Networks Hold the Key to a Decades-Old Problem About Waves
8 days ago
- #Mathematical Breakthrough
- #Graph Theory
- #Fourier Analysis
- Joseph Fourier introduced the Fourier transform, a technique to express functions as sums of simple waves, widely used in various fields.
- Sarvadaman Chowla's 1965 problem questioned the minimum value of a simple Fourier transform (sum of cosine waves), highlighting gaps in mathematical understanding.
- For decades, progress on Chowla's cosine problem was minimal, with the best result from 2004 by Imre Ruzsa being significantly weaker than Chowla's conjecture.
- In September, four mathematicians (Zhihan Jin, Aleksa Milojević, István Tomon, Shengtong Zhang) made the first significant advance in 20 years, using graph theory rather than traditional Fourier analysis.
- The problem was reformulated using Cayley graphs, where eigenvalues correspond to values of the cosine sum, linking graph theory to Fourier analysis.
- The team's work on MaxCut in graph theory provided tools to analyze eigenvalues, leading to a breakthrough in understanding Chowla's problem.
- Their result showed that for any set of N integers, the cosine sum's minimum is less than −N^(1/10), marking the first power-law bound aligned with Chowla's conjecture.
- Shortly after, Benjamin Bedert improved the bound to −N^(1/7) using traditional Fourier analysis, further closing the gap to Chowla's conjecture.
- The breakthrough suggests deeper connections between graph theory and Fourier analysis, potentially opening new avenues for research in both fields.