Applied mathematics: The left image is an example of the polar data gap (dark blue disc) on August 30, 2007, with shading outside the disc indicating concentration. The middle and right images show the data fill presented here; the color shading at right is similar to that used by the National Snow and Ice Data Center (http://nsidc.org).

Applied mathematicians in the service of pressing global issues

Pure mathematicians, the truths of mathematics, and their “mystery”

When shaping the profile of a mathematician, we get a kind of atypical scientist: one who is concerned more with questions than answers, more with puzzles and conjectures than simple problems, who is more creative than applicative, and usually is more or less inclined toward philosophy. This individual is one whose work is driven by criteria of beauty, symmetry, and aesthetics. (These words in fact characterise pure mathematics itself.) Despite those aesthetic criteria, mathematicians, like scientists, of course search for adequacy, rigour, and truth, but their truth is different from the truths of science. The mathematician has more confidence in mathematical truths than scientists have in their truths or in mathematical truths applied in science.

Golden ratio in the Whirlpool Galaxy (M51). Credit: NASA and The Hubble Heritage Team (STScI/AURA)

Mathematicians enjoy mathematical “play.” They can lay down a set of axioms (or modify an existing one) and create a new mathematical theory just to satisfy their intellectual need to play with mathematics. Whether the new theory finds applications in mathematics or sciences seems not to be the concern of its creator. However, the history of science has proved that such theories do eventually find their application, perhaps after decades, and any set of axioms describes a seen or unseen part of nature. This “mysterious” aspect of applied mathematics is known in the philosophy of science as ‘Wigner’s puzzle’.

Applied mathematics is unifying sciences

Mathematics has become the foundation of modern physics, and nowadays it is hard to draw a border between pure and applied mathematics, as well as between physics and mathematical physics. Mathematics is currently being applied in domains where we never imagined that it could be. Particularly exciting are developments in the biological sciences, where cell life, tumour growth, body fluid dynamics, neurophysiology, and many other processes and phenomena are described and investigated through precise mathematical models.

Effects of dimension on the left and right pathways.
Effects of dimension on the left and right pathways. Credit: Mitchel J. Colebank on SIAM News

The border is also becoming more and more diffuse between domains of physics and biology, with the “help” of the underlying mathematics. Unavoidably, pure mathematicians have become applied mathematicians. Despite their “lone-wolf” abstract-inclined profile, they are capable whenever needed of making this switch from the pure math on paper to the applied math of reality and science.

Space missions made possible by mathematicians

When astronauts landed on the Moon with the Apollo 11 mission, mathematicians were present in the Houston crew to assist NASA scientists and the astronauts. In fact, they contributed to all space missions from the stage of preparation, calculation and prediction of trajectories, and design of the aerospace vehicles to monitoring the data processing systems and assisting with any engineering problems. They did not do this job under stress-free conditions, but under the pressure of time and circumstances. They were on standby, ready to resolve within minutes any condition that could result in the unwelcome “Houston, we have a problem….”

The Apollo 11 lunar module, the Moon, and the Earth
The Apollo 11 lunar module, the Moon, and the Earth. Credit: NASA

Of particular note in that endeavour is the now-famous NASA mathematician Katherine Johnson, (turning 100 in 2018), who brought a fundamental contribution to space missions. Ms. Johnson manually computed trajectories, launch windows, and emergency return paths for manned space missions, including those of astronauts Alan Shepard, the first American in space, and John Glenn, the first American in orbit. Her contributions were essential to the beginning of the Space Shuttle Program and all subsequent Apollo missions. Her greatest contribution to space exploration – according to her own testimony – are the calculations that helped sync Project Apollo’s Lunar Lander with the moon-orbiting Command and Service Module (Lee Shetterly, 2018).

NASA research mathematician Katherine Johnson is photographed at her desk at NASA Langley Research Center with a globe, or "Celestial Training Device," in 1962.
NASA research mathematician Katherine Johnson is photographed at her desk at NASA Langley Research Center with a globe, or “Celestial Training Device,” in 1962. Credit: NASA

In point of fact, the Moon landing was first realised as a viable project through the crucial contribution of another mathematician, Richard Arenstorf. In 1963, Arenstorf solved a special case of the three-body problem, giving the mission the necessary information as the initial conditions for the celestial mechanics to be applied successfully. The conjecture was over 300 years old. The three bodies in question were the Earth, the Moon and the spacecraft in motion. NASA’s problem was to determine when to launch and how much thrust would be required to reach the Moon and then return to Earth. To arrive at these calculations, it was necessary to take into account the influence of the gravitational fields of the Earth and the Moon on the spacecraft (Vanderbilt University, 2009). Arenstorf reduced the eight interactions between the three objects to four and proved the existence of a family of near rotating Keplerian ellipses with the appropriate parameters for being a stable orbit. Thus, Arenstorf found an eight-shaped stable orbit for a spacecraft orbiting between the Earth and Moon, (known as ‘Arenstorf Periodic Orbits’).

