Researchers Solve First Problem From Mathematical Physics Wish List

For every popular list of unsolved problems, there are scholars and students dreaming of — and working towards — solving the puzzles they contain. Many search for creative solutions, only to reach dead ends. Yet, rarely, knowledge and dedication align, giving birth to a solution to one of these open mysteries. Such a solution was recently reached by the mathematical physics community.

The newly solved problem relates to the quantum Hall effect. The Hall effect “was discovered in a groundbreaking experiment by Edwin Hall in 1879 that showed, for the first time, that electric currents in a metal can be deflected in the presence of a magnetic field perpendicular to the surface,” a news release notes.

101 years later, Klaus von Klitzing, a German physicist, modified Hall’s original experiment, conducting it at a lower temperature and in the presence of a stronger magnetic field.  Through this experiment he discovered the quantum Hall effect. What’s more, he was awarded a 1985 Nobel Prize in physics for this breakthrough.

Comparing these two effects

The Hall effect happens when a magnetic field deflects the movement of electrons. This can be observed when a magnetic field is placed at a right angle to a metal band that has electrical current flowing through it. In contrast, the quantum Hall effect happens at places where a conductor, such as copper, meets a semiconductor, such as silicon. “In this effect, changes in the magnetic field result in changes in what is known as Hall conductance that vary in steps of whole-number multiples of a constant,” the Nobel Prize website notes.

The recently solved problem

Michael Aizenman, professor of physics and mathematics at Princeton University and the former president of the International Association of Mathematical Physics, maintains what could be called a “wish list” of unsolved problems in mathematical physics. To date, there have been two major points of progress related to the problems on the list. One is a partial solution, yet that progress still resulted in two Fields Medals. (Those high honors are awarded every four years to a maximum of four mathematicians under the age of 40 in recognition of their “outstanding mathematical achievement for existing work and for the promise of future achievement,” the website for the International Mathematical Union notes.)

The second major point of progress? The quantum Hall conductance problem that was recently solved by Matthew Hastings, a quantum information theorist and mathematical physicist at Microsoft, and Spiros Michalakis, staff researcher and outreach manager in the Institute for Quantum Information and Matter at Caltech.

Their work, which began in 2008 when Michalakis was a postdoctoral researcher at Los Alamos National Laboratory under the advisement of Hastings, builds on other results following the discovery of the quantum Hall effect. While von Klitzing’s experiments showed that Hall conductance varies in steps that are integer multiples of a constant, later follow-up work showed something even more surprising.

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