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| Spiros Michalakis. Photo: Caltech |
The problem, related to the "quantum Hall effect," was first proposed in 1999 as one of 13 significant unsolved problems to be included on a list maintained by Michael Aizenman, a professor of physics and mathematics at Princeton University and the former president of the International Association of Mathematical Physics.
Like the "millennium" math challenges put forth by the Clay Mathematics Institute in 2000, the idea behind these problems was to record some of the most perplexing unsolved puzzles in mathematical physics—a field that uses rigorous mathematical reasoning to address physics questions. So far, the problem undertaken by Michalakis is the only one fully solved, while another has been partially solved. Progress made on the partially-solved problem has resulted in two Fields Medals, the highest honor in mathematics.
A Mathematical Approach
Michalakis started working on the problem back in 2008 at Los Alamos National Laboratory, where he was a postdoctoral scholar in mathematics. He built his research on pioneering work by Hastings, his advisor at the time, who had developed new mathematical tools for scrutinizing the quantum Hall effect, based on decades of research by others. Michalakis says that reading through all the previous literature proved almost as challenging as solving the problem itself.
"There was a mountain of research that already existed," he says. "And most of it required advanced knowledge of physics. Coming from a math background, I had to break the problem down into small pieces, each of which I could solve. Basically, I decided to dig under that mountain of knowledge to get to the other side."
A key to the ultimate solution is topology, which is a way of mathematically describing objects by their shapes.
"Topology is the study of properties of shapes that don't change when the shape is bent or stretched," says Hastings. "For example, a donut can be stretched into the shape of a coffee cup, but it cannot be turned into a sphere without tearing. Something like this is behind the Hall effect: the conductance isn't changed even though there are impurities in the material."...
The Communications in Mathematical Physics paper describing the solution is titled "Quantization of Hall Conductance for Interacting Electrons on a Torus."
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Additional resources
Matthew B. Hastings et al. Quantization of Hall Conductance for Interacting Electrons on a Torus, Communications in Mathematical Physics (2014).
DOI: 10.1007/s00220-014-2167-x
Source: Phys.org
