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Jordana Cepelewicz on two new topographical results that bring some order to the confoundingly difficult study of four-dimensional shapes…

The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions.

But in four dimensions, “everything goes a bit crazy,” said Sam Hughes, a postdoctoral researcher at the University of Oxford. Tools stop working; exotic behavior emerges. As Tom Mrowka of the Massachusetts Institute of Technology explained, “There’s just enough room to have interesting phenomena, but not so much room that they fall apart.”

In the early 1990s, Mrowka and Peter Kronheimer of Harvard University were studying how two-dimensional surfaces can be embedded within four-dimensional manifolds. They developed new techniques to characterize these surfaces, allowing them to gain crucial insights into the otherwise inaccessible structure of four-dimensional manifolds. Their findings suggested that the members of a broad class of surfaces all slice through their parent manifold in a relatively simple way, leaving a fundamental property unchanged. But nobody could prove this was always true.

In February, together with Daniel Ruberman of Brandeis University, Hughes constructed a sequence of counterexamples — “crazy” two-dimensional surfaces that dissect their parent manifolds in ways that mathematicians had believed to be impossible. The counterexamples show that four-dimensional manifolds are even more remarkably diverse than mathematicians in earlier decades had realized. “It’s really a beautiful paper,” Mrowka said. “I just keep looking at it. There’s lots of delicious little things there.”

Late last year, Ruberman helped organize a conference that created a new list of the most significant open problems in low-dimensional topology. In preparing for it, he looked at a previous list of important unsolved topological problems from 1997. It included a question that Kronheimer had posed based on his work with Mrowka. “It was in there, and I think it was a little bit forgotten,” Ruberman said. Now he thought he could answer it…

Read on for the details: “Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions,” from @jordanacep in @QuantaMagazine.

* Rene Thom

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As we savor surprising shapes, we might send carefully-modeled birthday greetings to William Bowie; he was born on this date in 1872. A geodetic engineer who joined the United States Coast and Geodetic Survey in 1895, he investigated isostasy (a principle that dense crustal rocks to tend cause topographic depressions and light crustal rocks cause topographic elevations).

Bowie was the first President of the American Geophysical Union from 1920 to 1922 and served as president a second time from 1929 to 1932. The William Bowie Medal, the highest honor of the AGU, is named in his honor.

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