Twenty-seven years ago, a group of physicists made an accidental discovery that flipped mathematics on its head. The physicists were trying to work out the details of string theory when they observed a strange correspondence: Numbers emerging from one kind of geometric world matched exactly with very different kinds of numbers from a very different kind of geometric world.

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Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

To physicists, the correspondence was interesting. To mathematicians, it was preposterous. Theyâ€™d been studying these two geometric settings in isolation from each other for decades. To claim that they were intimately related seemed as unlikely as asserting that at the moment an astronaut jumps on the moon, some hidden connection causes his sister to jump back on earth.

â€œIt looked totally outrageous,â€� said David Morrison, a mathematician at the University of California, Santa Barbara, and one of the first mathematicians to investigate the matching numbers.

Nearly three decades later, incredulity has long since given way to revelation. The geometric relationship that the physicists first observed is the subject of one of the most flourishing fields in contemporary mathematics. The field is called mirror symmetry, in reference to the fact that these two seemingly distant mathematical universes appear somehow to reflect each other exactly. And since the observation of that first correspondenceâ€”a set of numbers on one side that matched a set of numbers on the otherâ€”mathematicians have found many more instances of an elaborate mirroring relationship: Not only do the astronaut and his sister jump together, they wave their hands and dream in unison, too.

Recently, the study of mirror symmetry has taken a new turn. After years of discovering more examples of the same underlying phenomenon, mathematicians are closing in on an explanation for why the phenomenon happens at all.

â€œWeâ€™re getting to the point where weâ€™ve found the ground. Thereâ€™s a landing in sight,â€� said Denis Auroux, a mathematician at the University of California, Berkeley.

The effort to come up with a fundamental explanation for mirror symmetry is being advanced by several groups of mathematicians. They are closing in on proofs of the central conjectures in the field. Their work is like uncovering a form of geometric DNAâ€”a shared code that explains how two radically different geometric worlds could possibly hold traits in common.

### Discovering the Mirror

What would eventually become the field of mirror symmetry began when physicists went looking for some extra dimensions. As far back as the late 1960s, physicists had tried to explain the existence of fundamental particlesâ€”electrons, photons, quarksâ€”in terms of minuscule vibrating strings. By the 1980s, physicists understood that in order to make â€œstring theoryâ€� work, the strings would have to exist in 10 dimensionsâ€”six more than the four-dimensional space-time we can observe. They proposed that what went on in those six unseen dimensions determined the observable properties of our physical world.

â€œYou might have this small space that you canâ€™t see or measure directly, but some aspects of the geometry of that space might influence real-world physics,â€� said Mark Gross, a mathematician at the University of Cambridge.

Eventually, they came up with potential descriptions of the six dimensions. Before getting to them, though, itâ€™s worth thinking for a second about what it means for a space to have a geometry.

Mark Gross, a mathematician at the University of Cambridge, and a colleague are putting the finishing touches on a proof that establishes a universal method for constructing one mirror space from another.

Consider a beehive and a skyscraper. Both are three-dimensional structures, but each has a very different geometry: Their layouts are different, the curvature of their exteriors is different, their interior angles are different. Similarly, string theorists came up with very different ways to imagine the missing six dimensions.

One method arose in the mathematical field of algebraic geometry. Here, mathematicians study polynomial equationsâ€”for example, x^{2} + y^{2} = 1â€”by graphing their solutions (a circle, in this case). More-complicated equations can form elaborate geometric spaces. Mathematicians explore the properties of those spaces in order to better understand the original equations. Because mathematicians often use complex numbers, these spaces are commonly referred to as â€œcomplexâ€� manifolds (or shapes).

The other type of geometric space was first constructed by thinking about physical systems such as orbiting planets. The coordinate values of each point in this kind of geometric space might specify, for example, a planetâ€™s location and momentum. If you take all possible positions of a planet together with all possible momenta, you get the â€œphase spaceâ€� of the planetâ€”a geometric space whose points provide a complete description of the planetâ€™s motion. This space has a â€œsymplecticâ€� structure that encodes the physical laws governing the planetâ€™s motion.

