area of mathematics I am trying to elucidate by way of an example rather than general explanation.

Since quite a while ago, all physics has been mathematical physics. Galileo famously proclaimed in his book The Assayer (1623) that the book of the universe is written in the language of mathematics, and that those who do not know it are doomed to wander in a dark labyrinth. Still, someone who calls themselves a mathematical physicist has concerns that are somewhat different from most physicists.

Firstly, they will usually not have direct contact with experiments. So to the extent that they are physicists, they will be theoretical physicists. Secondly, they are typically much more concerned than even theoretical physicists with rigorous argumentation. What does this mean? A simplified version of the work of physicists might go like this. Some natural phenomena require explanation. The physicist proposes that there is a theory that explains it. This means that if you assume some simple rules about nature--the formulation of such rules is what require's genuine insight--a sequence of logical deductions from them will lead to the observed phenomena, that is, predict them. I should say that the term 'predict' is being used a bit casually here. This is because many such predictions are observed before the theory. Thus, they become consistency checks, obviously important. But it's most impressive when the theory predicts unknown phenomena that are subsequently observed. In any case, it's convenient to call both kinds of deductions 'predictions from theory.' For example, the prediction of black holes from the general theory of relativity is a genuine prediction of the latter kind. On the other hand, a classic example of prediction of known phenomena comes out of Newton's law of motion and his theory of gravity. Starting from them, one can use calculus to deduce that the orbits of the planets about the sun must be ellipses as had been observed by Kepler. Because such predictive power is so astounding, when this happens in a fairly reliable way, people say the theory is correct. It is at this stage that mathematical physics comes in. You see, unfortunately, in most of the cutting-edge theories of fundamental physics today, the logical steps from the theory to the predictions are carried out quite loosely. So some people feel nervous about concluding at this point that the theory is correct. To illustrate the distinction, I remark that the predictions of general relativity can, by and large, be deduced by way of rigorous argumentation.

On the other hand, the predictions of the main theory of particle physics, the so-called standard model, are based on fairly shaky, even if intuitively plausible, logic. So many mathematicians work very hard on the still unsolved problem of making this process of deduction rigorous. These are examples of what I would call 'hard-core' mathematical physicists. I find this activity admirable, but do not have the precise thinking power and stamina necessary to engage in it myself. Instead, I'm a breed of mathematical physicist that emerged en masse only in the late 1980s. We try to use ideas of physics to make interesting discoveries in mathematics. In other words, instead of the well-known process of using mathematics for physics, we use physics for mathematics. So sometimes, a person like me is called a 'physical mathematician.' There have been truly spectacular advances in the areas of algebraic geometry and topology that came out of physical mathematics. For the past 5 years or so, I have been advocating the view that physics is also useful for arithmetic geometry. However, arithmetic geometers are a tough crowd. It's not so easy to convince them this idea has substance.

When I was studying for my Ph.D., in fact, I worked on mathematical physics for the first three-and-half years or so, advised mostly by Igor Frenkel, a great expert on the algebraic structures used in string theory. Then, in the last one-and-half years, I switched to arithmetic geometry, supervised by Serge Lang and Barry Mazur. This didn't stop me from continuing to learn physics, mostly through conversations with my old friend John Baez, who is the most patient explainer of anything under the sun. (I really don't know why he puts up with me.) Then, as the years went by, I got help from people like Dan Freed, Jeff Harvey, Sergei Gukov, Philip Candelas, Xenia de la Ossa, Chris Beem, Tudor Dimofte, Theo Johnson-Freyd, Philsang Yoo, Kevin Costello, Brian Williams, Ingmar Saberi, Alex Schekochihin, Yang-Hui He, Bernd Schroers, Neil Turok,... even some experimental physicists like Alan Barr, Sangwook Cheong, and Philip Kim. Anything non-trivial I know about the subject is entirely thanks to them. Quite recently, I started publishing in actual physics journals. However, my goal is not to write papers. Truly, I would just like to understand physics.

Max Tegmark's Mathematical Universe

You may have heard about the cosmologist Max Tegmark's idea that the entire universe is a mathematical structure. I tend to agree, even though I don't know if he and I mean the same thing by this. For me, it's quite simple-minded. If you would like to explain reasonably precisely what the basic constituents of the universe are, for example, electrons or quarks, it's essentially impossible to describe them as anything but quite abstract mathematical structures, e.g., solutions to the Dirac equation of some sort that take values in a 'group representation'. Since the usual things we see and feel are built out of them, also according to mathematical rules for the assemblage (with the fancy name of tensor product), it's hard to know what else the world could be but a mathematical structure.