11,293 reputation
22752
bio website math.umn.edu/~garrett
location US
age 63
visits member for 4 years, 1 month
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.


1d
comment Resolvent estimate of hyperbolic Laplacian
Doesn't such an estimate hold for any non-positive self-adjoint operator, without using the specifics?
May
19
answered Intertwining Operators Associated to Simple Reflections
May
17
comment Old books still used
Let's not forget Iwasawa's ICM report on the same topic, at the same time, which might have inhibited Tate from publication... until Cassels-Frohlich's 1967. So I myself find "Iwasawa-Tate theory" a more accurate descriptor...
May
17
comment Old books still used
Apart from the details of its revision, I think historically "navigation" was very much a topic in mathematics: spherical trigonometry and all that! Bowditch was certainly a mathematician by U.S. standards, such as they were! :)
May
15
comment Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$
@johnmangual, the earlier parts of my notes do aim at an economical set-up of enough basics to (supposedly) make the later things reasonably intelligible. But/and, as always, a variety of sources is helpful in getting perspective.
May
15
answered Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$
May
14
comment Langlands reciprocity for C*-algebras
It seems to me that the indicated preprint gets so far off the rails so quickly that it's hard to visualize specific details of a plausible, corrected version. E.g., the "convolution" product at the bottom of the first page certainly does not converge, ... but there are many related integrals which do, but, ... And, yes, as @Corbennick comments, not all reductive groups have arithmetic quotients which are algebraic varieties... So the notion of a "corrected" version is too ambiguous to discuss, I fear.
May
13
revised about subgroup of general linear group
deleted 4 characters in body
May
13
answered about subgroup of general linear group
May
13
answered Connection between the two definitions of Siegel Upper Half Space
May
11
comment Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$
@JochenWengenroth's edit/addition explains why these maps factor through limitands...
May
11
answered When does a function space allow for point evaluations?
May
8
comment Meromorphic continuation of Eisenstein series
@GHfromMO, sorry to be so late in responding, but, indeed, if I could usefully clarify, what might it be?
May
8
comment Primer on Eisenstein series
I hasten to comment that E. Lapid has written many wise notes about Eisenstein series... and has noted some crazy subtleties... but/and there is no trivial resolution/outcome.
May
8
answered Primer on Eisenstein series
May
7
comment Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$
Did you mean to write the colimit of the duals, rather than just the $H_i$ themselves? If so, then this would be correct, because a continuous linear map to a normed TVS (such as the scalars) from a projective limit of Banach spaces factors through some limitand, for nearly definitional, easy reasons. Are $H_i$'s Banach? Hilbert? It would not be good to identify every Hilbert space with its dual.
May
5
awarded  Revival
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...