Revealing the secrets of 'atomic geometry': machine learning is driving the development of mathematics

#An algebraic variety is a set defined by multiple polynomial equations. It is an important concept in algebraic geometry and studies the properties of the set of solutions to polynomial equations in geometric space. The equations of algebraic varieties can be of any dimension, and can be equations in the field of real numbers or equations in the field of complex numbers. Studying the properties of algebraic varieties can help us understand the distribution and geometric form of the roots of polynomial equations

Algebraic geometry is a discipline that integrates the two branches of mathematics, algebra and geometry. On the one hand, it involves algebra, that is, the study of the properties and solutions of equations; on the other hand, it also involves geometry, that is, the study of the properties and characteristics of shapes. The goal of algebraic geometry is to apply abstract algebraic methods to geometry to solve problems related to complex and specific shapes, surfaces, spaces and curves

The basic problem of algebraic geometry is to Classifying the solution sets of polynomial equations is simply to classify the space. The basic object of its research is called algebraic variety, which is the geometric representation of the solution set of polynomial equations.

The Fano variety is an important type of algebraic variety. In a sense, they are "atomic pieces" of mathematical shapes. Fano varieties also play an important role in string theory.

Rewritten content: Fano clusters are the basic building blocks of geometric shapes. They are "atomic blocks" of mathematical shapes. The latest research into the classification of Fano clusters includes the analysis of a type of invariance known as quantum periodicity. A quantum period is a sequence of integers used to provide a numerical fingerprint for a Fano cluster. It is speculated that the geometric properties of the Fano cluster can be recovered directly from its quantum period, if this hypothesis holds true

Recently, mathematicians from the University of Nottingham and Imperial College London have used for the first time Machine learning to expand and accelerate the study of "atom shape." These "atomic shapes" are the building blocks that make up the basic geometric shapes of higher dimensions

Specifically, the researchers applied machine learning to a question: does the quantum period of Dimensions? Note that there is no theoretical understanding of this. Research shows that a simple feedforward neural network can determine the dimensions of X with 98% accuracy. On this basis, the researchers established strict asymptotic properties within the quantum period of a class of Fano clusters. These asymptotic properties determine the dimensions of the quantum period of X . The results show that machine learning can pick out structures from complex mathematical data in the absence of theoretical understanding. They also provide positive evidence for the conjecture that the quantum period of the Fano cluster determines the diversity.

The research is titled "Dimensions of Fanno Diversity in Machine Learning" and was published in "Nature Communications" on September 8, 2023

Paper link: https://www.nature.com/articles/s41467-023-41157-1

A few years ago, The research team began work on creating a periodic table of shapes. They called the atomic fragments Fano clusters. The team associated a sequence of numbers called quantum cycles with each shape to provide a "barcode" or "fingerprint" that describes the shape. Recently, they succeeded in quickly sifting through these barcodes by using a new machine learning method that allowed them to identify shapes and their properties, such as the dimensions of each shape. Alexander Kasprzyk said: "For mathematicians, the key step is to identify the patterns in a given problem. This can be very difficult, and some mathematical theories can take years to discover."

Tom Professor Coates said: "This is where artificial intelligence can really revolutionize mathematics, as we have shown that machine learning is a powerful tool for discovering patterns in complex areas such as algebra and geometry."

Sara Veneziale said: "We are very excited about the fact that we can use machine learning in pure mathematics. This will accelerate new insights in the entire field."

Overall, this Research shows that machine learning can discover previously unknown structures in complex mathematical data and is a powerful tool for developing rigorous mathematical results. It also provides evidence for the basic conjecture in the Fano variety program: the regular quantum period of the Fano variety determines this change

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