Science & Technology

MIT Mathematicians Solve an Old Geometry Problem on Equiangular Lines

In an everyday icosahedron (purple), six important inside diagonals (purple strains) make equal angles with one another. Credit score: Picture: Zilin Jiang

What number of strains might be pairwise separated by the identical angle in excessive dimensions? Geometry breakthrough offers new insights into spectral graph principle.

Equiangular strains are strains in area that move via a single level, and whose pairwise angles are all equal. Image in 2D the three diagonals of an everyday hexagon, and in 3D, the six strains connecting reverse vertices of an everyday icosahedron (see the determine above). Mathematicians will not be restricted to a few dimensions, nonetheless. 

“In excessive dimensions, issues actually get fascinating, and the chances can appear limitless,” says Yufei Zhao, assistant professor of arithmetic.

However they aren’t limitless, based on Zhao and his workforce of MIT mathematicians, who sought to resolve this drawback on the geometry of strains in high-dimensional area. It’s an issue that researchers have been puzzling over for not less than 70 years.  

Their breakthrough determines the utmost doable variety of strains that may be positioned in order that the strains are pairwise separated by the identical given angle. Zhao wrote the paper with a bunch of MIT researchers consisting of undergraduates Yuan Yao and Shengtong Zhang, PhD scholar Jonathan Tidor, and postdoc Zilin Jiang. (Yao just lately began as an MIT math PhD scholar, and Jiang is now a school member at Arizona State College). Their paper shall be revealed within the January 2022 problem of Annals of Arithmetic.

“The proof labored out cleanly and superbly,” says Yufei Zhao (middle). “We had a lot enjoyable working on this drawback collectively.” Left to proper: Zilin Jiang, Jonathan Tidor, Zhao, Yuan Yao, and Shengtong Zhang. Credit score: Picture: Sandi Miller/MIT Division of Arithmetic

The arithmetic of equiangular strains might be encoded utilizing graph principle. The paper supplies new insights into an space of arithmetic often called spectral graph principle, which supplies mathematical instruments for finding out networks. Spectral graph principle has led to vital algorithms in pc science similar to Google’s PageRank algorithm for its search engine. 

This new understanding of equiangular strains has potential implications for coding and communications. Equiangular strains are examples of “spherical codes,” that are vital instruments in info principle, permitting totally different events to ship messages to one another over a loud communication channel, similar to these despatched between NASA and its Mars rovers.

The issue of finding out the utmost variety of equiangular strains with a given angle was launched in a 1973 paper of P.W.H. Lemmens and J.J. Seidel.

“This can be a stunning outcome offering a surprisingly sharp reply to a well-studied drawback in extremal geometry that obtained a substantial quantity of consideration beginning already within the ’60s,” says Princeton College professor of arithmetic Noga Alon.

The brand new work by the MIT workforce supplies what Zhao calls “a satisfying decision to this drawback.”

“There have been some good concepts on the time, however then folks bought caught for almost three a long time,” Zhao says. There was some vital progress made just a few years in the past by a workforce of researchers together with Benny Sudakov, a professor of arithmetic on the Swiss Federal Institute of Know-how (ETH) Zurich. Zhao hosted Sudakov’s go to to MIT in February 2018 when Sudakov spoke within the combinatorics analysis seminar about his work on equiangular strains.

Jiang was impressed to work on the issue of equiangular strains based mostly on the work of his former PhD advisor Bukh Boris at Carnegie Mellon College. Jiang and Zhao teamed up in the summertime of 2019, and had been joined by Tidor, Yao, and Zhang. “I needed to discover a good summer season analysis undertaking, and I assumed that this was an awesome drawback to work on,” Zhao explains. “I assumed we’d make some good progress, nevertheless it was undoubtedly past my expectations to utterly remedy all the drawback.”

The analysis was partially supported by the Alfred P. Sloan Basis and the Nationwide Science Basis. Yao and Zhang participated within the analysis via the Division of Arithmetic’ Summer time Program for Undergraduate Analysis (SPUR), and Tidor was their graduate scholar mentor. Their outcomes had earned them the arithmetic division’s Hartley Rogers Jr. Prize for one of the best SPUR paper.

“It is among the most profitable outcomes of the SPUR program,” says Zhao. “It’s not each day {that a} long-standing open drawback will get solved.”

One of many key mathematical instruments used within the answer is called spectral graph principle. Spectral graph principle tells us the best way to use instruments from linear algebra to know graphs and networks. The “spectrum” of a graph is obtained by turning a graph right into a matrix and its eigenvalues.

“It’s as in the event you shine an intense beam of sunshine on a graph after which look at the spectrum of colours that come out,” Zhao explains. “We discovered that the emitted spectrum can by no means be too closely concentrated close to the highest. It seems that this basic reality in regards to the spectra of graphs has by no means been noticed.”

The work offers a brand new theorem in spectral graph principle — {that a} bounded diploma graph will need to have sublinear second eigenvalue multiplicity. The proof requires intelligent insights relating the spectrum of a graph with the spectrum of small items of the graph.

“The proof labored out cleanly and superbly,” Zhao says. “We had a lot enjoyable working on this drawback collectively.”

Reference: “Equiangular strains with a hard and fast angle” by Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao, Accepted, Annals of Arithmetic.

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