Symplectic and complex geometries are as different from one another as beeswax and steel. They make very different kinds of spaces. Complex shapes have a very rigid structure. Think again of the circle. If you wiggle it even a little, itâ€™s no longer a circle. Itâ€™s an entirely distinct shape that canâ€™t be described by a polynomial equation. Symplectic geometry is much floppier. There, a circle and a circle with a little wiggle in it are almost the same.

â€œAlgebraic geometry is a more rigid world, whereas symplectic geometry is more flexible,â€� said Nick Sheridan, a research fellow at Cambridge. â€œThatâ€™s one reason theyâ€™re such different worlds, and itâ€™s so surprising they end up being equivalent in a deep sense.â€�

In the late 1980s, string theorists came up with two ways to describe the missing six dimensions: one derived from symplectic geometry, the other from complex geometry. They demonstrated that either type of space was consistent with the four-dimensional world they were trying to explain. Such a pairing is called a duality: Either one works, and thereâ€™s no test you could use to distinguish between them.

Physicists then began to explore just how far the duality extended. As they did so, they uncovered connections between the two kinds of spaces that grabbed the attention of mathematicians.

In 1991, a team of four physicistsâ€”Philip Candelas, Xenia de la Ossa, Paul Green and Linda Parkesâ€”performed a calculation on the complex side and generated numbers that they used to make predictions about corresponding numbers on the symplectic side. The prediction had to do with the number of different types of curves that could be drawn in the six-dimensional symplectic space. Mathematicians had long struggled to count these curves. They had never considered that these counts of curves had anything to do with the calculations on complex spaces that physicists were now using in order to make their predictions.

The result was so far-fetched that at first, mathematicians didnâ€™t know what to make of it. But then, in the months following a hastily convened meeting of physicists and mathematicians in Berkeley, California, in May 1991, the connection became irrefutable. â€œEventually mathematicians worked on verifying the physicistsâ€™ predictions and realized this correspondence between these two worlds was a real thing that had gone unnoticed by mathematicians who had been studying the two sides of this mirror for centuries,â€� said Sheridan.

The discovery of this mirror duality meant that in short order, mathematicians studying these two kinds of geometric spaces had twice the number of tools at their disposal: Now they could use techniques from algebraic geometry to answer questions in symplectic geometry, and vice versa. They threw themselves into the work of exploiting the connection.

### Breaking Up Is Hard to Do

At the same time, mathematicians and physicists set out to identify a common cause, or underlying geometric explanation, for the mirroring phenomenon. In the same way that we can now explain similarities between very different organisms through elements of a shared genetic code, mathematicians attempted to explain mirror symmetry by breaking down symplectic and complex manifolds into a shared set of basic elements called â€œtorus fibers.â€�

A torus is a shape with a hole in the middle. An ordinary circle is a one-dimensional torus, and the surface of a donut is a two-dimensional torus. A torus can be of any number of dimensions. Glue lots of lower dimensional tori together in just the right way, and you can build a higher dimensional shape out of them.

To take a simple example, picture the surface of the earth. It is a two-dimensional sphere. You could also think of it as being made from many one-dimensional circles (like many lines of latitude) glued together. All these circles stuck together are a â€œtorus fibrationâ€� of the sphereâ€”the individual fibers woven together into a greater whole.

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Torus fibrations are useful in a few ways. One is that they give mathematicians a simpler way to think of complicated spaces. Just like you can construct a torus fibration of a two-dimensional sphere, you can construct a torus fibration of the six-dimensional symplectic and complex spaces that feature in mirror symmetry. Instead of circles, the fibers of those spaces are three-dimensional tori. And while a six-dimensional symplectic manifold is impossible to visualize, a three-dimensional torus is almost tangible. â€œThatâ€™s already a big help,â€� said Sheridan.

A torus fibration is useful in another way: It reduces one mirror space to a set of building blocks that you could use to build the other. In other words, you canâ€™t necessarily understand a dog by looking at a duck, but if you break each animal into its raw genetic code, you can look for similarities that might make it seem less surprising that both organisms have eyes.

Here, in a simplified view, is how to convert a symplectic space into its complex mirror. First, perform a torus fibration on the symplectic space. Youâ€™ll get a lot of tori. Each torus has a radius (just like a circleâ€”a one-dimensional torusâ€”has a radius). Next, take the reciprocal of the radius of each torus. (So, a torus of radius 4 in your symplectic space becomes a torus of radius Â¼ in the complex mirror.) Then use these new tori, with reciprocal radii, to build a new space.

