Science & Nature

MIT Mathematicians Solve an Old Geometry Problem on Equiangular Lines

Regular Icosahedron

In a daily icosahedron (purple), six principal inside diagonals (crimson traces) make equal angles with one another. Credit: Image: Zilin Jiang

How many traces could be pairwise separated by the identical angle in excessive dimensions? Geometry breakthrough provides new insights into spectral graph idea.

Equiangular traces are traces in area that move by a single level, and whose pairwise angles are all equal. Picture in 2D the three diagonals of a daily hexagon, and in 3D, the six traces connecting reverse vertices of a daily icosahedron (see the determine above). Mathematicians aren’t restricted to 3 dimensions, nevertheless. 

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

But they aren’t limitless, in accordance with Zhao and his staff of MIT mathematicians, who sought to resolve this drawback on the geometry of traces in high-dimensional area. It’s an issue that researchers have been puzzling over for at the very least 70 years.  

Their breakthrough determines the utmost doable variety of traces that may be positioned in order that the traces are pairwise separated by the identical given angle. Zhao wrote the paper with a gaggle of MIT researchers consisting of undergraduates Yuan Yao and Shengtong Zhang, PhD scholar Jonathan Tidor, and postdoc Zilin Jiang. (Yao lately began as an MIT math PhD scholar, and Jiang is now a college member at Arizona State University). Their paper might be printed within the January 2022 difficulty of Annals of Mathematics.

Mathematicians Solve Old Geometry Problem

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

The arithmetic of equiangular traces could be encoded utilizing graph idea. The paper gives new insights into an space of arithmetic referred to as spectral graph idea, which gives mathematical instruments for learning networks. Spectral graph idea has led to vital algorithms in laptop science comparable to Google’s PageRank algorithm for its search engine. 

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

The drawback of learning the utmost variety of equiangular traces with a given angle was launched in a 1973 paper of P.W.H. Lemmens and J.J. Seidel.

“This is an exquisite consequence offering a surprisingly sharp reply to a well-studied drawback in extremal geometry that acquired a substantial quantity of consideration beginning already within the ’60s,” says Princeton University professor of arithmetic Noga Alon.

The new work by the MIT staff gives what Zhao calls “a satisfying decision to this drawback.”

“There have been some good concepts on the time, however then folks received caught for almost three many years,” Zhao says. There was some vital progress made a number of years in the past by a staff of researchers together with Benny Sudakov, a professor of arithmetic on the Swiss Federal Institute of Technology (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 traces.

Jiang was impressed to work on the issue of equiangular traces primarily based on the work of his former PhD advisor Bukh Boris at Carnegie Mellon University. Jiang and Zhao teamed up in the summertime of 2019, and have been joined by Tidor, Yao, and Zhang. “I needed to discover a good summer time analysis venture, and I believed that this was an important drawback to work on,” Zhao explains. “I believed we’d make some good progress, however it was undoubtedly past my expectations to utterly resolve the whole drawback.”

The analysis was partially supported by the Alfred P. Sloan Foundation and the National Science Foundation. Yao and Zhang participated within the analysis by the Department of Mathematics’ Summer Program for Undergraduate Research (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 on daily basis {that a} long-standing open drawback will get solved.”

One of the important thing mathematical instruments used within the resolution is called spectral graph idea. Spectral graph idea tells us how you can use instruments from linear algebra to grasp graphs and networks. The “spectrum” of a graph is obtained by turning a graph right into a matrix and taking a look at its eigenvalues.

“It is as when you shine an intense beam of sunshine on a graph after which study 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 concerning the spectra of graphs has by no means been noticed.”

The work provides a brand new theorem in spectral graph idea — {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 engaged on this drawback collectively.”

Reference: “Equiangular traces with a set angle” by Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao, Accepted, Annals of Mathematics.
arXiv:1907.12466

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