18,663 reputation
23070
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 6 months
seen 5 hours 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.


Dec
12
comment Genius mathematicians who never published anything
In those days there was nothing resembling "publication" in the sense of the last 250+ years.
Dec
9
answered The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.
Dec
9
awarded  Nice Answer
Dec
9
answered Primality of number 1
Dec
9
awarded  Caucus
Dec
7
comment What is wrong with this proposed proof of the twin prime conjecture?
(An aside: the question of whether $1$ should be a prime or not is a sort of red herring, insofar as it doesn't really matter: other definitions and assertions would simply have to be modified accordingly. Indeed, until late in the 19th century, $1$ was usually considered prime, apparently. The point is that we understand that specific aspect, and it's not really the issue at all...)
Dec
1
comment “Honest” introductory real analysis book
@littleO, to invest one's time and energy in the proposed fashion is, to my mind, suboptimal in at least two ways. First, surely one has caught on to the general pattern of the low-level details after a certain number of example-proofs, without having to continue and see every other idea accompanied by all those low-level details... which tend to swamp the main idea. Second, getting into the habit of conceiving of the activity of mathematics as essentially involving writing out all possible details, rather than choosing the most-relevant, critical details, is simply bad practice.
Dec
1
comment “Honest” introductory real analysis book
I can't help but remark/claim that your friend should not really study from such a thing, etc. That is, carrying out every detail, and doing so "properly" (code for exaggerated-formal) is substantially misguided.
Nov
30
comment Mathematical structures
Indeed, there is no apparent reason to think that there is a simplifying viewpoint to understand the vast and varied activities in modern mathematics. No meta- "royal road". It is true that various meta-notions do help organize various things, but a good suitcase does not express meaning of its contents, etc., however useful it is.
Nov
20
awarded  Nice Answer
Nov
11
comment Bernoulli Numbers and radius of convergence
@GeraldEdgar, indeed! The power series of the function $f(z)=1/(1+z^2)$ at $0$ powerfully illustrates this point, to my mind: the function is well-behaved on $\mathbb R$, (rhetorically) why should the power series fail to converge beyond $\pm 1$? :) Well, because its behavior in the complex plane is a genuine phenomenon that constrains its behavior on the real line, etc.
Nov
11
answered Bernoulli Numbers and radius of convergence
Nov
10
comment Some issues for solving differential equations using Fourier transform
To add emphasis: indeed, it is necessary to think of $\hat{x}$ as a generalized function (=tempered distribution, if one prefers) in order to "catch/detect" all reasonable solutions.
Nov
7
comment Meaning of the adjoint representation of a Lie group
Maybe it's just me, but the question... and this answer... blur possibly-unhelpfully the distinctions between the various "adjoint" representations. Indeed, "Ad" of $G$ on $\mathfrak g$ is the derivative of (what I'd call) the conjugation action of $G$ on itself, at $1_G$, and then lowercase-ad is the further derivative of that. Just a fretful paraphrase...
Nov
7
answered If the normalizer of a subgroup in a group is equal to the subgroup then is the subgroup abelian?
Nov
2
comment Stitching two analytic functions?
@DanielFischer, I'm not attempting to assert anything too substantive: only, first, that the same Morera-thm argument as in reflection principle's proof applies to the form of the question I saw here; second, that one might wonder (though this was not asked) to what extent all such questions reduce to a single one... thinking of Riemann Mapping. The examples of mapping polygons to disks illustrates (to me) that such questions do not reduce to a single case. Just free-associating, perhaps...
Nov
1
comment Stitching two analytic functions?
@DanielFischer, indeed, one must have an involutive anti-holomorphic map that fixes the "boundary" curve, as many of us know. So, yes, there must be an anti-analytic map (akin to complex conjugation for the real line, or $1/\overline{z}$ for the unit circle) that fixes pointwise the desired boundary arc. The Riemann mapping theorem does non-constructively make a simply-connected, bounded region into the disk... so we have an example-existence result. But in any case the Morera's theorem argument that proves the usual reflection principle can apply.
Nov
1
comment Stitching two analytic functions?
Reflection Principle (and perhaps Riemann Mapping theorem).
Nov
1
comment Widespread, persistent mathematical disagreement?
The verifiability is true only in principle, and might be very difficult, essentially functionally impossible, in practice.
Nov
1
comment Errata in Algebra Demystified book?
Yes, MathJax, which is operationally LaTeX. The FAQ has some discussion of this.