10,495 reputation
22548
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 6 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.


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 Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?
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
Oct
6
answered Fourier approximation error in L^2 for piecewise continuous functions
Oct
5
answered Inequality for a gamma function
Oct
2
revised algebraic groups over non-archimedean local fields acting on buildings
edited tags; edited tags
Oct
2
answered algebraic groups over non-archimedean local fields acting on buildings
Oct
1
answered On the reductive group
Sep
28
comment Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?
Hilbert's student Blumenthal treated examples of this for $SL_2$ very early in the 20th century, to study Hilbert-Blumenthal modular forms. Siegel's arguments of the 1930s and 1940s for computing volumes of the quotients attached to $SL_n(\mathbb Z)$ and $Sp_n(\mathbb Z)$ immediately apply to number fields, imitating some aspects of Hecke's 1910s and 1920s treatments of $L$-functions attached to number fields.
Sep
28
revised Topology on the space of Schwartz Distributions
added 852 characters in body
Sep
22
comment What justification can you give for the fact that “most ODEs do not have an explicit solution”?
As is often the case, "explicit" is a misnomer for "elementary", where the latter more literally refers to polynomials, exponentials-and-logs, trig functions, roots, ... As in is implicit in Loic Teyssier's answer, the coefficients of a differential equation can be elementary, while the solutions are demonstrably not.
Sep
21
awarded  Nice Answer
Sep
20
awarded  Necromancer
Sep
16
reviewed Leave Open Eigenstates of Fourier transformation