Create a classification of manifolds, describe any graph and predict how the orbit of the planet changes - everyone
Topology is a science, it consists in studying the properties of an object that will not change when it is deformed.
“Sullivan repeatedly changed our view oftopology. He introduced new concepts, proved iconic theorems, answered old hypotheses, and formulated new problems that moved the field forward,” the text of the 2022 Abel Prize speech. His reward was $854,000.
Throughout his work, Sullivan moved from one area of mathematics to another and solved problems using a wide variety of tools, organizers say.
Since the Abel Prize was firstawarded in 2003, it has become a reward for a person's lifetime achievements, says Hans Munthe-Kaas, chairman of the committee from the University of Bergen. The last 24 Abel Prize winners are all famous mathematicians. “It’s great to be part of this distinguished list,” says Sullivan.
About the winner
Sullivan was born in Port Huron, Michigan, in1941 and grew up in Texas. He began his mathematical career in the 1960s. At that time, topology was developing rapidly, the researchers were trying to classify all existing manifolds.
A manifold in mathematics is a topologicala space that is linear on small scales is indistinguishable from a plane. But the global shape of a manifold does not always resemble the shape of a flat space, just as the surface of a sphere differs from the surface of a two-dimensional sheet. In this case, these objects are said to be topologically distinct.
What is the essence of the work of the laureate?
- Topology
In the mid-20th century, mathematicians realized that the topology of manifolds works in completely different ways. It all depends on the number of dimensions an object has, Sullivan says.
Studying manifolds with four or fewer dimensions was very similar to geometry, he says. The methods were also geometric: figures were cut into pieces and then put back together.
But for objects with a large number of dimensions - fiveand more - we managed to move much further. Sullivan, together with other researchers, created a classification of varieties. He broke down problems into ones that could also be solved using algebraic calculations, says Niels Baas, a mathematician at the Norwegian University of Science and Technology in Trondheim.
Sullivan stated that he was most proud ofresults he got in 1977. He was able to determine the most important properties of space using a tool called rational homotopy. This has become one of his most cited works and most widely used techniques.
Homotopy is a family of continuous mappings,whose preimage and image are topological spaces. That is, what we can do with an object through tension, compression and viscous displacement is when some of its neighborhood is displaced along with the point. This is necessary for the algorithmic description of processes, and building the logic of the architecture using a mathematical model.
- Dynamic systems
In the 1980s, Sullivan became interesteddynamic systems. These are systems that evolve over time—the orbits of planets or cyclical changes in a population. Here, too, Sullivan made contributions worthy of the Abel Prize, says Munthe-Kaas.
In particular, Sullivan proved the fact thatUsing computer modeling, it was discovered by the American mathematical physicist Mitchell Feigenbaum. Certain numbers, now called Feigenbaum constants, appear in many types of dynamical systems, and Sullivan's work explained why.
“It is one thing to know this from an experiment oncomputer, and it's quite another to know it as an exact mathematical theorem, ”says Sullivan. Other mathematicians have tried to prove this with existing tools, but have failed.
- Graph classification
Sullivan and his colleague Bill Parry were able to formulate for the first time the Parry-Sullivan invariant, a number that can be used to create classifications and partially one-dimensional dynamical systems.
They declared that from any graph one can makean incidence matrix where rows are vertices and columns are edges. As a result of the work, it turned out that the determinant of the difference in the unit and square incidence matrices cannot be topologically measured. The authors decided to use it to describe the structure of a graph.
- Turbulent behavior of liquids
In the decades that followed, Sullivan developed a passion forturbulent behavior of liquids, such as water flows. He said his dream is to discover patterns that could make such a move predictable on a large scale.
According to the chairman of the Abel Committee, Hans Munthe-Kaas, one of the important qualities of the laureate is the ability to find relationships and analogies between different and rather distant areas of mathematics.
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