MIT mathematicians solve an ancient geometric problem about contemporary lines-scientific research-cnBeta.COM

At high dimensional spacing, how many lines can be divided into pairs at the same angle? Advances in geometry gave people a new understanding of spectrogram theory.An isometric line is a line passing through a point in space whose diagonals are equal. Imagine the three diagonal lines of a two-dimensional regular hexagon and the connecting lines of six pairs of perpendiculars of a three-dimensional regular icosahedron (see figure). However, mathematicians do not limit the hypothesis to three-dimensional space.

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Assistant Professor of Mathematics Zhao Yufi said: “At high dimensions, things are really interesting, and the possibilities seem endless. But according to Zhao and his team of MIT mathematicians, they are not infinite. The problem was confusing.Their progress research determined the maximum number of lines that could be placed so that these lines could be divided into pairs.

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Xiaodong Zhao co-authored the study with a team of researchers at the Massachusetts Institute of Technology, including undergraduate students Yuan Yao and Shengdong Zhang, PhD student Jonathan Didor and postgraduate Dr. Jillin Jiang. Yao recently became a PhD student in mathematics at MIT, and Jiang is now a professor at Arizona State University). ) Their article will be published in the January 2022 “Mathematical Yearbook”.

Isometric mathematics can be coded with graph theory. This article presents new insights into the field of mathematics known as spectrogram theory, which provides mathematical tools for studying networks. Spectrogram theory brings important algorithms such as Google’s PageRank algorithm in computer science to its search engine.

This new understanding of isometric has a potential impact on coding and communications. The isometric line is an example of a “spherical code” that is an important tool in information theory, allowing information to be transmitted to each other over a quiet communication channel, such as information sent between NASA and its rover.

The problem of reading maximum isometric lines with a given angle was proposed in 1973 by PWH Lemmens and JJ Seidel in a paper.

Noka Allen (Noka Allen), a professor of mathematics at Princeton University, said: “This is a beautiful answer to a question that has been carefully studied in extreme geometry since the 1960s. It has received a lot of attention.”

“There were some good ideas at the time, but then people stumbled for almost 30 years,” Zhao said. A few years ago, a research team, including Benny Sudakov, a professor of mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich, made some significant progress when Sudakov spoke about his work on isometric lines at an integrated research seminar.

Jiang Carnegie was inspired by the work of Book Boris, a former PhD supervisor at Mellon University, and began to study the problem of isometric lines. Jiang and Zhao teamed up in the summer of 2019 and invited Didor, Yao and Zhang to join.

This research was conducted by Alfred B. Somewhat supported by the Sloan Foundation and the National Science Foundation. Yao and Zhang participated in this research through the Bachelor of Research Summer Program (SPUR) in the field of mathematics, and Didor was their graduate teacher. Their results won them the Hartley Rogers Jr. Outstanding SPUR Paper Award in Mathematics. “This is one of the most successful outcomes of the SPUR project. The long-term open problem is not solved every day.”

One of the main mathematical tools used in solution is called the spectrogram theory. Spectrogram theory tells how to use linear algebraic tools to understand maps and networks. The “spectrum” of the map is obtained by converting the map into a matrix by looking at its eigenvalues.

“It’s like you put strong light on an image and then check the spectrum of colors,” Zhao explained. “We found that the emitted spectrum can never accumulate near the peak. This kind of true spectrum has not been observed.”

This work presents a new theorem in the theory of spectrograms — the boundary degree diagram must have a subdivision second eigenvalue multiplication. This proof requires a unique insight that connects the spectrum of the map with the spectrum of small fragments of the map.

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