# When bus schedules and particle physics collide

**Imagine taking a time-lapse photo of a clear sky at night. **The photograph will be filled with circular arcs of light that reflect the movement of stars across the sky as the Earth rotates around its axis. These paths have been the object of human wonderment since the time of ancient civilizations, and our precise mathematical knowledge of the position of the stars has allowed us to find our way home after long journeys across vast oceans.

Now imagine that you are on a distant planet, the rotation of which is not as regular or predictable. She can have a shorter or longer rotation cycle, depending on which of her suns is near. How do we construct mathematical tools to navigate the way back to such a planet? This question might sound like the beginning of a science fiction novel. But in reality, the same questions arise in many contexts, including our daily life on Earth.

If you’ve ever waited for a bus, you know one of these parameters. The exact arrival times of the buses (at least in my home town) are unpredictable. The published timetable may suggest that the bus you’re waiting for arrives in four minutes, but you have no way of knowing if it has already arrived and left until you get to the bus stop. In most cases, it will be delayed and you will not know how long it has been delayed. In my experience, often the next bus overtakes it and both buses arrive simultaneously at the stop where everyone is waiting.

In Cuernavaca, Mexico, a private bus system evolved to overcome some of these problems. A conductor pays a small fee to a spotter at each stop to find out when a previous bus has departed on the same route. If the departure time was recent, the driver waits. If the departure time was some time ago, the driver leaves. They adjust wait times and speeds so they don’t get too late or get too close to the next bus. This is an example of a system designed to maximize the number of passengers on each bus while minimizing the waiting time between buses.

The beauty of the math is that the description of Cuernavaca bus arrival times also works in other situations where there is attraction and repulsion between objects. Instead of buses, think of subatomic particles interacting with each other in a particle collider deep underground in Switzerland. Instead of particles, think of large primes and how they are spaced apart on the number line.

A prime number is a positive integer that is only divisible by itself and the number one – some small prime numbers are 2, 3, 5, 7, 11, 13, …, while some larger ones are 88,969, 200,023. Large prime numbers form the basis of the RSA algorithm, which is a widely used public-key cryptographic system for secure data transmission. There is no predictive algorithm for prime numbers. Pairs of successive primes differ by as little as 1 or as much as 1,113,106. The search for larger and larger primes continues. As of December 2020, the largest known prime number is 2.^{82,589,933 _ }1, a number that has 24,862,048 digits (in base 10).

Prime numbers are related to the zeros of a function called the Riemann zeta function. There is a famous unsolved problem in mathematics called the Riemann hypothesis, which states that the zeros of this zeta function must lie on some vertical line in the complex plane. (This is the subject of a Millennium Prize problem, the verified proof of which will earn you a $1 million prize.) Thus, the study of the large zeros of the zeta function is a very active field in mathematics. The spacing of these large zeros appears to follow a law that also describes how subatomic particles are repelled in a scattering experiment and how bus system statistics behave in Cuernavaca, Mexico.

The statistics of these spacings are worked out using mathematical models that I study. But there is a border that has not yet been crossed. In the settings I have described, the weather is continuously changing. What would happen if the clock we have changed time in discontinuous steps of variable length?

In our time-lapse photographs of stars in the sky, the weather was continually changing. If we took a picture once an hour, instead of leaving the camera aperture open for a long time, we would always get the same star information (we can smoothly interpolate between shots to describe paths borrowed by the stars, because we know how the Earth rotates). But the problem is more difficult if our camera is only allowed to take snapshots at varying times, sometimes as short as one minute apart or as long as two hours apart. The problem may not be solved on the alien planet whose rotation is unpredictable.

I work on problems from physics where the timestamps on the clock are not evenly spaced. For example, they change multiplicatively (i.e., a timestamp *you* changes to *qty*, or *q* is a non-zero number not equal to unity), or they change according to a more complicated pattern given by some functions called elliptical functions.

I am thrilled by the prospect of discovery, the possibility that the mathematics I develop will lead to new connections and new patterns that will describe how the world’s most elusive structures change over time.

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