In the final months of 2011, Brian White would occasionally hear a tap on his Stanford University office door. Waiting outside would be two younger mathematicians, Fernando Codá Marques and André Neves, always with the same rough question: Did White have a few minutes to help them understand some confusing part of an obscure, several-hundred-page doctoral dissertation written three decades earlier?
The dissertation, by Jon Pitts, presented powerful machinery for constructing minimal surfaces — structures akin to soap films and bubbles — within a wide variety of shapes. Minimal surfaces, when they can be constructed, offer a lens through which to study the geometry of the surrounding space, and they turn up in a range of scientific settings, from the study of black holes … Read the rest
One of the biggest and most basic questions in physics involves the number of ways to configure the matter in the universe. If you took all that matter and rearranged it, then rearranged it again, then rearranged it again, would you ever exhaust the possible configurations, or could you go on reconfiguring forever?
Physicists don’t know, but in the absence of certain knowledge, they make assumptions. And those assumptions differ depending on the area of physics they happen to be in. In one area they assume the number of configurations is finite. In another they assume it’s infinite. For now, at least, there’s no way to tell who’s right.
But over the last couple years, a select group of mathematicians and computer scientists has been busy … Read the rest
In the ancient Egyptian rubbish heap of Oxyrhynchus, fragments of Euclid’s Elements were found on papyrus. Some of these salvaged scraps, such as one now at the University of Pennsylvania’s Penn Museum, were illustrated with diagrams expressing the influential work of the 4th-century BCE Greek mathematician. Centuries later, an Irish professor of mathematics named Oliver Byrne would transform Euclid’s mathematical proofs and propositions into one of the first multicolor printed books. His 1847 publication — The First Six Books of The Elements of Euclid in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners — used original illustrations to present the 300 BCE geometry treatise with diagrams … Read the rest
Have you ever sat in a math classroom and wondered, “When will I ever use this?” You might have asked yourself this question when you first encountered “imaginary” numbers, and with good reason: What could be less practical than a number described as imaginary?
But imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful. They have a far-reaching impact in physics, engineering, number theory and geometry. And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our physical world. Let’s take a look at how these unfamiliar numbers are rooted in the numbers we know, but at the same time, are unlike anything … Read the rest
Success for Robert Zimmer is defined differently these days. As the president of the University of Chicago since 2006, he’s made headlines for landing nine-figure financial gifts and writing op-eds in defense of campus free speech. But before Zimmer was a university president he was a mathematician. And long after he left serious research behind, the research plan he set in motion is finally paying off.
A year ago a trio of mathematicians solved what’s called Zimmer’s conjecture, which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries. Their proof stands as one of the biggest mathematical achievements in recent years. It settles a question that emerged for Zimmer during a period of intense intellectual activity in the … Read the rest
Physics contains equations that describe everything from the stretching of space-time to the flitter of photons. Yet only one set of equations is considered so mathematically challenging that it’s been chosen as one of seven “Millennium Prize Problems” endowed by the Clay Mathematics Institute with a $1 million reward: the Navier-Stokes equations, which describe how fluids flow.
Last month I wrote a story about an important new result related to those equations. If anything, the new work suggests that progress on the Millennium Prize will be even harder than expected. Why are these equations, which describe familiar phenomena such as water flowing through a hose, so much harder to understand mathematically than, say, Einstein’s field equations, which involve stupefying objects like black holes?