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IN THE NEW York Times, February 14, 2021, Julie Rehmeyer observed the passing, at age 96, of mathematician Isadore Singer. The article caught my eye when she wrote that Singer “bridged a gulf from math to physics.”
Singer’s mathematical “discoveries” (as mathematicians like to call them) include the Kadison-Singer conjecture (formulated in 1959 and proved only in 2013), the Ambrose-Singer theorem, the McKean-Singer formula, and the concept of Ray-Singer torsion.
The Atiyah-Singer Index. “But,” Rehmeyer notes, “all those were dwarfed by his singular contribution, the Atiyah-Singer Index theorem. Together with [British mathematician Michael] Atiyah, he created an unimagined link between the mathematical subfields of analysis and topology—and then united those fields with theoretical physics.”
Rehmeyer continues, “Dr. Singer was the expert in analysis, which is the study of differential equations, used to describe physical phenomena in the language of calculus.
“Dr. Atiyah, meanwhile, specialized in topology,” Rehmeyer continues, “which … considers shapes to be elastic, so that objects can be pulled or squished without changing their fundamental nature.”
The Abel Prize. In the early 1960s, Singer and Atiyah began applying topological tools to differential equations. Their work eventually earned them the 2004 Abel Prize, one of mathematics’ equivalents of the Nobel Prize. Fields Medals are another, specifically awarded to those mathematicians under age 40.
As described in their Abel citation, Singer and Atiyah were honored “for their discovery and proof of the index theorem, bringing together topology, geometry, and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.”
For a deep dive on the subject, Professor John Rognes of the University of Oslo offered insight on the Atiyah-Singer Index theorem in 2004. For even more details, see the Atiyah-Singer Index article in The Princeton Companion to Mathematics, edited by Timothy Gowers, June Barrow-Green, and Imre Leader, Princeton University Press, 2008. Tidbits from this book: “The Atiyah-Singer index theorem is concerned with the existence and uniqueness of solutions to linear partial differential equations of elliptic type.” And “… there is a nine-dimensional manifold that is homeomorphic to the sphere despite not being positively curved in even the weakest sense. (By contrast, the usual sphere is positively curved in the strongest possible sense.)” That is, there’s a surface in nine-space that “looks just like a ball” in one sense, but utterly unlike it in another.
Dynamical Systems Theory. My own mathematical speciality (apart from enjoying teaching undergraduates) was dynamical systems theory, another blending of topology and analysis. I’ve described the area as, sort of, “differential equations without the dirty bits.”
These “dirty bits” arise because many equations of motion inherently admit only approximate solutions.
By contrast, as its name implies, dynamical systems theory studies systems in motion, but in a theoretical way. It asks, “What mathematical structure is required to support a study of things in motion?”
The beauty of this minimalist approach is in broadening potential applications beyond ordinary three-dimensional space (or four-dimensional space-time, or the higher-dimensional whatevers of modern physics).
Living Life to the Fullest Through Math. Julie Rehmeyer quotes mathematician/economist Eric Weinstein on “Is,” as Isadore Singer was known to colleagues: “He drove a sports car when everyone else drove a Volvo,” Dr. Weinstein said. “He showed up with a cravat when no one wore neck gear. He had a very romantic vision of living life to the fullest through math, with style and panache, and he wielded his taste and style as a tool in mathematics like no one else.” ds
© Dennis Simanaitis, SimanaitisSays.com, 2021