Totally beautiful and fascinating: A Unified Theory of Randomness
Humans are odd creatures. You get Donald Trump, but then you also get the findings and explorations described in this article by Kevin Hartnett in Quanta:
Standard geometric objects can be described by simple rules — every straight line, for example, is just y = ax + b — and they stand in neat relation to each other: Connect two points to make a line, connect four line segments to make a square, connect six squares to make a cube.
These are not the kinds of objects that concern Scott Sheffield. Sheffield, a professor of mathematics at the Massachusetts Institute of Technology, studies shapes that are constructed by random processes. No two of them are ever exactly alike. Consider the most familiar random shape, the random walk, which shows up everywhere from the movement of financial asset prices to the path of particles in quantum physics. These walks are described as random because no knowledge of the path up to a given point can allow you to predict where it will go next.
Beyond the one-dimensional random walk, there are many other kinds of random shapes. There are varieties of random paths, random two-dimensional surfaces, random growth models that approximate, for example, the way a lichen spreads on a rock. All of these shapes emerge naturally in the physical world, yet until recently they’ve existed beyond the boundaries of rigorous mathematical thought. Given a large collection of random paths or random two-dimensional shapes, mathematicians would have been at a loss to say much about what these random objects shared in common.
Yet in work over the past few years, Sheffield and his frequent collaborator,Jason Miller, a professor at the University of Cambridge, have shown that these random shapes can be categorized into various classes, that these classes have distinct properties of their own, and that some kinds of random objects have surprisingly clear connections with other kinds of random objects. Their work forms the beginning of a unified theory of geometric randomness.
“You take the most natural objects — trees, paths, surfaces — and you show they’re all related to each other,” Sheffield said. “And once you have these relationships, you can prove all sorts of new theorems you couldn’t prove before.”
In the coming months, Sheffield and Miller will publish the final part of a three-paper series that for the first time provides a comprehensive view ofrandom two-dimensional surfaces — an achievement not unlike the Euclidean mapping of the plane.
“Scott and Jason have been able to implement natural ideas and not be rolled over by technical details,” said Wendelin Werner, a professor at ETH Zurich and winner of the Fields Medal in 2006 for his work in probability theory and statistical physics. “They have been basically able to push for results that looked out of reach using other approaches.”
A Random Walk on a Quantum String
In standard Euclidean geometry, objects of interest include lines, rays, and smooth curves like circles and parabolas. . .
This reminds me somehow of the recent articles on the use of immunology to fight cancer effectively: use what is already in place to deal with it, but just intervene so that the cancer cannot block the T-cells: free the T-cells to do their work.
That is showing a deeper understanding of nature and how to work within it with the least possible disruption (unlike radiation therapy and chemotherapy, neither of which use the tools already in place in the body, the very tools that evolved with the body and thus extremely well attuned to what is required without hurting the body. Chemotherapy and radiation therapy are both extremely hard on the body.
And we are getting mathematical tools to deal with this more complex world of finding out how better to describe/know nature and thus learn how better to work with(in) it.