The words “best” and “optimization” come from the Latin “Optimus” or “Best” as in “Make the Best of Things”. Alessio Figalli, a mathematician at ETH Zurich University, is studying the best transfer: the most effective distribution of start points in the end points. The field of research is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, who was awarded the Fields Medal in 2018, likes mathematics motivated by specific 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 the mathematics will be around for “quite some time”) “I like the fact that if you prove a theorem, you prove it,” he said. “There is no doubt, it is true or false. In a hundred years, you can rely on it, no matter what. ”
The study of the optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and political incentives for problems in military engineering. His ideas have found broader applications solving problems during the Napoleonic era – for example, identifying the most effective way to build fortifications in order to minimize the cost of transporting materials throughout Europe.
In 1975, Russian mathematician Leonid Kantorovich shared Nobel in economic science to refine a strict mathematical theory of optimal allocation of resources. “He had an example with bakeries and cafes,” said Dr. Figalli. The goal of optimizing in this case was to ensure that on a daily basis each bakery handed all the croissants and each cafeteria received all the croissants they want.
“It is called a problem of global wellness optimization in the sense that there is no competition between bakeries, no competition between cafes,” he said. “It’s not like optimizing a player’s utility. It optimizes the global utility of the population. And so it’s so complicated: because if a bakery or a café does something different, it will affect everyone else.”
The following conversation with Dr. Figalli – held at an event in New York organized by the Simons Laufer Mathematical Sciences and before and after interviews – has been concentrated and edited for clarity.
How would you complete the proposal “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 people realized that it was the right way to model the earth and what they observed. What is exciting is that it works so well.
Does nature always seek to optimize?
Nature is of course an optimizer. It has a principle of minimum energy-nature alone. Then, of course, it becomes more complicated when other variables enter the equation. It depends on what you study.
When I applied the optimal transfer to meteorology, I was trying to understand the movement of the clouds. It was a simplified model where some natural variables that may affect the motion of the clouds were neglected. For example, you may ignore the friction or wind.
The movement of water particles in the clouds follows an optimal transport route. And here you are transferring billions of points, billions of water particles, to billions, so it is a much larger problem than 10 bakeries in 50 cafes. The numbers grow too much. So you need math to study it.
What about the optimal transfer that has taken your interest?
I was very excited about the applications and the fact that the mathematics were very beautiful and came from very specific problems.
There is a constant exchange between what mathematics can do and what people need in the real world. As mathematicians, we can imagine. We like to increase the dimensions – we work in an infinite space that people always think is a little crazy. But it is what now allows us to use mobile phones and google and all the modern technology we have. Everything would not exist if the mathematicians were not crazy enough to get out of the formal boundaries of the mind, where we only live in three dimensions. The reality is much more than that.
In society, the danger is always that people only see mathematics as important when they see the connection to applications. But it is important beyond that – the thought, the developments of a new theory that came through mathematics over time that led to major changes in society. Everything is math.
And often mathematics came first. It’s not that you wake 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 great. The other way around this can work, but in a more limited way, problem than the problem. Great changes usually occur because of free thinking.
Optimization has its limits. Creativity cannot really be optimized.
Yes, creativity is the opposite. Suppose you are doing very good research in one area. The optimization shape will stay there. But it is best to take risks. Failure and frustration is the key. Great discoveries, big changes, always come because at some point you take yourself out of your comfort zone, and this will never be a optimization process. Optimizing everyone leads to missing opportunities sometimes. I think it’s important to really appreciate and be careful with what you optimize.
What are you working on these days?
A challenge is to use the optimal transfer to mechanical learning.
From a theoretical point of view, mechanical learning is just a optimization problem where you have a system and want to optimize certain parameters or features so that the machine can do a certain number of tasks.
To sort the images, the optimal metaphor measures the way the two images are by comparing features such as colors or textures and placement of these features in alignment – by conveying them – between the two images. This technique helps to improve accuracy, making models more powerful in changes or distortions.
These are very high -dimensional phenomena. You try to understand the objects that have many features, many parameters and each feature corresponds to one dimension. So if you have 50 features, you are in a 50 -dimensional space.
The higher the dimension where the object lives, the more complicated the optimal transport problem – it takes a lot of time, too many data to solve the problem and you will never be able to. This is called a curse of dimensions. Recently people are trying to consider ways to avoid the curse of dimensions. One idea is to develop a new type of optimal transport.
What is its essence?
With the collapse of certain features, I reduce my optimal transfer to a lower -dimensional space. Let’s say three dimensions are too big for me and I want to make it a one -dimensional problem. I take a few points in my 3D space and project them on a line. Solve the optimal transfer to the line, I calculate what I have to do and repeat this for many lines. Then, using these results in the dimension one, I try to rebuild the original 3-D space from a kind of gluing together. It is not an obvious process.
It sounds like the shadow of an object-a two-dimensional, square shade that provides some information about the 3D cube that throws the shade.
It’s like shadows. Another example is X-rays, which are 2-D images of your 3-D body. But if you do x -rays in several directions, you can essentially compose the images and rebuild your body.
Would the conquest of the dimensional curse help with AI’s weaknesses and restrictions?
If we use some optimal transport techniques, perhaps this could make some of these optimization problems in mechanical learning more powerful, more stable, more reliable, less biased, safer. This is the beginning of the post -start.
And, in the interaction of pure and applied mathematics, the practice here is the real need to motivate the new mathematics?
Exactly. The mechanics of mechanical learning is far ahead. But we don’t know why it works. There are few theorems. Comparing what it can achieve with what we can prove, there is a huge gap. It is impressive, but mathematically it is very difficult to explain why. So we can’t trust it enough. We want to improve it in many directions and we want to help mathematics.
