9,765 reputation
22143
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 4 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.


3h
comment Is real analytic function good enough (see problem)?
@GHfromMO, I think people "believed in" a "principle of permanence" for some time before the notion of holomorphic function was made explicit. And, more often these days I would call it "the identity principle"...
6h
comment Existence of real modular function with specific behavior as $q\to 0$
@fernando, this construction wouldn't be able to give you a function vanishing along $q=\overline{q}$, anyway, because then the corresponding $f$ would be of absolute value $1$ on the imaginary axis, so couldn't go to $0$ at $i\infty$. But on the compactified modular curve $\isom \mathbb P^1$, giving up the Riemannian structure inherited from the upper half-plane, the $q$ is a local coordinate at the image of $i\infty$, and the fundamental solution for the (spherical) Laplacian on that $\mathbb P^1$ will have a log-like singularity, yes. Is this the sort of thing you want?
9h
answered Existence of real modular function with specific behavior as $q\to 0$
1d
reviewed Approve suggested edit on Real-world applications of mathematics, by arxiv subject area?
Aug
24
comment If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?
A quibble: The way the first example is worded may give a misleading impression, blurring "taking a limit" with "evaluation of meromorphic continuation at a point". If the meromorphic continuation has the desired point at the edge of the circle of convergence, then a non-tangential limit should give the same outcome, yes, but the idea is not restricted to that case...
Aug
21
comment On the pole of local L-function
Another, similar, common misconception is that "local functional equations" should combine to give the "global functional equation", and this is not the case.
Aug
21
comment Where to find (personal) motivation
@William, yes, in various possible wordings, there are two different things, or at least two different extremes on a spectrum, and accidentally trying (in one's mind) to make them be the same thing surely creates stress of several sorts. At the very least, one can try to remind oneself that, for example, there's "the art/science of mathematics", and then there's "the business of mathematics".
Aug
20
answered Where to find (personal) motivation
Aug
20
comment Where to find (personal) motivation
Also a rather mixed bag to "appear in Wikipedia"... :)
Aug
20
comment When is a local subring of a number field a valuation ring?
How is your non-Dedekind ring described, then? Is it an "order" of a ring of algebraic integers, for example? If so, then localizing at any prime not dividing the "conductor" will make the localization be the same as the localization of the full ring of algebraic integers... Or what context do you have?
Aug
20
answered When is a local subring of a number field a valuation ring?
Aug
14
reviewed Approve suggested edit on Vanishing of Motivic Cohomology
Aug
14
comment When is a cubic polynomial a cube?
Because it's cubic, barring various degeneracies it'll be an elliptic curve, and many things are known about rational points (Mordell-Weil), while integral points are subtler. Searching on "elliptic curve" should give you a broader context.
Aug
8
reviewed Approve suggested edit on citations tag wiki excerpt
Aug
4
reviewed Reject suggested edit on Why are they called Specht Modules?
Aug
3
comment Multiplicity of automorphic representation
The only discrete spectrum is cuspidal or made from residues of cuspidal-data Eisenstein series. In the continuous spectrum, the same continuous spectrum can occur over and over, but not as genuine subrepresentations. What do you mean to ask about "multiplicities" in the latter case?
Jul
30
revised Whittaker models for $GL_n$ and Fourier coefficients
added 899 characters in body
Jul
30
comment Whittaker models for $GL_n$ and Fourier coefficients
@GHfromMO, ah, I hadn't noticed the edit in the question. Thanks...
Jul
30
answered Whittaker models for $GL_n$ and Fourier coefficients
Jul
30
comment Whittaker models for $GL_n$ and Fourier coefficients
Maybe the transition from character of $\mathbb Q\backslash \mathbb A$ to "generic" character on $N$ could bear some amplification... and, as the questioner anticipates, a one-dimensional repn must factor through a quotient group, in this case $N$ by its derived group, etc.