11,033 reputation
22651
bio website math.umn.edu/~garrett
location US
age 63
visits member for 3 years, 11 months
seen 13 mins ago

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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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.