The paper was posted in March and was a major extension of one of the biggest advances in geometry in recent decades. The work they built on relates to a well-known conjecture from the 1960s. Arnold was studying classical mechanics and wanted to know when the planets would return to their original configuration after a set period.
Arnold worked in an area of mathematics that looked at all the different configurations of a physical system, like bouncing billiard balls or planets. These configurations are found in phase spaces, which are part of a flourishing mathematical field called symplectic geometry.
Arnold predicted that every phase space of a certain type has a minimum number of configurations in which the system it describes returns to where it started. This would be similar to billiards balls coming to occupy the same positions. He thought that the minimum number would be equal to the number of holes in the overall phase space, which can be used to make objects like a sphere or a doughnut.
Two different ways of thinking about a shape were linked by the Arnold conjecture. It suggested that mathematicians could get information about the motion of objects in a given shape by looking at how many configurations return the object to where it started.
symplectic things are more difficult than purely topological things. Being able to tell something from the data is the main interest, according to a researcher.
The first major advancement on the Arnold conjecture took place in the 1980's when a young mathematician named Andreas Floer developed a new way of counting holes. One of the central tools in symplectic geometry was Floer's theory. Even though mathematicians used Floer's ideas, they thought it would be possible to develop other theories in light of the new perspective that Floer opened up.
It has been a long time since Abouzaid and Blumberg did it. In their March paper, they remake another important theory in terms of the techniques for counting holes. The new theory is used to prove a version of the Arnold conjecture. The early proof-of-concept result has mathematicians anticipating that they will find many more uses for the ideas.
Ailsa Keating of the University of Cambridge said that it was an important development for the field.
The motion of the object.
Imagine a planet moving through space and imagine how configurations of a physical system can be used to build a geometric object.
The planet's position and momentum can be described by six numbers. The phase space of the system can be created by representing each of the different configurations of the planet's position and momentum as a point with six coordinates. It has the shape of a flat six-dimensional space. A line can be represented as the motion of a single planet.
Phase spaces can take on many different shapes. The phase space of a pendulum can be represented as a point on a circle, so that the pendulum's position and momentum can be seen as points on a circle.
The properties of general phase spaces are studied by sphyllic geometry. Some paths loop back and forth on the manifolds. It is a challenging problem to describe closed orbits. It's often difficult to answer a simple question about a physical system.
In the 1960s, Vladimir Arnold wanted to make the task of counting closed orbits simpler.
There are some holes that are counted.
Shapes have different dimensions. The inside of a rubber band has one-dimensional holes. Two-dimensional holes are like the inside of a balloon. Higher-dimensional holes are nearly impossible to see.
In lower dimensions, our intuition about holes is shaky. How many holes does a straw have? The formal way to count holes is called homology. Homology associates to each shape an object, which can be used to find out the number of holes in the object.
To perform the association, mathematicians first break down the shape into component pieces that look like triangles in different dimensions. Topologists use a sort of algebra shapes to figure out which parts of a hole form a loop.
Floer's work was revolutionary. For the way one would look at the field as a whole, not just for this problem.
The University of Cambridge has an associate professor named Ivan Smith.
The computations are done using whole numbers. They can be done with other number systems, like the rational numbers, which can be expressed as fractions, or the cyclical number systems, which count in circles.
Depending on which number system you use to count the holes, the question of relating the number of closed loops to the number of holes comes out differently.
A recent paper by Abouzaid and Blumberg proves the conjecture when the homology is computed with a number system. They had to build on the ideas of Floer, who created a new theory in the 1980's that would eventually make it possible to compute the homology with rational numbers.
The work of Frog was revolutionary. Ivan Smith of Cambridge said that it was not just for this problem but for the way one would look at the field as a whole.
Floer has a perspective.
Floer had to count closed orbits to prove the Arnold theory. He started by drawing loops through the phase space and then combined neighboring loops to form geometric objects. The smallest of the geometric objects came about when the loops that formed them were closed. Critical points correspond to these objects.
