bio | website | math.umn.edu/~garrett |
---|---|---|
location | US | |
age | 63 | |
visits | member for | 3 years, 11 months |
seen | 13 mins ago | |
stats | profile views | 5,045 |
I am interested in the application of modern analysis to the spectral theory of Eisenstein series and other automorphic forms, L-functions, and number theory. This includes representation theory of (mostly reductive) Lie groups and p-adic groups, and harmonic analysis on these groups and their homogeneous spaces. Necessarily Schwartz' distributions, Sobolev spaces, and similar things play important supporting roles. I find that a mildly categorical viewpoint is very helpful in keeping things straight.
Mar 27 |
comment |
$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?
As in @PeterMichor's answer, the nuclearity means that genuine tensor products exist, as opposed to the more typical situation that one can have one half or the other, but not both, of the characterizing properties of a tensor product. Some of the notes on my functional analysis page math.umn.edu/~garrett/m/fun talk about the impossibility of having a "genuine" tensor product of Hilbert spaces, and also about nuclear Frechet spaces occuring as suitable (proj) limits of Hilbert spaces. |
Mar 20 |
awarded | Quorum |
Mar 15 |
answered | Tensor product of two elements of the Selberg class |
Mar 7 |
comment |
Averages over integer points of the sphere
In my modular forms course 2013-14 I did the much-easier analogous case of dimension 8n... which illustrates many of the principles without the delicacies and difficulties of the 3-D case. (math.umn.edu/~garrett/m/mfms) |
Mar 3 |
answered | Representing quaternions as matrices |
Feb 28 |
revised |
Unitary representation with fixed Casimir
added 325 characters in body |
Feb 28 |
comment |
Unitary representation with fixed Casimir
Unfortunately I don't remember which H-C paper it was, but probably you can find it by perusing Wallach's or Knapp's books on representation theory. |
Feb 28 |
answered | Unitary representation with fixed Casimir |
Jan 29 |
comment |
Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorenstein variety Calabi-Yau?
Spelling? "Gorenstein"? |
Jan 27 |
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Who first talked about “holes” in homology?
Please pardon yet-one-more anecdotal-personal comment: as a kid, I found the cutting-up of "handle bodies" completely unpersuasive. Rather, the idea that a ("closed") thing that was not a "boundary" of something corresponded to a missing "something", that is, a "hole". The bar (etc) constructions in group (co)homology struck me as making simplicial complexes whose "physical" (co)homology reflected whatever-the-heck-it-was about groups that was desired. It was only later (esp. Nick Katz' lectures on Weil II c. 1974) that is became clear (to me) that "hom-things" did much more... |
Jan 27 |
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Who first talked about “holes” in homology?
One more dim recollection, possibly irrelevant to questions about more-professional history of the terminology: speaking of "what would a 2-D hole be?", I think by the 1960s (or earlier) the sort of "Flatland" sci-fi explicitly suggested to kids (including me) to think that a sphere would look to 4-D entities like a loop looks to us... etc... justifying thinking of the inside of a sphere as a "hole", perhaps. Maybe the sci-fi writers made it up? |
Jan 27 |
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Who first talked about “holes” in homology?
Colin, I don't necessarily disagree at all, but, nevertheless, somehow in the late 1960s, I did manage to convince myself that a 2-sphere has a 2-dimensional hole inside it, and that, indeed, a compact, connected, oriented surface has 2-D hole in it. By now, I don't know whether this was rationalization or suggested by some external source... What would a higher-dimensional hole be, after all? :) |
Jan 27 |
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Who first talked about “holes” in homology?
Yes, certainly people talked that way by the mid-1970s there, and/but I was trying to think of earlier precedents. Certainly my perceptions in the late 1960s were naive, but I do have a vivid recollection of the "holes" business by that point. It may have been a forced interpretation in terms intelligible to a naive kid, given the flimsy excuse of that off-hand comment in Alexandroff. |
Jan 27 |
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Who first talked about “holes” in homology?
My dim recollection of my own reaction was that, once mentioned, the notion of "holes" counted by homology didn't need to be repeated! I still cannot recall where I got the idea that the $n$-th Betti number counted the $n$-dimensional holes... and the "paradox" of having torsion, etc. It would have been late 1960s, whatever the source. But I guess one sees what one wants to see. |
Jan 27 |
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Who first talked about “holes” in homology?
An inexpert comment: I recall getting the impression that homology counted "holes" in the late 1960s from Alexandroff's little book (essentially on combinatorial alg top over coefficients in a field with 2 elements, to avoid wrangling over signs, I suppose). Other sources (now forgotten by me) gave a similar impression late-1960s. Also, I dimly recall a comment that it was Emmy Noether who recommended that homology groups be groups, as opposed to "mere" Betti numbers... |
Jan 25 |
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Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
I think it has been established in various ways that linear combinations of nice $L$-functions do not reliably have non-vanishing properties that the individuals might have, although they'd obviously have the same growth properties. (E.g., the Bombieri-Hejhal paper about linear combinations of the two ideal-class characters for $\sqrt{-5}$.) That is, RH-type results are not at all stable under linear combinations, but Lindelof-type results, or subconvexity results, would be. So, on general considerations already, I'd be surprised if such a thing were true... |
Jan 25 |
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Shortest/Most elegant proof for $L(1,\chi)\neq 0$
In terms of "clearest causation", I still do think that the spectral argument using the constant term of Eisenstein series for $GL_2$ is the most memorable, the most explanatory, and the most suggestive of the broader situation... |
Jan 24 |
revised |
Rankin-Selberg convolution and product of degrees
edited body |
Jan 24 |
answered | Rankin-Selberg convolution and product of degrees |
Jan 24 |
comment |
Rankin-Selberg convolution and product of degrees
This is conjectured to be true, by "Langlands functoriality", but proven in very, very few cases. |