This April, as undergraduates strolled along the street outside his modest office on the campus of the University of California, Los Angeles, the mathematician Terence Tao mused about the possibility that water could spontaneously explode. A widely used set of equations describes the behavior of fluids like water, but there seems to be nothing in those equations, he told me, that prevents a wayward eddy from suddenly turning in on itself, tightening into an angry gyre, until the density of the energy at its core becomes infinite: a catastrophic ‘‘singularity.’’ Someone tossing a penny into the fountain by the faculty center or skipping a stone at the Santa Monica beach could apparently set off a chain reaction that would take out Southern California.

This doesn’t tend to happen. And yet, Tao explained, nobody can say precisely why. It’s a decades-old conundrum, and Tao has recently been working on an approach to a solution — one part fanciful, one part outright absurd, like some lost passage from ‘‘Alice’s Adventures in Wonderland.’’

Imagine, he said, that someone awfully clever could construct a machine out of pure water. It would be built not of rods and gears but from a pattern of interacting currents. As he talked, Tao carved shapes in the air with his hands, like a magician. Now imagine, he went on, that this machine were able to make a smaller, faster copy of itself, which could then make another, and so on, until one ‘‘has infinite speed in a tiny space and blows up.’’ Tao was not proposing constructing such a machine — ‘‘I don’t know how!’’ he said, laughing. It was merely a thought experiment, of the sort that Einstein used to develop the theory of special relativity. But, Tao explained, if he can show mathematically that there is nothing, in principle, preventing such a fiendish contraption from operating, then it would mean that water can, in fact, explode. And in the process, he will have also solved the Navier-Stokes global regularity problem, which has become, since it emerged more than a century ago, one of the most important in all of mathematics.

Tao, who is 40, sat at a desk by the window, papers lying in drifts at the margins. Thin and unassuming, he was dressed in Birkenstocks, a rumpled blue-gray polo shirt and jeans with the cuffs turned up. Behind him, a small almond couch faced a glyph-covered blackboard running the length of the room. The couch had been pulled away from the wall to accommodate the beat-up Trek bike he rides to work. At the room’s other end stood a fiberboard bookcase haphazardly piled with books, including ‘‘Compactness and Contradiction’’ and ‘‘Poincaré’s Legacies, Part I,’’ two of the 16 volumes Tao has written since he was a teenager.