The method for studying these critical points was already used by mathematicians. Imagine a torus suspended in a bucket filled with water. When the water first touches the bottom of the torus, the bottom of the hole, the top of the hole and the top of the torus are when the water changes shape.
The rising water gives important information which can be used to derive a shape. The number of holes in a shape is connected to the critical points of the shape in this way.
You look at the top of the object.
It was almost enough to solve the Arnold conjecture, but it only works in finite dimensions. Floer found a way to apply the theory to loops that were infinite-dimensional. Floer's modified construction became the bridge to solve the Arnold conjecture, because the closed orbits in the Arnold conjecture become critical points in a space of loops, which are tied to the homology.
The Institute for Advanced Study said that Floer's insight was incredible.
Dividing by nothing.
Floer theory was very useful in the study of knots and mirror symmetry.
Manolescu said that it was the central tool in the subject.
Floer's method only worked on one type of manifold, so it didn't completely resolve the Arnold conjecture. Over the next two decades, the community worked to overcome this obstruction. The work led to a proof of the Arnold conjecture where the homology is computed using rational numbers. The Arnold conjecture was not resolved when holes are counted using other number systems.
The proof involved dividing by the number of symmetries of a specific object is the reason that the work didn't extend to cyclical number systems. This is possible with rational numbers. Division is more difficult with numbers. The numbers 5 and 10 are equivalent to zero if the number system cycles back after five. This is similar to the way 1 p.m. is the same as 1 pm, so dividing by 5 is the same as dividing by zero. Someone was going to have to come up with new ways to circumvent this issue.
The first thing that comes to mind is the fact that we have to introduce these denominators, because technical things are preventing Floer theory from developing.
To expand Floer's theory and prove the Arnold conjecture with numbers, Abouzaid and Blumberg needed to look beyond homology.
The Topologist's Tower is being climbed.
A specific recipe can be applied to a shape. During the 20th century, topologists began to look at homology on its own terms.
Let's not think about the recipe. What comes out of the recipe? What properties did this group have?
Topologists looked for theories that satisfied the same fundamental properties. These were known as generalized theories. Topologists built up a tower of complicated generalized homology theories, all of which can be used to classify spaces.
The ground-floor theory of homology is mirrored by Floer homology. Floer's original theory connected the generalized homology with specific features of a space in an infinite-dimensional setting, and so it has been wondered if it is possible to develop Floer versions of topological theories higher up on the tower.
Floer died in 1991 at the age of 34 and never had a chance to attempt this work himself. The mathematicians continued to look for ways to expand their ideas.
Benchmarking a new theory.
After nearly five years of work, the vision has been realized. The Floer version of Morava K-theory is the one they use to prove the Arnold conjecture.
Keating said there was a sense in which this completes a circle for us which ties back to Floer's original work.
The Morava K-theory was created to expand the tower of theories. It had no obvious connection to the geometry at the time. Morava K-theory is an invariant, which means that it captures some essential and unchanging feature of an underlying shape. A Floer version of Morava K-theory was the key to proving a new version of the Arnold conjecture.
The original method involved dividing by a certain number of symmetries, a requirement that resulted in over counting certain objects. The Floer version of Morava K-theory only counts the objects once. It can now be used to count holes that are higher-dimensional and to prove the Arnold conjecture using number systems.
The Floer Morava K-theory or Floer Homotopy theory is not about the Arnold theory, according to the authors.
We did not do this in order to solve the Arnold question, said Blumberg. It is a sanity check to make sure you are doing the right things.
The new Floer Morava K-theory will be useful for many problems, not just the Arnold conjecture, according to mathematicians. The new paper answers a 25-year-old question in symplectic geometry and was written by Abouzaid, Smith and Mark McLean.
As mathematicians stand at the threshold of a new theory, other applications are almost certain to follow, and in ways that are hard to anticipate.
Jack Morava, a mathematician at the University of Baltimore and the inventor of Morava K-theory, said that it was exciting to see that. You can go through a door and end up in a different universe. It is very similar to Alice in Wonderland.
