A century-old puzzle that remained unsolved for 78 years was cracked in just three days!
The question was whether a complex structure exists on the 6-dimensional sphere, a problem dating back to 1948. The answer is now: yes.
The solution was presented by Harvard mathematician Levent Alpöge and Claude.
What's even more remarkable is the method they used to break through.
They bypassed the well-trodden paths followed for decades and directly constructed the object, then pointed to it and declared, 'This is the complex structure on S6.'
When Alpöge announced it on X, his opening line sounded more like announcing the arrival of a new life—
Welcome this beautiful new geometric object into the world.
Over 70 Years, All Hanging on This One Sphere
Among all spheres, only two qualify to discuss complex structures: S2 and S6; others were ruled out long ago.
S2 is no surprise; it's the most fundamental building block in complex geometry.
So, for over 70 years, all eyes were fixed on this sole candidate, S6.
Some claimed it existed, while others swore it absolutely did not, but both sides stumbled.
For example, Michael Francis Atiyah, one of the greatest living mathematicians and a Fields Medalist, claimed to have solved it in 2016, but his proof was found to have gaps.
Master mathematician Shiing-Shen Chern also studied this problem in his later years.
What Alpöge presented this time is a hefty 108-page document.
Every matrix, coordinate chart, and gluing method used in the construction is laid out in black and white.
Mathematician Qiaochu Yuan deliberately tried to find flaws using GPT-5.6 Sol. After scrutinizing it for 6 minutes, he found none.
Undeterred, he examined it for another 15 minutes but still couldn't find any issues; instead, he gained a deeper understanding of the argument.
Sol concluded that if these 108 pages hold their ground, this could be the most significant AI achievement in mathematics to date.
If this work were purely human, it would likely win a Fields Medal, if not for its difficulty or fame, then at least for its impact.
Three Integers That Decided the Fate of the Problem
So, how exactly was this new object constructed?
Step 1: Lay the foundation.
Take a (3,4,∞) triangle group and use it to tessellate the upper half-plane. The resulting shape, intuitively, is a sphere.
However, this sphere has three special points pinned to it: a point of order 3, a point of order 4, and a cusp, located at t = 0, t = 1, and t = ∞, respectively.
Alpöge didn't hide his favoritism, stating outright that the triangle group and the family of tori attached to it were his proudest touch.
Step 2: Attach tori to the foundation.
At every point on the base except for the three special ones, attach a complex 2-torus, a structure that is 2-dimensional over the complex numbers (4-dimensional over the reals).
This object attached above a point is called the fiber at that point. The whole X is constructed fiber by fiber in this manner.
After this step, the spots above the three special points remain empty, effectively punching three holes in the sphere.
Step 3: Fill the three holes.
Filling these holes means assigning a fiber to each of these vacant points and stitching them together to form a complete compact manifold.
The clever part is that the three holes are not filled using the same method; each hole precisely falls within the scope of a classical filling technique.
The cusp at t = ∞ is filled using Mumford's toric degeneration; the fiber inserted there is called W, obtained by taking a hexagon boundary of a degree 6 del Pezzo surface and gluing opposite edges in pairs.
The remaining points at t = 0 and t = 1 are filled using Kodaira's logarithmic transformation, with multiplicities 3 and 4, matching the orders 3 and 4 of the base points.
The moment the three holes are filled, a compact complex threefold named X comes into existence.
The Object Is Constructed, But Is It S6?
At this point, X is a perfectly legitimate complex manifold, but the interrogation isn't over.
Section 7 of the paper rigorously computes the fundamental group of X: π1(X) ≅ Z / |12l0 − 4l1 − 3l2|.
Roughly speaking, the fundamental group tells you if there are unavoidable holes in the space. Any loop drawn on a sphere can be contracted to a point, so the sphere's fundamental group is trivial.
The three integers (l0, l1, l2) in the formula record the twisting of the fibers when filling the three holes.
Substituting (0, 1, −1) gives 12×0 − 4×1 − 3×(−1) = −1, whose absolute value is firmly 1.
Z modulo 1 is the trivial group. The fundamental group vanishes here, satisfying the first condition matching X with the sphere.
Since X itself is simply connected, its integral homology matches S6 exactly. Applying the Hurewicz and Whitehead theorems, it is indeed a homotopy 6-sphere. Combined with Smale's generalized Poincaré conjecture from 1961, it is homeomorphic to S6.
Finally, only the smooth structure remains to be verified.
In topology, homeomorphism does not imply diffeomorphism. Two objects might look identical, but the ways calculus works on them might not align. Such imposters have a special name: exotic spheres.
Fortunately, as early as 1963, Kervaire and Milnor settled this account: the six-dimensional world is clean, having exactly zero exotic spheres. In contrast, seven dimensions can produce up to 28 of them.
Thus, homeomorphism here directly upgrades to diffeomorphism. X's true identity is S6.
The Solver Wasn't Originally a Complex Geometer
Alpöge is a Junior Fellow at the Harvard Society of Fellows and concurrently a postdoctoral researcher at Anthropic. His primary field is number theory and arithmetic geometry; he doesn't typically wade into the waters of complex geometry.
Yuan revealed that just three days before solving this problem, he had discussed this deadlock with him.
From Finding Answers to Creating Answers
Solving a 70-year-old problem in three days is already absurd enough.
But this was actually the third time within 35 days.
July 20: Alpöge, using Claude Fable 5, produced a fatal counterexample to the Jacobian conjecture, a problem posed in 1939 that had remained unsolved for 87 years.
Only three weeks later, on August 10: An unreleased research version of Claude, whose identity remains undisclosed, increased the proven proportion of zeros of the Riemann zeta function lying on the critical line from 41.6% to 67.2%.
In that computational battle, it mobilized about 60 sub-agents, executed roughly 2400 shell commands, and consumed 31 million output tokens.
Then came August 24: this S6 result.
For the first two instances, you could still argue that the AI was merely a powerful search engine—one exploring known solution spaces for a counterexample, the other forcibly stitching together two existing papers already in the literature.
This time, however, the nature fundamentally changed.
This geometric object did not previously exist; the model literally created it.
Justin Curry, Associate Professor of Mathematics and Statistics at the University at Albany, SUNY, stated plainly: If the proof holds, this is certainly the most impressive AI achievement in recent memory.
For the past 78 years, everyone was asking the same question: does a complex structure exist on S6?
From this moment on, the question might already be a different one.
How many more are hidden there?
Reference: https://alpo.ge/s6.pdf
This article is from the WeChat public account "Xinzhiyuan"; author: ASI Apocalypse; editor: Moses David
