10,624 reputation
22648
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 7 months
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.


Nov
20
awarded  Good Answer
Nov
17
comment Are there non-reflexive abelian topological groups isomorphic to their second dual?
No, unfortunately, I did not, but/and I've become ever more distrustful of Banach spaces, in contrast to Hilbert spaces (and families of Hilbert spaces), so I might have not been paying attention... Sorry to be unhelpful.
Nov
16
comment Are there any books that take a 'theorems as problems' approach?
@RyanRiech, yes, I'm afraid that many people, in good faith, somehow misinterpret, or over-interpret, suffering as genuinely virtuous activity. In fact, I don't think there's much moral virtue, and certainly not professional virtue, in re-inventing really-crappy wheels... given that one's time and energy are finite, especially. But various sado-masochistic relationship themes have an enduring popularity in the species... so there-we-are.
Nov
16
comment Are there any books that take a 'theorems as problems' approach?
The point is that "understanding" this text is a diagnostic, an exam, a test, in itself. The book is not a learning device, in many regards, but a test, in itself, ... as are (too...) many.
Nov
11
comment Non-trivial global solution for Dirichlet eigenvalue problem
@AlexandreEremenko, ... that was playing on the non-condition(s) on the unit circle, nothing more. Indeed, this possibility was meant to be evidence for the dys-formulation. That is, on a not-connected open, a real-analytic function can be $0$ on one connected component, and something else on another. Nothing subtle. Probably you are inadvertently presuming that the problem was better posed than it really was, at least in the original.
Nov
11
comment Non-trivial global solution for Dirichlet eigenvalue problem
@AlexandreEremenko, I am not asserting anything you surely don't know... only that the original formulation allowed many solutions without boundary constraints and other things, probably unintended. E.g., yes, locally solutions are real-analytic, certainly, but/and without suitable constraints we'd not have a self-adjoint operator, for example, in case that were accidentally wanted/presumed. E.g., if the equation is to be satisfied off the unit circle with no boundary conditions on the unit circle... E.g., on the real line, $e^{-c|x|}$ is an eigenfunction for the Laplacian off ${0}$, etc.
Nov
11
comment Non-trivial global solution for Dirichlet eigenvalue problem
Some sort of tighter constraints are needed... For example, for $D$ the unit circle, $(\Delta-\lambda)f=0$ has solutions non-zero outside the circle and $0$ inside, for essentially all complex $\lambda$...
Nov
7
comment A question about pointwise convergence of Fourier transform in $N$-dimensions
A too-short answer, but it seems that any comment would likely get lost in the others... Also, not responding directly to the literal question, but to the context: the notion of "wave-front set" would seem to me to be one of the concepts the questioner might find useful in refining the formulation of the issue (e.g., refining to the point that the assertions are not easily shown faulty in various ways, e.g., coordinate-(in)dependence as @WillieWong comments).
Nov
2
comment Regularity of random Fourier series
@YemonChoi, it was not clear to me whether "random" was meant in a colloquial sense or some of several possible technical senses. Baire category arguments give one sort of argument about "generic", but maybe not "random" in a serious sense. I think I have no simple, immediate opinion about a more serious "random" notion, though I think something can be said...
Nov
2
answered Regularity of random Fourier series
Nov
2
comment Regularity of random Fourier series
Don't you need $x^{-k-n-\epsilon}$ in your second? The sawtooth in one dimension has $\ell$th coefficient $\ell^{-1}$, etc. Similarly, isn't Sobolev imbedding's index-shift essentially sharp?
Oct
30
comment Weyl group actions on 0-weight spaces
Very useful scholarship... bravo.
Oct
15
comment Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)
@JoelFine and Jose'F.-O'F, it seems to me very funny that at this moment in history quaternions could conceivably be "less elementary" than Lie algebras (although, I hasten to add, I am a very great fan of Lie algebras, etc, etc, etc, ...!!!) Pity that this example of-so-many-things has become obscure...
Oct
14
comment Has uniform ellipticity implications on the spectrum?
Isn't the usual Laplacian on $\mathbb R^n$ "uniformly elliptic", but with purely continuous spectrum?
Oct
11
comment Non invertibility of certain integral arising from group action
Ah, ok, in light of your edits, I'll remove my earlier comments in a little while...
Oct
10
answered The periodic architecture underlying the natural numbers
Oct
9
reviewed Close Largest eigenvalue of the sum of hermitian matricies
Oct
7
comment on the Rankin-Selberg L-function
Yes, I'd be entirely willing to believe that there is some delicacy in that situation...
Oct
7
comment on the Rankin-Selberg L-function
With cuspidal data and $m\not=n$ it is relatively straightforward to prove that there is no pole in the right half-plane from the critical line. For $m=n-1$ the form of the integral representation (the "Hecke" form) also makes clear that the $L$-function is entire, for cuspidal data. For $m=n$ but non-contragredient cuspidal data, again the form of the integral repn makes clear the entire-ness. All other cases are more complicated...
Oct
6
revised Fourier approximation error in L^2 for piecewise continuous functions
edited body