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The rungs of mathematical discovery

by Angela Herring of news@Northeastern

More than a decade ago, math­e­matics pro­fessor Valerio Toledano Laredo was puz­zling over the rela­tion­ship between the sym­me­tries of macro­scopic and micro­scopic sys­tems when he dis­cov­ered a brand new set of dif­fer­en­tial equa­tions. Toledano Laredo named them the Casimir equa­tions in honor of the physi­cist Hen­drick Casimir, who, in the 1930s, dis­cov­ered a key ingre­dient in their construction.

Toledano Laredo recently received a grant from the National Sci­ence Foun­da­tion to con­tinue exploring the Casimir equa­tions’ unex­pected prop­er­ties. Since their dis­covery, the equa­tions have been turning up in the far­thest cor­ners of math­e­matics — from rep­re­sen­ta­tion theory (the math­e­mat­ical study of sym­metry), to analysis (the field of cal­culus) and alge­braic geometry.

They are also rel­e­vant to both string theory, which seeks to unify Einstein’s theory of gen­eral rel­a­tivity with quantum physics, and sta­tis­tical mechanics, which explores large assem­blies of small par­ti­cles such as mol­e­cules in a gas or pho­tons in a laser beam.

The Casimir equa­tions also govern a phe­nom­enon in alge­braic geom­etry known as wall-​​crossing, wherein slight vari­a­tions in the equa­tions gov­erning the shape of a curve or a sur­face cause it to dra­mat­i­cally morph—like a bubble pop­ping off the sur­face of the bath­water and drifting away.

Toledano Laredo said he and his stu­dents have grad­u­ally begun to under­stand these mul­ti­fac­eted equa­tions better. Building on work from a pre­vious NSF col­lab­o­ra­tive grant with researchers at Brown, Columbia and MIT, Toledano Laredo will explore in par­tic­ular a prop­erty that he dis­cov­ered in con­junc­tion with Columbia Uni­ver­sity assis­tant pro­fessor Sachin Gautam, who grad­u­ated from North­eastern in 2011 under Toledano Laredo.

“The equa­tions seem to exist in more and more sophis­ti­cated guises that are arranged like the rungs of a ladder,” said Toledano Laredo.

One com­po­nent of the new project will involve studying the equa­tions on the higher rungs. A second com­po­nent of the project is a spin-​​off from the first and has taken on a life of its own. As Toledano Laredo put it, “It is con­cerned with exactly under­standing the rela­tion between the rungs.”

Sur­pris­ingly, the less sophis­ti­cated equations—those on the lower rungs—seem to con­tain all the tools nec­es­sary to under­stand the more sophis­ti­cated ones, he explained. “People know very well how to go up the ladder and think that what’s below is actu­ally sim­pler,” he said. But as he and Gautam have shown, this is not actu­ally the case. The new grant will allow Toledano Laredo to probe that fact in greater depth.

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