Optimal Transport: The Pursuit of Efficiency in Nature and Mathematics
The Origins of Optimal Transport
The words "optimal" and "optimize" derive from the Latin "optimus," or "best," as in "make the best of things." Alessio Figalli, a mathematician at the University of ETH Zurich, studies optimal transport: the most efficient allocation of starting points to end points. The scope of investigation is wide, including clouds, crystals, bubbles, and chatbots.
Dr. Figalli’s Insights on Math and Nature
Dr. Figalli, who was awarded the Fields Medal in 2018, likes math that is motivated by concrete problems found in nature. He also likes the discipline’s "sense of eternity," he said in a recent interview. "It is something that will be here forever." (Nothing is forever, he conceded, but math will be around for "long enough.") "I like the fact that if you prove a theorem, you prove it," he said. "There’s no ambiguity, it’s true or false. In a hundred years, you can rely on it, no matter what."
The Study of Optimal Transport
The study of optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found broader application solving logistical problems during the Napoleonic Era — for instance, identifying the most efficient way to build fortifications, in order to minimize the costs of transporting materials across Europe.
Applications of Optimal Transport
In 1975, the Russian mathematician Leonid Kantorovich shared the Nobel in economic science for refining a rigorous mathematical theory for the optimum allocation of resources. "He had an example with bakeries and coffee shops," Dr. Figalli said. The optimization goal in this case was to ensure that on a daily basis every bakery delivered all its croissants, and every coffee shop got all the croissants desired.
The Complexity of Optimal Transport
"It’s called a global wellness optimization problem in the sense that there is no competition between bakeries, no competition between coffee shops," he said. "It’s not like optimizing the utility of one player. It is optimizing the global utility of the population. And that’s why it’s so complex: because if one bakery or one coffee shop does something different, this will influence everyone else."
The Beauty of Math
For me, math is a creative process and a language to describe nature. The reason that math is the way it is is because humans realized that it was the right way to model the earth and what they were observing. What is fascinating is that it works so well.
Optimization and Creativity
Nature is naturally an optimizer. It has a minimal-energy principle — nature by itself. Then, of course, it gets more complex when other variables enter into the equation. It depends on what you are studying.
The Future of Optimal Transport
When I was applying optimal transport to meteorology, I was trying to understand the movement of clouds. It was a simplified model where some physical variables that may influence the movement of clouds were neglected. For example, you might ignore friction or wind.
The Curse of Dimensionality
The movement of water particles in clouds follows an optimal transport path. And here you are transporting billions of points, billions of water particles, to billions of points, so it’s a much bigger problem than 10 bakeries to 50 coffee shops. The numbers grow enormously. That’s why you need mathematics to study it.
Conclusion
Optimal transport is a fundamental concept in mathematics, with applications in a wide range of fields, from logistics to machine learning. The study of optimal transport has been motivated by concrete problems found in nature, and has led to significant advances in our understanding of the world around us. As Dr. Figalli notes, "Math is a creative process and a language to describe nature. The reason that math is the way it is is because humans realized that it was the right way to model the earth and what they were observing."
FAQs
Q: What is optimal transport?
A: Optimal transport is the study of the most efficient allocation of starting points to end points, with applications in a wide range of fields, from logistics to machine learning.
Q: Who introduced the concept of optimal transport?
A: Gaspard Monge, a French mathematician and politician, introduced the concept of optimal transport almost 250 years ago, motivated by problems in military engineering.
Q: What are some of the applications of optimal transport?
A: Optimal transport has applications in a wide range of fields, including logistics, machine learning, and meteorology.
Q: What is the curse of dimensionality?
A: The curse of dimensionality is the problem of dealing with high-dimensional data, where the number of dimensions can become exponentially large, making it difficult to analyze and interpret the data.
Q: How can we overcome the curse of dimensionality?
A: One way to overcome the curse of dimensionality is to use techniques such as dimensionality reduction, where we collapse some of the features to reduce the number of dimensions.
