11,103 reputation
22751
bio website math.umn.edu/~garrett
location US
age 63
visits member for 4 years
seen 1 hour 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.


Apr
20
comment Atkin-Lehner theory for nonholomorphic Eisenstein series
For $GL_2$, all Eisenstein series generate principal series (possibly ramified at some places) everywhere locally. So any questions are not only local, but should be explicitly answerable, since there is no supercuspidal stuff interfering.
Apr
17
comment Primes as uncorrelated random variables
@JonMarkPerry, :) the point is that the notion of "probability" of primality is not well-founded, although an extremely intuitive/engaging idea. :) That it is not (at this time in history) something that generates causality, that is, that gives true proof, is disappointing, but a contingent fact.
Apr
17
comment Primes as uncorrelated random variables
@JonMarkPerry, it's not that I have an objection to your remark, but that the terseness of it might mislead naive people. That is, the ramifications of "true facts" are often subtler than people generally understand, etc. That is, the limits of heuristics are not-at-all widely understood, and, in fact, I do strongly claim, there is no wide-spread procedure for distinguishing good heuristics from bad, etc. Rhyming is not truth. Catchiness is not truth. Yet we do hope (as humans) that such stuff does correctly suggest truth, etc. That's all I meant...
Apr
17
comment Primes as uncorrelated random variables
@JonMarkPerry, I hope you are being a little bit facetious, or else you are possibly misleading the questioner or other naive souls.
Apr
9
comment What's the minimum amount of knowledge to start doing research?
@JoelDavidHamkins, your example might be about belief impeding one, rather than knowledge, ... if we can distinguish these. E.g., belief that "mathematics" (whatever that is) is captured by the formal, orthodox litany of school-mathematics, etc. I'd sincerely claim that school-math, and much cliched formal mathematics, is indeed an excellent approximation of ... something... but can be expected to fail in "edge cases".
Apr
9
answered What's the minimum amount of knowledge to start doing research?
Apr
4
awarded  Yearling
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.