What is the best, according to the Italian mathematician Alessio Figalli?

The “optimal” and “optimize words are derived from the Latin” optimus “or” better “, as in” take full advantage of things. ” Alessio Figalli, a mathematician from eth Zurich University, studies optimal transport: the most efficient assignment of the starting points to the end points. The scope of the investigation is wide, including clouds, crystals, bubbles and chatbots.

Dr. Figalli, who received the Fields Medal in 2018, likes mathematics that are motivated by concrete problems found in nature. He also likes the “sense of eternity” of discipline, he said in a recent interview. “It's something that will be here forever.” (Nothing is forever, he admitted, but mathematics will be “enough”). “I like the fact that if you show a theorem, you show it,” he said. “There is no ambiguity, it is true or false. In a hundred years, you can trust it, whatever happens. “

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. Their ideas found broader applications that solved logistics problems during the Napoleonic era, for example, identifying the most efficient way of building fortifications, to minimize material transport costs in Europe.

In 1975, the Russian mathematician Leonid Kantorovich shared the Nobel in Economic Sciences to refine a rigorous mathematical theory for the optimal allocation of resources. “I had an example with bakeries and coffee shops,” said Dr. Figalli. The objective of optimization in this case was to make sure that each bakery delivered all its crossings, and each cafeteria had all the desired crossings.

“It is called a global well -being optimization problem in the sense that there is no competition between bakeries, there is no competition between coffee shops,” he said. “It's not like optimizing the usefulness of a player. It is optimizing the global utility of the population. And that is why it is so complex: because if a bakery or a cafeteria does something different, this will influence everyone else. “

The following conversation with Dr. Figalli, held in an event in New York City, organized by the Institute of Mathematical Sciences Simons Laufer and in interviews before and after, has been condensed and edited by clarity.

How would you finish prayer “Mathematics are …”? What is mathematics?

For me, mathematics is a creative process and a language to describe nature. The reason why mathematics is the way it is because humans realized that it was the correct way to model the earth and what they were observing. The fascinating thing is that it works very well.

Is nature always looking to optimize?

Nature is naturally an optimizer. It has a minimal energy principle, nature by itself. Then, of course, it becomes more complex when other variables enter the equation. It depends on what you are studying.

When he applied optimal transport to weather, he was trying to understand the movement of clouds. It was a simplified model where some physical variables were neglected that can influence the movement of clouds. For example, you can ignore friction or wind.

The movement of water particles in the clouds follows an optimal transport route. And here it is transporting billions of points, billions of water particles, to billions of points, so it is a problem much greater than 10 bakeries to 50 coffee shops. The numbers grow greatly. That is why you need mathematics to study it.

What about optimal transport captured your interest?

I was more excited about applications and the fact that mathematics was very beautiful and came from very specific problems.

There is a constant exchange between what mathematics can do and what people require in the real world. As mathematicians, we can fantasize. We like to increase dimensions: we work in an infinite dimensional space, that people always think they are a little crazy. But it is what allows us to use cell and google and all the modern technology we have. Everything would not exist if mathematicians were not crazy enough to get out of the standard limits of the mind, where we only live in three dimensions. Reality is much more than that.

In society, risk is always that people simply see mathematics as important when they see the connection with applications. But it is important beyond that: thought, the developments of a new theory that occurred through mathematics with the time that led to great changes in society. Everything is mathematics.

And often mathematics were first. It is not that he wakes up with an applied question and find the answer. Usually, the answer was already there, but it was there because people had the time and the freedom to think big. On the contrary it can work, but in a more limited way, the problem by problem. The big changes generally happen due to free thinking.

Optimization has its limits. Creativity cannot really be optimized.

Yes, creativity is the opposite. Suppose you are doing a very good investigation in an area; Your optimization scheme would make you stay there. But it is better to take risks. Failure and frustration are key. Great advances, great changes, always come because at some point you get from your comfort zone, and this will never be an optimization process. The optimization of everything results in missing opportunities sometimes. I think it is important to really value and be careful with what optimizes.

What are you working on these days?

A challenge is to use optimal transport in automatic learning.

From a theoretical point of view, automatic learning is only an optimization problem in which it has a system, and wishes to optimize some parameters or characteristics, so that the machine makes a certain number of tasks.

To classify the images, the optimal transport measures how similar are two images comparing characteristics such as colors or textures and putting these characteristics in alignment, transporting them, between the two images. This technique helps improve precision, making the models more robust to changes or distortions.

These are high -dimension phenomena. It is trying to understand objects that have many characteristics, many parameters and each characteristic corresponds to a dimension. So, if you have 50 features, you are in a 50 dimensional space.

The higher the dimension where the object lives, the more complex the optimal transport problem: it requires too much time, too many data to solve the problem, and you can never do it. This is called the curse of dimensionality. Recently, people have been trying to find ways to avoid the curse of dimensionality. An idea is to develop a new type of optimal transport.

What is the essence?

When collapsing some characteristics, I reduce my optimal transport to a lower dimensional space. Let's say three dimensions is too big for me and I want it to be a one -dimensional problem. I take some points in my three -dimensional space and the projects in a line. I solve the optimal transport on the line, I calculate what I must do, and I repeat this for many, many lines. Then, using these results in Dimension One, I try to rebuild the original 3-D space through a kind of attached together. It is not an obvious process.

It sounds like the shadow of an object: a two -dimensional and square shadow provides information about the three -dimensional cube that throws the shadow.

It's like shadows. Another example is the radiographs, which are 2-D images of your body 3-D. But if it makes radiographs in sufficient addresses, you can essentially get the images and rebuild your body.

To conquer the curse of dimensionality would help with the deficiencies and limitations of ai?

If we use some optimal transport techniques, perhaps this could make some of these optimization problems in automatic learning more robust, more stable, more reliable, less biased, safer. That is the goal principle.

And, in the interaction of pure and applied mathematics, here the practical need of the real world is motivating new mathematics?

Exactly. Automatic learning engineering is far ahead. But we don't know why it works. There are few theorems; Comparing what you can achieve with what we can try, there is a great gap. It is impressive, but mathematically it is still very difficult to explain why. Then we cannot trust enough. We want to do better in many directions, and we want mathematics to help.