Dec 13, 2025 11:54 PM
https://www.quantamagazine.org/string-th...-20251212/
INTRO: In August, a team of mathematicians posted a paper claiming to solve a major problem in algebraic geometry — using entirely alien techniques. It instantly captivated the field, stoking excitement in some mathematicians and skepticism in others.
The result deals with polynomial equations, which combine variables raised to powers (like y = x or x2 - 3xy = z2). These equations are some of the simplest and most ubiquitous in mathematics, and today, they’re fundamental to lots of different areas of study. As a result, mathematicians want to study their solutions, which can be represented as geometric shapes like curves, surfaces and higher-dimensional objects called manifolds.
There are infinitely many types of polynomial equations that mathematicians want to tame. But they all fall into one of two basic categories — equations whose solutions can be computed by following a simple recipe, and equations whose solutions have a richer, more complicated structure. The second category is where the mathematical juice is: It’s where mathematicians want to focus their attention to make major advances.
But after sorting just a few types of polynomials into the “easy” and “hard” piles, mathematicians got stuck. For the past half-century, even relatively simple-looking polynomials have resisted classification.
Then this summer, the new proof appeared. It claimed to end the stalemate, offering up a tantalizing vision for how to classify lots of other types of polynomials that have until now seemed completely out of reach.
The problem is that no one in the world of algebraic geometry understands it. At least, not yet. The proof relies on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
Some researchers trust the reputation of one of the paper’s authors, a Fields medalist named Maxim Kontsevich. But Kontsevich also has a penchant for making audacious claims, giving others pause. Reading groups have sprung up in math departments across the world to decipher the groundbreaking result and relieve the tension.
This review may take years. But it’s also revived hope for an area of study that had stalled. And it marks an early victory for a broader mathematical program that Kontsevich has championed for decades — one that he hopes will build bridges between algebra, geometry and physics.
“The general perception,” said Paolo Stellari, a mathematician at the University of Milan who was not involved in the work, “is that we might be looking at a piece of the mathematics of the future.” (MORE - details)
INTRO: In August, a team of mathematicians posted a paper claiming to solve a major problem in algebraic geometry — using entirely alien techniques. It instantly captivated the field, stoking excitement in some mathematicians and skepticism in others.
The result deals with polynomial equations, which combine variables raised to powers (like y = x or x2 - 3xy = z2). These equations are some of the simplest and most ubiquitous in mathematics, and today, they’re fundamental to lots of different areas of study. As a result, mathematicians want to study their solutions, which can be represented as geometric shapes like curves, surfaces and higher-dimensional objects called manifolds.
There are infinitely many types of polynomial equations that mathematicians want to tame. But they all fall into one of two basic categories — equations whose solutions can be computed by following a simple recipe, and equations whose solutions have a richer, more complicated structure. The second category is where the mathematical juice is: It’s where mathematicians want to focus their attention to make major advances.
But after sorting just a few types of polynomials into the “easy” and “hard” piles, mathematicians got stuck. For the past half-century, even relatively simple-looking polynomials have resisted classification.
Then this summer, the new proof appeared. It claimed to end the stalemate, offering up a tantalizing vision for how to classify lots of other types of polynomials that have until now seemed completely out of reach.
The problem is that no one in the world of algebraic geometry understands it. At least, not yet. The proof relies on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
Some researchers trust the reputation of one of the paper’s authors, a Fields medalist named Maxim Kontsevich. But Kontsevich also has a penchant for making audacious claims, giving others pause. Reading groups have sprung up in math departments across the world to decipher the groundbreaking result and relieve the tension.
This review may take years. But it’s also revived hope for an area of study that had stalled. And it marks an early victory for a broader mathematical program that Kontsevich has championed for decades — one that he hopes will build bridges between algebra, geometry and physics.
“The general perception,” said Paolo Stellari, a mathematician at the University of Milan who was not involved in the work, “is that we might be looking at a piece of the mathematics of the future.” (MORE - details)