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I first met Dennis Sullivan during a conference in Liverpool in 1970, and then we worked on similar issues. He came to Liverpool as the new hippies from Berkeley and impressed everyone with his intuition and engineering arguments.
A few years later we were at the same time in Paris at the IHES.
Pioneer The oldest mathematician in Paris at the time was Henri Cartin who took a course in geometry without a single digit – everything had to be official. It was the opposite of Sullivan’s sentiments, once he stated that any mathematical proof should consist only of numbers. As a result, one of his best articles was rejected in a famous magazine. Space plays an important role in our thinking and our work.
What is the room? We have lines, surfaces, and three-dimensional space in which we live. In mathematics, a space of random dimensions is studied. To do this, one needs to break it down into simpler parts. For example, divide a surface into triangles, similarly in higher dimensions. This is called triangulation, and it is one area in which Sullivan has made fundamental contributions.
The core of Sullivan’s work is to understand how space is constructed and recognizable.
In mathematics meet Often one is non-geometric objects, but one imposes on them a geometric structure so that you can use geometric intuition for further study. This has been a success in many applications.
At IHES in Paris was also the great French mathematician René Thom who also had a wonderful geometric intuition about space. He and Sullivan agreed on their spatial ideas despite their outward differences: an American inspired by the hippie Sullivan, Tom resembles a French village priest on his way to church.
At this time, there was another great mathematician at IHES who revolutionized the concept of space from an algebraic point of view, Alexander Grothendieck. He was at that time about to quit math when he felt he could no longer work with math 16 hours a day.
This was it its minimum requirements. I remember he stopped at IHES and then put on a black monk-like suit and sort of a shepherd’s staff. In this environment the concept of space flourished!
Diversity is a type of space that is very important in both mathematics and physics. It’s made up of smaller parts that are glued together, and each of these parts is just like our regular room.
Diversities are there In all dimensions, its classification was a major problem. It went well in dimensions 1, 2, 5 and larger, but in dimensions 3 and 4 it got tricky. It’s packed here! Dimension 4 is still a huge problem, but through the deep insight of many mathematicians, dimension 3.
It was the last decisive contribution of the Russian Grigory Perelman. For this work he was awarded the Outstanding High Fields Medal and the Millennium Prize (one million dollars), but refused to receive both!
Alexander Grothendik also rejected the Swedish Crafford Prize on the grounds that flag prices are corrupt. No one has yet rejected the Abel Prize!
In mathematics As mentioned, we use our geometric spatial intuition to illuminate situations and problems in new ways. Where does our spatial intuition come from? Maybe it comes from the brain. but how?
In the brain there are millions of neurons that turn on and off. How can the concept of space arise in this chaos? May-Britt and Edvard Moser discovered a new type of cell in the brains of mice called “reticular cells” that form a kind of spatial navigation system. Called special combinations of these modules, they collectively describe the movement of a rat on a “hoop” – an area that corresponds to the surface of a donut or “donut”.
It’s amazing That such a low-dimensional, simple space comes from the chaotic activity pattern of individual cells. This is a completely new result based on experiments in Moser’s lab and published in Nature in January (2022). The point here is that geometric (topological) spaces are produced in the brain! The same brain that produced the abstract concept of space. So here we are biting our tails the natural way!
I’m going to assume that’s just the tip of the iceberg. More and better brain data will only require new geometric structures and spaces. They need to be identified and marked in order to study how they play together. Then we can come close to understanding cognitive processes as well.
Rooms and numbers They are basic concepts in our daily life and our thinking. To distinguish room types, we often count their gaps in each dimension. It has proven very useful in a new field called topological data analysis where one looks for a structure with complex amounts of data in the same way that tone appears in brain data.
It is likely that the concept of space will develop strongly in the future, and it will definitely become more and more abstract. Then it is important to maintain the engineering intuition that this year’s Abel Prize winner is the foundations of the first order.
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