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bio website math.umn.edu/~garrett
location US
age 63
visits member for 4 years, 3 months
seen 45 mins 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.


2d
comment History of Geometric Analogies in Number Theory
I think that Minkowski's "geometry of numbers" is not the sense of the question, which is more about the analogies between rings of algebraic integers and integral extensions of $\mathbb F_q[T]$.
Jul
1
answered Are spherical harmonics uniformly bounded?
Jun
23
comment Uniform convergence of Fourier (orthonormal) expansion of series
I don't know about a definitive result for "pointwise a.e." convergence in higher dimensions. But/and such a result by itself (even as difficult as the Carleson result is in one dimension) doesn't allow us to do much with the expansion without further hypotheses (hence, my mentioning uniform pointwise and Sobolev imbedding sorts-of-things).
Jun
22
comment When does analytic in the operator norm imply analytic in the trace class norm?
I'm sorry, I didn't mean to suggest that trace-norm continuity implies operator analyticity. Maybe I misunderstood what hypothesis you're taking and hoping to prove trace-norm analyticity...
Jun
22
comment When does analytic in the operator norm imply analytic in the trace class norm?
I'm confused by your opening remarks: if you've proven that continuity of $W(\lambda)$ in operator norm implies holomorphy in trace norm, then since that continuity is implied by analyticity in the operator norm, you'd be done... ? Is there a typo? Am I misunderstanding something?
Jun
22
answered Uniform convergence of Fourier (orthonormal) expansion of series
Jun
21
revised Spherical harmonics and ellipticity of the Laplacian
added 13 characters in body
Jun
21
reviewed Leave Open $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$
Jun
21
reviewed Close example of non computable implicit operations
Jun
21
reviewed Leave Open Representation Theory of $U(N)$
Jun
20
answered Spherical harmonics and ellipticity of the Laplacian
Jun
19
comment Does there exist a smooth version of Cohen's factorization theorem?
Not clear to me, either. The argument is not at all trivial, even after Casselman's discussion... Entire functions and such...
Jun
19
comment Does there exist a smooth version of Cohen's factorization theorem?
I don't really know about the latest on this. Bill Casselman has a nice essay about D-M and its proof on his web-site ... which probably you can find as quickly as I can. :)
Jun
19
comment Does there exist a smooth version of Cohen's factorization theorem?
It's not only approximated, but equal. (The weaker and much easier "approximation" assertion is essentially Garding's from 1947 or so, which would also cope directly with approximation of smooth by convolutions). But then, in any case, you can certainly approximate arbitrary smooth by test functions, if that's enough for you. To get exact equality with $C^\infty*C^\infty_c$ seems tricky... Is approximation good enough for you?
Jun
19
comment Does there exist a smooth version of Cohen's factorization theorem?
So the D-M result is not addressing some aspect of things relevant to your purposes? Yes, the D-M more directly addresses test functions, i.e., compactly-supported, not all smooth, but that's not a fatal problem. A technical point is that finite sums are necessary in that context, so not every test function is exactly a single convolution of two. Can you clarify your issue?
Jun
19
comment Does there exist a smooth version of Cohen's factorization theorem?
Isn't this exactly the Dixmier-Malliavin result?
Jun
19
comment different definitions of epsilon constants for representations of GL(2) from modular forms
If it's any consolation, I think that the issue of the compatibilities of various notions of epsilon-factor are not easy to certify. I seem to recall reading that Dwork and Langlands and others decided to not publish (!?!) certain of their investigations into such compatibilities because the write-up would simply "go on too long". In particular, any arrangement of an argument that would show that almost-all local epsilons are $1$, and certify the notion that a global epsilon-factor is well-defined, and so on, is apparently ... expensive.
Jun
17
comment Fourier series and transform related to Epicycles
The context confuses me, even if it confuses no one else: a "curve" as posed is exactly a complex-valued function on the circle, so certainly has a (complex-valued) Fourier series, and these things have been studied for a long time. I'd think that imbedding it in that context, or at least talking in a way that allows reference to it, would help people answer usefully...
Jun
13
comment Fourier expansion of Eisenstein Series
Your link to Zagier's paper is slightly mangled: it should have mpim-bonn rather than mpimbonn...
Jun
9
comment p-adic asymptotic analysis
I do not know of any intro on this... but/and, having thought about the archimedean case (!?!) in recent years, I wonder what things you're wanting...