A Mathematician Who Makes the Best of Things

The terms “optimal” and “optimize” are derived from the Latin word “optimus,” meaning “best,” as in “making the most of a situation.” Alessio Figalli, a mathematician at ETH Zurich, specializes in optimal transport, which involves finding the most efficient way to assign resources from one set of points to another. His research spans a wide range of areas, from the movement of clouds and the formation of crystals to the behavior of bubbles and the functionality of chatbots.

Dr. Figalli, who received the prestigious Fields Medal in 2018, is drawn to mathematics that addresses real-world problems. He also values the discipline’s “sense of eternity.” As he mentioned in a recent interview, “Mathematics will be around for a very long time.” He added that, while nothing is permanent, mathematics is reliable for a long period. “When you prove a theorem, it is final,” he explained. “There is no ambiguity—it’s either true or false. Even a century from now, it will still hold.”https://svtoday.org/

The concept of optimal transport was first introduced nearly 250 years ago by Gaspard Monge, a French mathematician and politician, who used it to solve problems in military engineering. Monge’s theories found further application in logistics during the Napoleonic Era, particularly in determining the most cost-effective way to build fortifications by optimizing the transportation of materials across Europe.

In 1975, Russian mathematician Leonid Kantorovich won the Nobel Prize in Economic Sciences for his work in developing a mathematical theory on the optimal allocation of resources. “He had a model involving bakeries and coffee shops,” said Dr. Figalli. In this case, the goal was to ensure that every bakery delivered the right amount of croissants to every coffee shop each day.

“This can be considered a global wellness optimization problem, where there is no competition between bakeries or coffee shops,” Figalli explained. “The goal isn’t to maximize the benefit of a single entity but rather to optimize the collective utility. The complexity lies in the fact that if one bakery or coffee shop deviates from the plan, it affects the entire system.”

The following conversation with Dr. Figalli, conducted at a New York City event organized by the Simons Laufer Mathematical Sciences Institute, has been edited for clarity.

What does the phrase “Math is…” mean to you?

For me, math is both a creative endeavor and a language to describe the world around us. It’s fascinating that mathematics is so effective because humans recognized it as the ideal way to model what we observe in nature.

Do you believe nature is constantly optimizing?

Nature is, in essence, an optimizer. It follows a minimal-energy principle. But the picture becomes more complex when other factors come into play. It really depends on the specific system you’re examining.

When I applied optimal transport to meteorology, for example, I tried to understand how clouds move. In that model, I simplified by neglecting certain variables that might influence cloud movement, such as friction or wind.

Cloud movement follows an optimal transport path. We’re talking about billions of particles being transported to billions of destinations, which makes it far more complex than the bakery-coffee shop problem. The scale grows exponentially, and that’s where mathematics comes in.

What drew you to optimal transport?

I was drawn to the practical applications and the inherent beauty of the mathematics, which stemmed from real-world problems.

There is an ongoing interaction between what mathematics can achieve and the real-world challenges people face. As mathematicians, we often push boundaries—working in infinite-dimensional spaces, which seems counterintuitive but allows us to develop technologies like cellphones and Google. Without mathematicians pushing beyond traditional limits, these innovations wouldn’t exist.

In society, people often recognize the importance of math only when they see its applications. However, its value goes beyond that. It’s the development of new theories and ideas through mathematics over time that sparks major societal changes. Ultimately, everything is rooted in math.

And often, the math comes first. It’s not that you start with a real-world problem and then find a solution. More often than not, the solution already exists, discovered by people who had the freedom to think big. Major breakthroughs typically happen when thinkers step outside of conventional boundaries.

Optimization has its limits—creativity can’t really be optimized, can it?

Exactly. Creativity is the opposite of optimization. If you’re conducting research in a specific field, your optimization strategy would keep you grounded in that area. But often, breakthroughs come from taking risks. Failure and frustration are essential. Big changes happen when you push yourself beyond your comfort zone, and that’s never an optimization process. Focusing too much on optimization can cause you to miss out on valuable opportunities. It’s important to recognize the role of risks and failures in innovation.

What are you currently working on?

At the moment, one challenge is applying optimal transport to machine learning.

From a theoretical perspective, machine learning is essentially an optimization problem: you have a system, and you want to optimize certain parameters or features so that the machine can perform specific tasks.

In image classification, for example, optimal transport measures the similarity between two images by comparing features like color or texture, aligning them by “transporting” these features. This improves the accuracy of the models, making them more resistant to distortions or changes.

These problems are high-dimensional, meaning that the features of objects can be represented as many parameters, with each feature corresponding to one dimension. If you have 50 features, you’re dealing with a 50-dimensional space.

The higher the number of dimensions, the more complex the optimal transport problem becomes. Solving it requires an enormous amount of time and data, which is why it’s often referred to as the “curse of dimensionality.” Recently, there has been an effort to develop new approaches to avoid this issue.

What does this new approach look like?

By reducing the number of features, I can simplify the optimal transport problem into a lower-dimensional space. For example, if three dimensions are too complex, I might project the points into a one-dimensional space. I then solve the optimal transport on this line, repeating the process for many lines. Once I have the results from these one-dimensional solutions, I try to reconstruct the original three-dimensional space by combining the information. This is not an easy task.

It’s somewhat akin to how a shadow of an object can provide information about the object itself, even though the shadow is only two-dimensional.

Another analogy is X-rays, which are two-dimensional images of your three-dimensional body. With enough X-ray images from different angles, you can reconstruct a three-dimensional model of your body.

Could this help with the limitations of AI?

If we can apply optimal transport techniques, it could make machine learning models more stable, robust, and reliable, reducing biases and improving safety. That’s the ultimate goal.

In the realm of pure and applied math, is real-world need driving new mathematical developments?

Exactly. While the engineering of machine learning has advanced significantly, we still don’t fully understand why it works. There’s a gap between what machine learning can achieve and what we can mathematically prove. It’s impressive, but there’s still a lot we don’t know. We want to improve it, and math is the key to bridging that gap.