most recent 30 from http://mathoverflow.net 2013-06-19T03:01:36Z http://mathoverflow.net/feeds/question/24090 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24090/what-is-so-spectral-about-spectra Martin Brandenburg 2010-05-10T12:08:05Z 2011-03-29T18:30:40Z

<p>What is the background of the terminology of <a href="http://en.wikipedia.org/wiki/Spectrum_%28homotopy_theory%29" rel="nofollow">spectra</a> in homotopy theory? In what extend does the name "spectrum" fit to the definition and the properties? Also, are there relations to other spectra in mathematics (algebraic geometry, operator theory)?</p> <p>PS: The title is an allusion to <a href="http://mathoverflow.net/questions/17357/what-is-so-spectral-about-spectral-sequences" rel="nofollow">this question</a> ;-)</p>

http://mathoverflow.net/questions/24090/what-is-so-spectral-about-spectra/24107#24107 Qfwfq 2010-05-10T14:06:51Z 2011-03-29T18:30:40Z

<p>It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb <em>spectare</em> (paradigm: <em>specto, -as, -avi, -atum, -are</em>), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.</p> <p><strong>Edit:</strong> after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: <em>spècere</em> (or interchangeably <em>spicere</em>)= "to see", from which comes the root <em>spec-</em> of the latin word <em>spectrum</em>= "something that appears, that manifests itself, vision". Furthermore, <em>spec-</em> = "to see", <em>-trum</em> = "instrument" (like in <em>spec-trum</em>). Also the term "spectrum" in astronomy and optics has the same origin.</p> <p>In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).</p> <p>I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from <em>in + spècere</em>...] step by step the deep properties of some cohomological constructions.</p> <p>Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?</p> <p><strong>Edit:</strong> The following paragraph from the wikipedia article on "primon gas" seems to support my guess:</p> <blockquote> <p>"The connections between number theory and quantum field theory can be somewhat further extended into connections between topological field theory and K-theory, where, corresponding to the example above, <strong>the spectrum of a ring takes the role of the spectrum of energy eigenvalues</strong>, the prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the Dirichlet characters, and so on"</p> </blockquote>