Trajectory transfer from Earth-Moon cycler to Mars departure asymptote, shown in Earth-Moon rotating frames.
Trajectory transfer from Earth-Moon cycler to Mars departure asymptote, shown in Earth-Moon rotating frames. Credit NASA

Applied mathematicians and global issues

But civic engagement of applied mathematicians manifests itself not only in outer space. Right here on Earth, humanity has faced and currently faces pressing global issues which threaten us, and science has moved its focus toward solving them. However, science can’t do this without mathematics. Given our initial description of mathematicians, we might expect little involvement, but this is not the case. Applied mathematicians are present in the front line of any battle concerning issues of Earth or human beings, not only to solve problems, but also to create powerful models and brand new mathematical theories specifically to address those global issues. And most importantly, they do it from duty rather than pleasure.

One example is global warming, currently the most pressing environmental issue. In 2017, a climate scientist and a mathematician collaborated to develop a powerful partial differential equations-based model with tuned stochastic spatial heterogeneity to estimate the sea ice concentration in the polar regions unobserved by satellites (Strong & Golden, 2017). The sea ice has been monitored for decades by satellites that detect microwave emission through clouds. However, the orbit inclination and instrument swath of the passive microwave satellites leave a “polar data gap” around the North Pole where sea ice is not observed. The proposed mathematical model makes objective estimation on the unobserved concentration.

The left image is an example of the polar data gap (dark blue disc) on August 30, 2007, with shading outside the disc indicating concentration. The middle and right images show the data fill presented here; the color shading at right is similar to that used by the National Snow and Ice Data Center (http://nsidc.org).
The left image is an example of the polar data gap (dark blue disc) on August 30, 2007, with shading outside the disc indicating concentration. The middle and right images show the data fill presented here; the colour shading at right is similar to that used by the National Snow and Ice Data Center (http://nsidc.org). Credit: Courteney Strong with authorisation from Siam News

The result is based on their past research on modelling atmospheric influence on Arctic marginal zone position and width in the Atlantic sector. This model allows approximation of the sea ice concentrations within the marginal ice zone where the concentration transitions from dense pack to open ocean. There is ongoing interdisciplinary research to improve their model and to develop other methods to fill the critical data gap that will impact our understanding of Earth’s changing climate.

Applied mathematics, evolution, and philosophy

The above example is neither singular nor isolated. Whether the bad news concerns a virus spreading, an asteroid collision, or risks to food security, applied mathematicians are ready to analyse the problem and develop mathematical models to represent, estimate, optimise, measure, assess risks, and provide solutions. And they fully take or accept these roles and challenges.

The complexity of such projects and the wide extent of mathematics from several of its branches (literally hard work) support the argument that mathematicians are perfectly capable of adapting to the priorities of the planet and using effectively the tools and truths of mathematics in the battle with global issues. They were here when men landed on the Moon, and they will be here when the time comes to leave this planet to colonise others.

Blackboard containing bits of the proof of the Four Vertex Theorem
Credit: CC BY 2.0 by Clayton Shonkwiler

Mathematicians don’t simply create their theories as pure mathematicians, but working as applied mathematicians, they constantly improve their theories to adequately describe the current reality and to follow their applicability to the ultimate goals. This transition would not be possible if we didn’t trust the truths of applied mathematics – that is, accepting Wigner’s “unreasonable effectiveness of mathematics” without any explanation. We do more than accept it: we put our very lives in the “hands” of those truths. Is this an anthropocentric-evolutionary feature of applying mathematics? If so, it is a wonderful picture of humans linking effectively the abstract and the concrete much more simply than philosophers will ever do.

References

Arenstorf, R. F. (1963). Periodic Solutions of the Restricted Three Body Problem Representing Analytic Continuations of Keplerian Elliptic Motions. NASA Technical Note D-1859.
Lee Shetterly, M. (2018). Katherine Johnson Biography. NASA – From Hidden to Modern Figures.
Strong, C., & Golden, K. M. (2017). Filling the Sea Ice Data Gap with Harmonic functions: A Mathematical Model for Sea Ice Concentration Field I Regions Unobserved by Satellites. SIAM News, 50(3), 1, 3.
Vanderbilt University. (2009). VU Mathematician Played Key Role in Moon Landing. Department of Mathematics website.
Wigner, E. P. The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications on Pure and Applied Mathematics, 1960, 13 (1): 1-14.

Featured image credit: Courteney Strong with authorisation from Siam News

Catalin Barboianu

Catalin Barboianu

Dr. Catalin Barboianu is mathematician and philosopher of science. He is science writer and the editor of a website dedicated to probability theory and its applications.
Catalin Barboianu

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