Imagine taking a time-lapse photo of the clear sky at night. The picture is filled with arcs of light that reflect the movement of the stars in the sky as the Earth rotates around its axis. These paths have been objects of wonder to mankind since the days of ancient civilizations, and precise mathematical knowledge of the positions of the stars has allowed us to navigate our way home from long journeys across vast oceans. .
Now imagine you are on a distant planet. The planet’s rotation is not very regular or predictable. Depending on which sun is closer, the rotation period can be shorter or longer. How do we build the mathematical tools to navigate our way back to such a planet? This question might seem like the beginning of science fiction. But in reality, the same question arises in many situations, including our daily life on Earth.
Anyone who has waited for a bus will know one of these settings. The exact arrival time of the bus is unpredictable (at least in my hometown). A published timetable may indicate that the waiting bus will arrive four minutes after him, but there is no way of knowing if the bus has arrived and departed before it reaches the bus stop. . Most of the time I’m late and I don’t know how long I’m late. In my experience, often the next bus catches up and both buses arrive at the stop where everyone is waiting at the same time.
In Cuernavaca, Mexico, a dedicated bus system was developed to overcome some of these problems. Drivers pay observers a small fee at each stop to get information about when the previous bus on the same route left. If the departure time is close, the driver will wait. If the departure time is slightly earlier, the driver will depart. Adjust your wait time and speed so you don’t get too late or too close to the next bus. An example of a system designed to maximize the number of passengers on each bus while minimizing waiting times between buses.
The beauty of mathematics is that Cuernavaca’s description of bus arrival times works in other situations where there is attraction and repulsion between objects.
The beauty of mathematics is that Cuernavaca’s description of bus arrival times works in other situations where there is attraction and repulsion between objects. Instead of baths, think of subatomic particles interacting in particle colliders far underground in Switzerland. Instead of particles, think of large prime numbers and how they are separated from each other on the number line.
A prime number is a positive integer that is divisible only by itself and the number 1. The small primes are 2, 3, 5, 7, 11, 13, … and the large primes are 88,969, 200,023. Large prime numbers form the basis of the RSA algorithm. The RSA algorithm is a public-key cryptosystem widely used to ensure secure data transmission. There is no prime number prediction algorithm. Pairs of consecutive prime numbers differ by only 1 to 1,113,106. The search for larger and larger prime numbers continues. As of December 2020, the largest known prime number is 2.82,589,933_ 1, 24,862,048 digits (base 10).
Prime numbers are related to zero in a function called the Riemann zeta function. There is a famous unsolved problem in mathematics called the Riemann Hypothesis, which claims that the zero of this zeta function must lie on a particular vertical line in the complex plane. (This is the subject of the Millennium Prize problem, and a verified proof garners him a $1 million prize.) Therefore, the study of the big zero of the zeta function is a very active field in mathematics. The spacing of these large zeros was found to obey laws that explain how subatomic particles are repelled in scattering experiments and how the bus system statistics behave in Cuernavaca, Mexico.
Such interval statistics are computed through a mathematical model that I am working on. But there are still uncrossed frontiers. In the setup I described, the time changes continuously. What if the clocks available to us change the time in discrete steps of variable length?
What if the clocks available to us change the time in discrete steps of variable length?
Time changed continuously in the time-lapse photography of the stars in the sky. You can get the same information about the stars if you take a picture once an hour instead of leaving the camera aperture open for a long time (with smooth interpolation between snaps so that the stars are We can describe the path taken, because we know how the earth rotates). But the problem becomes more difficult if our cameras are only allowed to take snaps at varying times. This problem may not be solvable on alien planets with unpredictable rotations.
I’m working on a problem arising from the physics of non-constant clock timestamp intervals.For example, they vary multiplicatively (i.e. one timestamp t change to qtWhere q is a non-zero number not equal to 1), or through more complex patterns given by certain functions called elliptic functions.
I am excited about the potential for discovery, the possibility that the mathematics I develop will lead to new connections and new models that explain how the world’s most elusive structures change over time.
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