Dennis Sullivan has always been driven by mathematical insights that are general and beautiful and have the power to impress, like a piece of music. He switched his major from chemical engineering to mathematics after an encounter with one of these theorems in his second year of college.
One of the highest honors in mathematics has been awarded to Sullivan, a mathematician who has worked in topology and dynamical systems. Sullivan was recognized by the prize committee for his contribution to the study and classification of shapes, as well as his ability to decipher a wide range of mathematical problems by viewing them through a geometric lens.
Hans Munthe-Kaas, a mathematician at the University of Bergen, said that he has been one of the most influential characters in modern topology since the 1960s.
Sullivan has devoted a lot of his career to understanding manifolds, spaces that look flat but have a more complicated global structure. Both the surface of a sphere and the surface of a torus are two-dimensional manifolds.
Sullivan wanted to see these shapes in higher dimensions. He provided a complete classification of manifolds of a particular type in five or more dimensions, and made significant progress on a problem related to different ways of dividing manifolds into smaller triangular pieces. He helped to develop an area called surgery theory, which involves cutting and regluing pieces of something, in order to change it into something else.
Sullivan said that some of his most ground-breaking work involved cutting to the core of a space and looking at it from other perspectives.
Calculating a topological shape can be done by tying it with objects called homotopy groups that capture important properties of the shape. These groups might explain how loops can be arranged in the space. Sullivan helped develop a way to break the groups into boxes and separate them.
Sullivan invented a notion of division, where loops on the original manifolds were divided into two or three or more. He was able to deal with groups with fractions rather than numbers because he replaced the manifold with a more complicated object. He said that the introduction of fractions made life simpler and could be used to prove certain properties.
Shmuel Weinberger is a mathematician at the University of Chicago. Sullivan's innovations were compared to chemical or genetic analysis because they allowed you to break them up into fundamental pieces that look very different from the original object.
Sullivan said the part that stood out was not the technique itself. It allowed him to show that symmetry and other properties existed in this divided or local sense, which was crucial for proving certain statements that would have otherwise escaped him. He used this method to prove several important results.
It is a great moment in the history of mathematics.
Tienne Ghys.
In terms of both his ideas and his personality, Sullivan was like a fresh wind from the outside, according to a mathematician at the Norwegian University of Science and Technology.
Sullivan's efforts to capture the essence of a space didn't stop there. In the late 1970s, in parallel with the mathematician Daniel Quillen, he founded what is now called rational homotopy theory, a way to ignore certain information about the topology of a manifold so that the remaining information can be packaged in an easier, more tractable way.
The work was motivated by the fact that it can be difficult to compute a group. We don't understand what spheres are from a purely mathematical point of view, according to Ulrike Tillmann, a mathematician at the University of Oxford. Rational homotopy theory is geared toward simplification so that mathematicians can still say what they think about the spaces they are interested in and what those spaces are equivalent to.
Quillen and Sullivan used different models to understand their manifolds. Sullivan's approach was notable because it relied on tools from calculus to capture information.
The cole Normale Supérieure in Lyon says his invention is a wonderful piece of art.
Sullivan described it as an exercise and said that it became the key to answering other questions he had been wondering about.
Sullivan's work was almost entirely focused on manifolds from what he called a "outside perspective." There are no beauty marks on it. The place looks like a puddle of milk.
He wanted to find a local personality. He switched to the study of motion within or through a space.
There was renewed interest in studying the dynamics that could arise from the iteration of simple functions. A number that has a real and an imaginary part is called a complex number. Plug the answer back into the function and repeat. The result is a sequence of points. Sullivan proved a 60-year-old theory about how points in these systems eventually return to their starting place.
To demonstrate this and other results, he forged connections between two seemingly unrelated fields: the study of dynamical systems generated from iterating functions, and the study of certain groups of symmetries that act on a particular kind. Weinberger said that the Sullivan's dictionary changed both subjects.
Sullivan went back to the outside view of the early years of his career in the early 2000s. He and his wife, a mathematician, developed a new way to classify manifolds by studying loops and paths on their surfaces.
He is starting to combine his interests in the two. The equations that describe fluid flows are being investigated by him.
Munthe-Kaas said he wasn't sure if he saw the boundaries of different areas of mathematics the same as other people.
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