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Topology and physics: a new conjecture

When I was a student, we were all fascinated by first steps the topological approach made into physics. It took a while to understand that the winding of phase around a vortex is always whatever you do; and that had a sweet taste of intellectual victory. We all read review of Mermin : it was so enlighting to learn that even  π in topology has a different, much more profound meaning of homotopy group.

Topology has progressed in years gone, and overwhelms in recent years. There are no more insulators: we’ve topological insulators. They insulate as wet towels, yet are much more profound. The advent of topological superconductors has revolted the field of superconductivity that seemed so stable. And no quantum computing scheme would ever work if topology does not give its protective blessing.

In a kind of conservative rebellition I suggested on Saturday in my talk that perhaps there may be some interesting things in physics that are not based on topology of coordinate space. O boy, how wrong I was. My only consolation is that my wrongness let the truth prevail.

Today Charles Marcus, a Harward professor and most intellectual experimentalist I know, has responded to my talk with a seminal conjecture that I have a great honour to publish.

Marcus’s conjecture

Any result involving integers, including 1 + 1 = 2, can be represented geometrically as a statement of topology, since 1 + 1 = 2 cannot be continuously deformed into any other relation between integers.

Breathtaking. I could have suspected so, why was I so blind…

A note for non-specialist: physics originates from, and is based on counting fingers. For all practical purposes, the result of such counting can be approximated by an integer non-negative number. The conjecture therefore puts physics into the true context: it appears to be a practical exercise in homotopy theory.

A technical note: Charles insisted on presenting his conribution as a conjecture, while the proposition clearly has the status of a theorem. He said he would not present the proof yet. I wonder how he has actually done the proof yet concealed. Being a physist, he could use the traditional medium of this profession, the backside of an envelope. Having understood the importance of topology, he could turn to a margin of a Greek manuscript, the only proper medium for seminal mathematical discoveries since Fermat. Being a conservative rebel, I’m just exploring the consequences of occasional and absolutely unintended dissapearence of this envelop/manuscript. What a wonderful lost for science could it be!

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