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Harmonic analysis
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Networks Hold the Key to a Decades-Old Problem About Waves
Mathematicians are still trying to understand fundamental properties of the Fourier transform, one of their most ubiquitous and powerful tools. A new result marks an exciting advance toward that goal.
What Is the Fourier Transform?
Amid the chaos of revolutionary France, one man’s mathematical obsession gave way to a calculation that now underpins much of mathematics and physics. The calculation, called the Fourier transform, decomposes any function into its parts.
At 17, Hannah Cairo Solved a Major Math Mystery
After finding the homeschooling life confining, the teen petitioned her way into a graduate class at Berkeley, where she ended up disproving a 40-year-old conjecture.
Graduate Student Solves Classic Problem About the Limits of Addition
A new proof illuminates the hidden patterns that emerge when addition becomes impossible.
‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.
Monumental Proof Settles Geometric Langlands Conjecture
In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.
‘Sensational’ Proof Delivers New Insights Into Prime Numbers
The proof creates stricter limits on potential exceptions to the famous Riemann hypothesis.
Number of Distances Separating Points Has a New Bound
Mathematicians have struggled to prove Falconer’s Conjecture, a simple, but far-reaching, hypothesis about the distances between points. They’re finally getting close.
Merging Fields, Mathematicians Go the Distance on Old Problem
Mathematicians have illuminated what sets of points can look like if the distances between them are all whole numbers.