In 1996, Andrew Strominger, Shing-Tung Yau and Eric Zaslow proposed this method as a general approach for converting any symplectic space into its complex mirror. The proposal that itâ€™s always possible to use a torus fibration to move from one side of the mirror to the other is called the SYZ conjecture, after its originators. Proving it has become one of the foundational questions in mirror symmetry (along with the homological mirror symmetry conjecture, proposed by Maxim Kontsevich in 1994).

The SYZ conjecture is hard to prove because, in practice, this procedure of creating a torus fibration and then taking reciprocals of the radii is not easy to do. To see why, return to the example of the surface of the earth. At first it seems easy to stripe it with circles, but at the poles, your circles will have a radius of zero. And the reciprocal of zero is infinity. â€œIf your radius equals zero, youâ€™ve got a bit of a problem,â€� said Sheridan.

This same difficulty crops up in a more pronounced way when youâ€™re trying to create a torus fibration of a six-dimensional symplectic space. There, you might have infinitely many torus fibers where part of the fiber is pinched down to a point â€” points with a radius of zero. Mathematicians are still trying to figure out how to work with such fibers. â€œThis torus fibration is really the great difficulty of mirror symmetry,â€� said Tony Pantev, a mathematician at the University of Pennsylvania.

Put another way: The SYZ conjecture says a torus fibration is the key link between symplectic and complex spaces, but in many cases, mathematicians donâ€™t know how to perform the translation procedure that the conjecture prescribes.

### Long-Hidden Connections

Over the past 27 years, mathematicians have found hundreds of millions of examples of mirror pairs: This symplectic manifold is in a mirror relationship with that complex manifold. But when it comes to understanding why a phenomenon occurs, quantity doesnâ€™t matter. You could assemble an arkâ€™s worth of mammals without coming any closer to understanding where hair comes from.

â€œWe have huge numbers of examples, like 400 million examples. Itâ€™s not that thereâ€™s a lack of examples, but nevertheless itâ€™s still specific cases that donâ€™t give much of a hint as to why the whole story works,â€� said Gross.

Mathematicians would like to find a general method of constructionâ€”a process by which you could hand them any symplectic manifold and they could hand you back its mirror. And now they believe that theyâ€™re getting close to having it. â€œWeâ€™re moving past the case-by-case understanding of the phenomenon,â€� said Auroux. â€œWeâ€™re trying to prove that it works in as much generality as we can.â€�

Mathematicians are progressing along several interrelated fronts. After decades building up the field of mirror symmetry, theyâ€™re close to understanding the main reasons the field works at all.

â€œI think it will be done in a reasonable time,â€� said Kontsevich, a mathematician at the Institute of Advanced Scientific Studies (IHES) in France and a leader in the field. â€œI think it will be proven really soon.â€�

One active area of research creates an end run around the SYZ conjecture. It attempts to port geometric information from the symplectic side to the complex side without a complete torus fibration. In 2016, Gross and his longtime collaborator Bernd Siebert of the University of Hamburg posted a general-purpose method for doing so. They are now finishing a proof to establish that the method works for all mirror spaces. â€œThe proof has now been completely written down, but itâ€™s a mess,â€� said Gross, who said that he and Siebert hope to complete it by the end of the year.

Another major open line of research seeks to establish that, assuming you have a torus fibration, which gives you mirror spaces, then all the most important relationships of mirror symmetry fall out from there. The research program is called â€œfamily Floer theoryâ€� and is being developed by Mohammed Abouzaid, a mathematician at Columbia University. In March 2017 Abouzaid posted a paper that proved this chain of logic holds for certain types of mirror pairs, but not yet all of them.

And, finally, there is work that circles back to where the field began. A trio of mathematiciansâ€”Sheridan, Sheel Ganatra and Timothy Perutzâ€”is building on seminal ideas introduced in 1990s by Kontsevich related to his homological mirror symmetry conjecture.

Cumulatively, these three initiatives would provide a potentially complete encapsulation of the mirror phenomenon. â€œI think weâ€™re getting to the point where all the big â€˜whyâ€™ questions are close to being understood,â€� said Auroux.

*Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.*