In the 19th century, mathematicians began to study the fundamental properties of shapes in greater depth. At this time, a new field emerged in geometry in which if one of the two objects could become the other by stretching or squeezing without any tearing or bonding, then the two objects could be considered identical.
In a field known as "topology," the letters a and b are the same, squares and circles are the same, and teacups and donuts are the same. Today, the value of topology in mathematics and other fields is immeasurable, with important applications in fields ranging from physics to economics and data science.
In recognition of its contributions to the field of topology, the Norwegian Academy of Sciences and Letters has decided to award the 2022 Abel Prize to Dennis Parnell Sullivan "in recognition of his pioneering contributions to topology in the broadest sense, particularly in algebra, geometry and dynamics".
Dennis Parnell Sullivan. Sullivan's important achievements in topology were his proof of the Adams conjecture, as well as his proof of rational mapping of no wandering domains in the dynamic system, solving the conjecture of 60 years ago. In 1999, he and Moira Chas discovered a new invariate based on loop-based manifolds, forming chord topology, an area that has developed rapidly in recent years. | Image credit: John Griffin/Stony Brook University
Hans Munthe-Kaas, chairman of the Abel Prize Committee, said: "Dennis Sullivan continues to advance topology by introducing new concepts, proving landmark theorems, answering old conjectures, and asking new questions that drive the field forward." He added: "Sullivan is like a true master, seemingly effortlessly using algebraic, analytic, and geometric ideas to transform between fields. ”
Manifolds: one-dimensional, two-dimensional, three-dimensional...
A "manifold" is a fundamental concept in topology, which is a shape that is identical everywhere and has no endpoints, edge points, intersection points, or branch points. The classification of manifolds (i.e., how many different manifolds there are and what they look like) has always been one of the fundamental areas of topology research. This was exactly the area in which Sullivan began his career, the subject of his research and important early work.
Let's start with a simplified classification and first look at one-dimensional manifolds. One-dimensional shapes are perhaps most easily imagined as shapes made of strings. We can use the string to represent the letter a, but obviously, a is not a manifold because it has two endpoints, which are located at the top and bottom of the letter a. The same is true of the letters "b" and "c", which are also not manifolds. The letter "o" is a manifold: it has no endpoints, intersections, or branches. In fact , the closed loop " o " is the only one-dimensional manifold that can be composed of a finite number of strings.
Next, let's look at the manifold of two-dimensional space. Two-dimensional shapes are perhaps most easily thought of as shapes made of faces. A piece of paper is two-dimensional (if we ignore its thickness), but it is not a manifold because it has edges; and a sphere (mathematically, a sphere is the surface of a ball) is a manifold, and the surrounding environment looks the same no matter where you are on a sphere.
Torus (shaped like a donut) is also a manifold; a double torus (the surface that looks like a figure-of-8 pretzel) is also a manifold. Not only that, in fact, three rings, four rings... It's all manifolds. To sum up: in two-dimensional space, spheres and torus families are the only two-dimensional directional manifolds that can be composed of a finite number of faces.
Let's move on to three-dimensional. Three-dimensional shapes are probably most easily thought of as shapes made of dough. But in the case of three dimensions, the visual analogy fails, and we enter the world of abstraction. From the previous example, notice how one-dimensional chordal manifolds (such as the letter o) exist in two-dimensional space, and how two-dimensional torus exist in three-dimensional space. By the same token, three-dimensional dough manifolds exist in four or more dimensions of space, and these shapes cannot be constructed in the three-dimensional space in which we live.
The classification of dough manifolds is the subject of the Poincaré conjecture. The Poincaré conjecture was the most famous unsolved puzzle in mathematics until the Russian mathematician Grigori Perelman solved it in 2002 and 2003.
Next, in higher dimensions, the classification of four-dimensional manifolds is fraught with all sorts of unsolved puzzles and mysteries; however, curiously, once five dimensions and above are reached, the classification of manifolds becomes easy. Topologists applied "cut-and-complement theory" to these manifolds and constructed new manifolds. In simple terms, the higher the dimension, the greater the "space" that can be moved.
Sullivan's early work was on the theory of cedant complements. He figured out what could be added to the patchwork. One of his innovations was the use of "classified spaces" to organize the theory of cut-offs, which were the key to understanding all high-dimensional manifolds. His work gives us a comprehensive understanding of which manifolds exist in space in five dimensions and beyond, and what their characteristics are.
Chaos theory
In the mid-1970s, computers inspired many new mathematical studies. For example, it makes it possible to study the behavior of systems that rely on multiple repetitions, some of which reveal fascinating and beautiful fractals.
For example, mathematical biologists have designed models to describe the increase and decrease in animal numbers. The simple formula shown below, known as the Logical Si Tsti Mapping, can be used to describe the year-to-year changes in animal populations.
xn is a number between 0 and 1 that represents the ratio of the size of the animal population to the largest population in the nth year; the parameter r is the reproductive rate of the system.
The logical mapping is iterative, which means that we can calculate the total group size of year 2 starting from the population size of year 1, and then substitute this value back into the equation to get the values of years 3, 4, and so on. This equation reflects both how the population grows proportionally (rxn part) and how population size decreases when too large a population puts pressure on finite resources (1 - xn part).
Logical mapping reveals unusually complex behavior caused by r-values. As shown in the following figure, the graph plots the values of r along the horizontal axis and the extremum of the population along the vertical axis.
Image credit: Morn/wikimedia commons
For example, when r is between 2.4 and 3, it will eventually stabilize at a fixed value regardless of the initial size of the population, so it appears as a line in the plot. And when r reaches 3, the line forks, which means that the population size does not eventually stabilize at one value. In this case, the extremum of the population oscillates between the two values each year. As r continues to increase, the two branches fork again, at which point the population oscillates between 4 values.
Such a figure, also known as a bifurcation chart, was one of the most famous mathematical images of the 1970s. This cascading phenomenon of multiple periods is a classic example of well-known chaos theory. Another popular term for chaos theory is the "butterfly effect," in which small changes in initial conditions can produce very different results.
Physicist Mitchell Feigenbaum discovered an interesting feature of the logical cartography: the ratio of distances between bifurcation points converges to a fixed value, 4.6692..., a number known as the Fegenbaum constant. In fact , the Faigenbaum constant appears not only in the above iterative formula rxn ( 1 -xn ) , but also in other formulas. This is a universal feature of this type of system, independent of the specific details of the formula.
Sullivan proved that the cascading limit of the multiplier period is universal. His research in this area has led to a deeper understanding of the concept of "re-normalization". Now, "re-regularization" has also become part of the foundation of the field. He uses novel methods to reveal how rich theory of complex numbers can be used to understand the emergence of rigid phenomena in practical dynamics.
Topology and dynamical systems exist in different mathematical perspectives. Sullivan's work, on the other hand, can be seen as a mutually consistent visionary of the geometry of space, whether it is manifolded or fractal. Sullivan's relentless exploration of basic cognition, and his ability to discover similarities between different fields of mathematics and build bridges between them, changed the field forever.
#创作团队:
Author: Light rain
Typography: Wenwen
#参考来源:
https://abelprize.no/en/abel-prize-laureates/2022