“I had heard rumors that this was coming up, and I didn’t know exactly what to expect,” said Vesna Stojanoska, a mathematician at the University of Illinois, Urbana-Champaign who attended the conference.

It was soon clear the rumors were true. Beginning on Tuesday, and over the next three days, Levy and his co-authors — Robert Burklund, Jeremy Hahn and Tomer Schlank — explained to the crowd of some 200 mathematicians how they’d proved that the telescope conjecture was false, making it the only one of Ravenel’s original conjectures not to be true.

The disproof of the telescope conjecture has wide-ranging implications, but one of the simplest and most profound is this: It means that in very high dimensions (think of a 100-dimensional sphere), the universe of different shapes is far more complicated than mathematicians anticipated.

Mapping the Maps

To classify shapes, or topological spaces, mathematicians distinguish between differences that matter and those that don’t. Homotopy theory is a perspective from which to make those distinctions. It considers a ball and an egg to be fundamentally the same topological space, because you can bend and stretch one into the other without ripping either. In the same way, homotopy theory considers a ball and an inner tube to be fundamentally different because you have to tear a hole in the ball to deform it into the inner tube.

Homotopy is useful for classifying topological spaces — creating a chart of all the kinds of shapes that are possible. It’s also important for understanding something else mathematicians care about: maps between spaces. If you have two topological spaces, one way to probe their properties is to look for functions that convert, or map, points on one to points on the other — input a point on space A, get a point on space B as your output, and do that for all the points on A.

To see how these maps work, and why they illuminate properties of the spaces involved, start with a circle. Now map it onto the two-dimensional sphere, which is the surface of a ball. There are infinitely many ways of doing this. If you imagine the sphere as Earth’s surface, you could put your circle at any line of latitude, for example. From the perspective of homotopy theory, they’re all equivalent, or homotopic, because they can all shrink down to a point at the north or south pole.

Next, map the circle onto the two-dimensional surface of an inner tube (a one-holed torus). Again, there are infinitely many ways of doing this, and most are homotopic. But not all of them. You could place a circle horizontally or vertically around the torus, and neither can be smoothly deformed into the other. These are two (of many) ways of mapping a circle onto the torus, while there is just one way to map it onto a sphere, reflecting a fundamental difference between the two spaces: The torus has one hole while the sphere has none.

It’s easy to count the ways we can map from the circle to the two-dimensional sphere or torus. They’re familiar spaces that are easy to visualize. But counting maps is much harder when higher-dimensional spaces are involved.

Dimensional Differences

If two spheres have the same dimension, there are always infinitely many maps between them. And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map.

Partly for that reason, counting maps is most interesting when the space you’re mapping from has a higher dimension than the space you’re mapping to, like when you map a seven-dimensional sphere onto a three-dimensional sphere. In cases like those, the number of maps is always finite.

“The maps between spheres in general tend to be more interesting when the source has a larger dimension,” Hahn said.

Moreover, the number of maps depends only on the difference in the number of dimensions (once the dimensions get big enough compared to the difference). That is, the number of maps from a 73-dimensional sphere to a 53-dimensional sphere is the same as the number of maps from a 225-dimensional sphere to a 205-dimensional sphere, because in both cases, the difference in dimension is 20.

Mathematicians would like to know the number of maps between spaces of any difference in dimension. They’ve managed to compute the number of maps for almost all differences in dimension up to 100: There are 24 maps between spheres when the difference is 20, and  3,144,960 when it’s 23.