Professional mathematicians have been stunned by the progress amateurs have made in solving long-standing problems with the assistance of AI tools, and say it could lead to a new way of doing mathematics
Category: mathematics – Page 12
John Forbes Nash Jr
(June 13, 1928 – May 23, 2015), known and published as John Nash, was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. [ 1 ] [ 2 ] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Nobel Prize in Economics. [ 3 ] In 2015, Louis Nirenberg and he were awarded the Abel Prize for their contributions to the field of partial differential equations.
As a graduate student in the Princeton University Department of Mathematics, Nash introduced a number of concepts (including the Nash equilibrium and the Nash bargaining solution), which are now considered central to game theory and its applications in various sciences. In the 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in Riemannian geometry. This work, also introducing a preliminary form of the Nash–Moser theorem, was later recognized by the American Mathematical Society with the Leroy P. Steele Prize for Seminal Contribution to Research. Ennio De Giorgi and Nash found, with separate methods, a body of results paving the way for a systematic understanding of elliptic and parabolic partial differential equations.
Wormholes may not exist—we’ve found they reveal something deeper about time and the universe
Wormholes are often imagined as tunnels through space or time—shortcuts across the universe. But this image rests on a misunderstanding of work by physicists Albert Einstein and Nathan Rosen.
In 1935, while studying the behavior of particles in regions of extreme gravity, Einstein and Rosen introduced what they called a “bridge”: a mathematical link between two perfectly symmetrical copies of spacetime. It was not intended as a passage for travel, but as a way to maintain consistency between gravity and quantum physics. Only later did Einstein–Rosen bridges become associated with wormholes, despite having little to do with the original idea.
But in new research published in Classical and Quantum Gravity, my colleagues and I show that the original Einstein–Rosen bridge points to something far stranger—and more fundamental—than a wormhole.
The Math Behind Evo Devo (TMEB #3)
The math behind Evo-devo~
Uri Alon’s Book:
Jim Collins paper:
https://www.researchgate.net/publication/12654725_Constructi…ichia_coli.
https://www.nature.com/articles/s41467-017-01498-0
The math behind fly development:
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.
Music:
City Life – Artificial. Music (No Copyright Music)
Link: https://www.youtube.com/watch?v=caT3j… ure Water by Meydän Link: • Meydän — Pure Water [Creative Commons — CC… Forever Sunrise — by Jonny Easton Link:
• Forever Sunrise — Soft Inspirational Piano… Softwares used: Manim CE Keynote.
Pure Water by Meydän.
Link: https://youtu.be/BU85yzb0nMU
Forever Sunrise — by Jonny Easton.
Link: https://youtu.be/9Xndx7nhGAs.
Softwares used:
Physics of foam strangely resembles AI training
Foams are everywhere: soap suds, shaving cream, whipped toppings and food emulsions like mayonnaise. For decades, scientists believed that foams behave like glass, their microscopic components trapped in static, disordered configurations.
Now, engineers at the University of Pennsylvania have found that foams actually flow ceaselessly inside while holding their external shape. More strangely, from a mathematical perspective, this internal motion resembles the process of deep learning, the method typically used to train modern AI systems.
The discovery could hint that learning, in a broad mathematical sense, may be a common organizing principle across physical, biological and computational systems, and provide a conceptual foundation for future efforts to design adaptive materials. The insight could also shed new light on biological structures that continuously rearrange themselves, like the scaffolding in living cells.
Scientists use string theory to crack the code of natural networks
For more than a century, scientists have wondered why physical structures like blood vessels, neurons, tree branches, and other biological networks look the way they do. The prevailing theory held that nature simply builds these systems as efficiently as possible, minimizing the amount of material needed. But in the past, when researchers tested these networks against traditional mathematical optimization theories, the predictions consistently fell short.
The problem, it turns out, was that scientists were thinking in one dimension when they should have been thinking in three. “We were treating these structures like wire diagrams,” Rensselaer Polytechnic Institute (RPI) physicist Xiangyi Meng, Ph.D., explains. “But they’re not thin wires, they’re three-dimensional physical objects with surfaces that must connect smoothly.”
This month, Meng and colleagues published a paper in the journal Nature showing that physical networks in living systems follow rules borrowed from an unlikely source: string theory, the exotic branch of physics that attempts to explain the fundamental structure of the universe.