Mathematicians are using Srinivasa Ramanujan's century-old formulae to push the boundaries of high-performance computing and verify the accuracy of calculations.
Manifolds are spaces that look Euclidean when you zoom in on any one of their points. Introduced by Bernhard Riemann, they have become a mathematical staple in fields like geometry, physics, and data analysis.
For decades, mathematicians have struggled to understand matrices that reflect both order and randomness, like those that model semiconductors.
A new method could change that.
Recent research has made a significant mathematical advance in understanding Anderson localization, the phenomenon where disorder in a material (like impurities in silicon) can stop electron flow. Researchers proved that, for a simplified model called band matrices, electrons do become trapped ("localized") with enough disorder. This breakthrough, achieved by Yan Yau and Jun Yin’s team, uses a new mathematical technique and brings us closer to fully understanding Anderson’s original model and designing materials with specific electronic properties. It’s a key step in understanding systems between order and randomness.
A new method could change that.
Recent research has made a significant mathematical advance in understanding Anderson localization, the phenomenon where disorder in a material (like impurities in silicon) can stop electron flow. Researchers proved that, for a simplified model called band matrices, electrons do become trapped ("localized") with enough disorder. This breakthrough, achieved by Yan Yau and Jun Yin’s team, uses a new mathematical technique and brings us closer to fully understanding Anderson’s original model and designing materials with specific electronic properties. It’s a key step in understanding systems between order and randomness.
The Langlands programme has inspired and befuddled mathematicians for more than 50 years. A major advance has now opened up new worlds for them to explore.
The article details the recent proof of the geometric Langlands conjecture, a significant advancement in mathematics that validates a decades-old program aiming for a "grand unified theory" of the field. Led by Dennis Gaitsgory and Sam Raskin, the proof—spanning five papers and nearly 1,000 pages—is expected to open new avenues of research and potentially bridge connections between mathematics and theoretical physics, particularly in understanding symmetries in quantum field theory. While not a complete solution to the broader Langlands program, it provides strong evidence for its underlying principles and offers new tools for tackling complex mathematical problems.
The article details the recent proof of the geometric Langlands conjecture, a significant advancement in mathematics that validates a decades-old program aiming for a "grand unified theory" of the field. Led by Dennis Gaitsgory and Sam Raskin, the proof—spanning five papers and nearly 1,000 pages—is expected to open new avenues of research and potentially bridge connections between mathematics and theoretical physics, particularly in understanding symmetries in quantum field theory. While not a complete solution to the broader Langlands program, it provides strong evidence for its underlying principles and offers new tools for tackling complex mathematical problems.
Physicists are revisiting the chaotic region near singularities within black holes, utilizing new mathematical tools to potentially reconcile general relativity and quantum mechanics and gain a deeper understanding of space and time.
analogue I-K of the Laplacian in potential theory
