MIT mathematicians solve ancient geometric problem on contemporary lines – scientific research-cnBeta.COM

Assistant math professor Zhao Yufi said, “At high dimension things are really interesting and the possibilities seem endless. But according to Zhao and his team of mathematicians at MIT, they are not endless. The problem was confusing. Their research on progress determined the maximum number of rows that could be placed so that those rows could be split into pairs.

Mathematicians solved the problem of old scale geometry.  JPG

Xiaodong Zhao co-authored the study with a team of researchers from the Massachusetts Institute of Technology, including undergraduates Yuan Yao and Shengdong Zhang, doctoral student Jonathan Didor, and 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 Mathematical Yearbook in January 2022.

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

This new understanding of isometry has a potential impact on coding and communications. The isometric line is an example of a “spherical code” which is an important tool in information theory, allowing information to be transmitted to each other through a silent 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 an article.

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

Jiang Carnegie drew on the work of Book Boris, a former thesis 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 them.

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 as part of the Bachelor of Research (SPUR) Summer Program in Mathematics, and Didor was their graduate teacher. Their results earned them the Hartley Rogers Jr. Outstanding SPUR Paper Award in Mathematics. “This is one of the most successful results 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 spectrogram theory. Spectrogram theory explains 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 a strong light on an image and then check the color spectrum,” Zhao explained. “We have 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 spectrogram theory – the border degree diagram must have a second subdivision of eigenvalues. This evidence requires a unique insight that connects the specter of the map with the specter of small fragments of the map.

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