"There’s a chance, however, that the universe is ultimately curved and bounded. But even if it is, the allowable curvature is so slight that, according to some analyses, the Hubble volume we see is just one out of at least one thousand such volumes that must exist. And a recently launched space instrument, the Planck telescope, may reveal within a few years that there are at least one million Hubble volumes out there in the cosmos, only one of which we’ll ever have access to.
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At the other extreme, microscopes, particle accelerators, and various imaging devices continue to reveal the universe on a miniature scale…there may even be more dimensions, and possibly quite a few more than the three spatial dimensions we’re intimately acquainted with.
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My father died unexpectedly when I was fourteen, leaving our family not only grief-stricken but destitute, with a slew of debts to pay off and virtually no income. As I needed to earn some money to support the family, an uncle advised me to leave school and raise ducks instead. But I had a different idea: teaching mathematics to other stu dents. Given our financial circumstances, I knew there was just one chance for me to succeed and I placed my bets on math, double or nothing.
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So why did geometry, of all the areas of mathematics, come to occupy center stage for me, both in my waking thoughts and in my dreams? Primarily because it struck me as the field closest to nature and therefore closest to answering the kinds of questions I cared about most. Besides, I find it helpful to look at pictures when grappling with difficult concepts, and pictures are few and far between in the more abstruse realms of algebra and number theory. Then there was this fantastic group of people doing geometry at Berkeley ...
Dehn’s lemma, a geometric version of which was proven by William Meeks and the author (Yau), provides a mathematical technique for simplifying a surface that crosses, or intersects, itself into a surface with no crossings, folds, or other singularities. The lemma is typically framed in terms of topology, but the geometric approach taken by Meeks and Yau offers a more precise solution.
The argument is difficult to explain mainly because the Einstein field equation, which relates the physics in this situation to geometry, is a complicated, nonlinear formulation that is not intuitive. Basically, we started off by assuming that the mass of a particular space was not positive. Next we showed that you could construct an area-minimizing surface in such a space whose average curvature was non-negative. The surface, in other words, could have zero average curvature. That would be impossible, however, if the space in which the surface sat was our universe, where the observed matter density is positive. And assuming that general relativity is correct, a positive matter density implies positive curvature.
While this argument might seem circular, it actually is not. The matter density can be positive in a particular space, such as our universe, even though the total mass is not positive. That’s because there are two contributions to total mass—one coming from matter and the other coming from gravity."
