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Multiple Supercomputers Used to Advance Experimental Statistics Theory

Comet and Bridges-2 Put Mirzakhani Theorem to the Test

Published January 31, 2022

The difference between a non-separating loop (shown in blue) and a separating loop (shown in red).  Credit: Mark Bell

By Kimberly Mann Bruch and Cynthia Dillon, SDSC External Relations

Excelling in mathematics is not for everyone and even those who achieve it make mistakes along the way. For example, the late renowned mathematician Maryam Mirzakhani, the first woman and the first Iranian to receive a Fields Medal (the “Nobel Prize of mathematics”), made an error in applying the theorem that now bears her name.

Mirzakhani’s theorem states that the ratio of the frequency of two types of loops on a surface is always a rational number that is never 0 or 1 but always somewhere in between, Her 2008 breakthrough work linked two major areas of mathematics: hyperbolic geometry and Moduli spaces. In addition to proving this result, she also gave a procedure for calculating this probability.

To understand this concept, imagine drawing a loop on a surface and cutting along the loop. If you did that repeatedly then most of the time you would still end up with a single connected surface. But every now and then you might draw a separating loop that resulted in two disconnected surfaces. Mirzakhani illustrated the process for the case of a “genus two surface”. A torus, colloquially a donut, is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. A genus two surface is formed by joining two tori. Her prediction was that the probability that a random loop is separating is 1/7.

A decade later, researchers Vincent Delecroix, Université de Bordeaux I; Elise Goujard, Institut de Mathématiques de Bordeaux; Peter Zograf, St. Petersburg State University and Anton Zorich, Institute of Mathematics of Jussieu University Sorbonne Paris (DGZZ), developed a different approach to the problem. While their results matched Mirzakhani's in almost every way, when they applied their technique to the genus two surface example, they predicted that you would actually draw a separating loop only 1/49th of the time.

So, Mark Bell, a visiting fellow at the Warwick Mathematics Institute in the UK and formerly a J. L. Doob Research Assistant Professor at the University of Illinois, used the San Diego Supercomputer Center’s Comet, located at UC San Diego, and Bridges-2 at the Pittsburgh Supercomputing Center, to conduct complex calculations that helped to resolve this discrepancy between Mirzakhani’s Theorem and the DGZZ research. The results of Bell’s study are published in Experimental Mathematics and they confirm and correct a numerical error in Mirzakhani’s widely touted mathematics example.

According to Bell, after using the supercomputers to produce millions of loops, his team ended up with a sample where 1/49.07 of the loops were separating, with very small error bars. This boosted confidence in the DGZZ result and renewed focus in re-examining Mirzakhani's work, which ultimately led to the identification of two small errors in her original example calculation. By making these corrections, Bell was able to show that Mirzakhani’s example also predicted 1/49.

“I was surprised at just how accurate the computational simulation actually ended up being. As the computers worked, data streamed in and so we started with a very crude prediction for the probability, which became more precise as more data came in,” said Bell. “This initial guess (which had large error bars) started at an ugly fraction, like 715/35084, and as the error bars shrunk with more data converged on a beautifully simple 1/49.”

Why It’s Important and How Supercomputers Helped

“Thanks to the National Science Foundation (NSF) Extreme Science and Engineering Discovery Environment (XSEDE) allocations on Comet and Bridges-2, we were able to obtain a large enough sample of statistically significant data to allow us to confirm the correct fraction of curves on a genus two surface that separates,” said Bell during the study. “This further gave us what we needed to confirm and correct a numerical mistake in the work of Mirzakhani.”

This research was funded by NSF (grant no. ACI-1548562). Computational allocations were funded by XSEDE (TG-DMS180008).

It should be recognized that Maryam Mirzakhani lost her battle to breast cancer in 2017. Her brilliant mathematics work at Stanford University and beyond is beautifully described in a Forbes article written by Paul Halpern. Each year, female mathematicians are celebrated in Mirzakhani’s honor via the international event known as May 12. The event title commemorates Mirzakhani’s birthdate: May 12, 1977.

About SDSC

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About PSC

The Pittsburgh Supercomputing Center (PSC) is a joint computational research center of Carnegie Mellon University and the University of Pittsburgh. Established in 1986, PSC is supported by several federal agencies, the Commonwealth of Pennsylvania and private industry and is a leading partner in XSEDE, the National Science Foundation cyber infrastructure program. PSC provides university, government and industrial researchers with access to several of the most powerful systems for high-performance computing, communications and data storage available to scientists and engineers nationwide for unclassified research. PSC advances the state of the art in high-performance computing, communications and data analytics and offers a flexible environment for solving the largest and most challenging problems in computational science.