Impossible-Seeming Surfaces Confirmed Decades After Conjecture

Last July, two mathematicians from Durham University, Will Hide and Michael Magee, confirmed the existence of a much sought after sequence of surfaces, each more complicated than the last, ultimately becoming so intricately connected with themselves that they nearly reach the limits of what is possible.

It wasn't obvious at first. Since the question of their existence was first raised in the 1980s, mathematicians have come to realize that these surfaces may actually be commonplace, even if they are exceedingly difficult to identify. The new work is a step in the right direction to understand the many ways that surfaces can manifest.

Peter Sarnak is a mathematician at the Institute for Advanced Study in New Jersey.

The Mbius strip is a two-dimensional object. They are important to mathematics and physics. Although mathematicians have a relationship with surfaces for hundreds of years, they don't know much about them.

The problem isn't simple surfaces. It's simple if the surface has a small number of holes. A sphere has no holes, and thus has a zero in its name.

Intuition doesn't work when the genera is high. Alex Wright ends up with holes in a row when he tries to visualize a high-genus surface. I could wrap it around into a circle with many holes if you wanted me to be more inventive. He said it would be hard to come up with a different mental picture. In high-genus surfaces, holes overlap in complicated ways that are hard to grasp. Wright said that a simple approximation is not representative.

Laura Monk is a mathematician at the University of Bristol. It's possible to make things which are not good. It's more difficult to make things which are good, which we expect to be true.

To truly understand the space of surfaces, mathematicians need to discover objects that they don't know exist.

The existence of surfaces that mathematicians had pondered about for decades was confirmed in the July paper. Graph theory is a different field of math than the one they proved.

The Maximum Possible

Graphs are networks of points connected by lines. The network that imposes a cost on connecting two nodes was studied by mathematicians. An expander graph is a type of graph that keeps the number of edges low while maintaining high connections.

The use of expander graphs in math and computer science has grown in importance. The graphs make it easy to travel from one point to another without covering the whole graph. If you don't want more than a few highways in your town, mathematicians like to limit the number of edges to three.

When the graph is very large, most of the random graphs are good expanders. Human beings have failed many times to produce expander graphs by hand.

A mathematician at the Hebrew University of Jerusalem said that you shouldn't draw them yourself if you want to build one. Our imaginations don't understand what an expander is.

There are many ways to measure the idea of expansion. One way to cut a graph is to slice it into two pieces. You only need to cut one edge to split the graph into two clumps. You will have to slice through more edges if the graph is connected more than once.

One way to get at connections is to wander around the graph from the start to the end, choosing an edge at random. How much time will it take to visit all the neighborhoods? If you don't step across the lone connection to the other half, you'll be confined to one of the bubbles. If there are many ways to travel between different parts of the graph, you will be able to see it in a short amount of time.

The measure of connectedness can be quantified by a number. If the graph is made up of two groups that aren't connected to each other, the gap is zero. The graph will get bigger as it gets more connected.

The gap can only get so high. The two main features of an expander graph are seemingly at odds. In 1988, Gregory Margulis and two co- authors described graphs with a high gap between the theoretical maximum and the actual maximum. The person said it was shocking that they existed.

Most graphs are close to this maximum. Finding the right places to put edges wasn't the only thing that needed to be done with optimal expanders and random graphs. The use of strange and sophisticated techniques was needed.

It would take a different approach to apply those results to the surfaces.

Surface Tension

At the same time that graph theorists were mapping out expander graphs, a Swissgeometer named Peter Buser was playing with similar ideas in surfaces with a lot of holes. They could be connected in the same way expander graphs were.

If you want to estimate a surface's connectedness, you can either slice it in two or wander aimlessly around the surface and see how long it takes you to get to the whole thing. Wright said that everything is more sophisticated but that the intuition goes through.

Jeff Cheeger was the first to show how easy it is to cut the surface in half. Cheeger had a stronger result thanks to Buser. Buser wondered if surfaces with a lot of holes could still be connected.

He thought the gap would have to be near zero for high-genus "hyperbolic" surfaces, an important class of surfaces that are curved. He changed his thinking after learning about expander graphs. Buser is a professor at the Swiss Federal Institute of Technology Lausanne. I had to learn that your intuition isn't right.

Buser's adviser found a ceiling for aspectral gap on a surface with the exact number dependent on the number of holes on the surface. The number is affected by the amount of holes. Buser wants to know more about this limitation. Is there any real surface that reached the cutoff? I was wondering if there was an equivalent to optimal expander graphs.

Buser showed in 1984 that some hyperbolic surfaces have a gap of at least 3/16. He said in the paper that it was likely that 3/16 could be replaced by 1/2.

The statement has been proved true by mathematicians.

Probably a Good Idea

The project didn't start out as an attempt to get to 1/2. When Hide began as a PhD student in the fall of 2020, he was told that he could try to reproduce one of the results of another PhD student.

It's possible to make things which are not good. It is more difficult to make things which we expect to be true.

In the spring of 2020, Frédéric Naud of Sorbonne University and Doron Puder of Tel Aviv University found 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 888-353-1299 Spheres and doughnuts have important properties. Unlike the infinite plane, they have no edges. Buser had focused on compact surfaces in his studies of the 1970s and '80s. They wanted to move their result from 3/16 all the way up to the 4th. It was almost as obscure as it was in 1984.

Hide was asked to think of a surface with spears poking out of it. The surface area is finite even though they are infinitely long. These surfaces are not compact.

The two men thought they could update ideas from the earlier work. Random strategies have been used to good effect in mathematics. They help mathematicians understand the typical properties of objects. Bad examples can be thrown away because of probabilities. If you calculate the probability that a marble is round and get 99%, you will get a lot of information about marbles, without being distracted by the reality that some marbles are damaged.

The strategy wasn't easy to make work. They couldn't say much about the objects they were interested in. They gave up for a short time in the spring of 2021.

Buser and two co- authors came up with a way to manipulate a finite-area surface into something compact.

There was a new meaning to Hide's abandoned project. The compact problem would be solved if you could prove not just 3/16, but also 1/2.

The more ambitious goal was to find a sequence of finite-area non-compact surfaces whose spectral gap was not just 3/16, but 1/2.

They posted their paper online on July 12th.

I was surprised that this thing I gave my student turned out to be crucial. It's more important than I thought.

All the Surfaces

There are many ways for mathematicians to explore from here. One possibility is to try to prove that there are high-genus surfaces that have a maximal value of 1/2.

Hide and Magee's technique only works for a very specific type of surface, meaning any probabilities they calculate will only tell you about a tiny fraction of all the surfaces out there.

Researchers have already studied this model, which applies to the entire space of hyperbolic surfaces, and it is the one that is out there. Monk said that her work was a complete revolution. The approach is stuck at 3-16.

The Hide-Magee result is worth celebrating on its own. We want precise results for objects that are hard to find. Parlier is a mathematician at the University of Luxembourg. Thespectral gap is so precise that we should be more amazed.

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