11,720 reputation
22754
bio website math.umn.edu/~garrett
location US
age 63
visits member for 4 years, 3 months
seen 13 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.


21h
comment Zeta-Determinant for shifted Laplacians on the circle
Repaired the computational errors since the question got bumped upward anyway...
21h
revised Zeta-Determinant for shifted Laplacians on the circle
deleted 3 characters in body
2d
comment Zeta-Determinant for shifted Laplacians on the circle
Some minor computational boo-boos in the above: should be $e^{-\pi(n^2+c)y}$, of course, and pursuant... doesn't change it qualitatively.
2d
comment Is the twisted symmetric fifth power $L$-function holomorphic?
Following up on @Marty's comment: in the 80s, several people, including PiateskiShapiro-Rallis, myself, and others, tried to identify what sort of Rankin-Selberg situation could produce such "higher" L-functions... tentatively thinking in terms of "generalized groups"... but/and found that (at least it seemed at the time) there was no sane "generalized group" recipe that could produce a given L-function "at will". Maybe there has been progress, but it is already not so easy to make a reductive "group" with arbitrarily specified Coxeter group as "Weyl group"... Such obstacles.
2d
comment Zeta-Determinant for shifted Laplacians on the circle
Also, an Euler-MacLaurin summation approach might work as well as anything, depending, ...
2d
answered Zeta-Determinant for shifted Laplacians on the circle
Jul
27
comment Zeta-Determinant for shifted Laplacians on the circle
It seems to me that a variant of Riemann's integral representation of $\zeta$ using $\theta$, the latter's functional equation from Poisson summation, gives the meromorphic continuation... but, unlike zeta itself, superficially it appears difficult to get information about the derivative at $0$, but is that what's really desired, or is it more the residue at the pole at $s=1$? Can you clarify "in terms of what" you'd really like the outcome?
Jul
22
comment Analytic Number Theory without Pigeonhole Principle
It is still not clear to me what you mean by "analytic number theory". Please clarify? There is much potential ambiguity in that label. E.g., do you mean Iwasawa-Tate theory of zeta functions? Jacquet-Langlands? Sieves? The Zhang-Maynard-Tao business? Subtle things about moment estimates? Subconvexity? "The thing is", most of these are not really "combinatorial" or "discrete" in any operational sense, so there's some element of mis-reference, to my mind. Please clarify?
Jul
22
answered Analytic Number Theory without Pigeonhole Principle
Jul
21
comment Make mathematical sense of the Dirac well Potential Equation
@ChristianRemling, yes, indeed, thanks... and to give an example-objection: one way to interpret the (mathematical) difficulty in interpreting $\delta'$ as a "potential" in analogous fashion is that it is not inside $H^{-1}$, and it itself cannot be "legally" applied to solutions to $(\Delta-\lambda)u=\delta'$, since these solutions will only be in $H^{1/2-\epsilon}$, not $H^{3/2+\epsilon}$, etc.
Jul
21
revised Make mathematical sense of the Dirac well Potential Equation
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Jul
21
revised Make mathematical sense of the Dirac well Potential Equation
added 2744 characters in body
Jul
21
comment Make mathematical sense of the Dirac well Potential Equation
@JeanDuchon, yes, what I wrote was careless. But/and I think the issue is not truly about literal pointwise multiplication, for a reason I will try to clarify in the edit I will do just now...
Jul
20
comment Make mathematical sense of the Dirac well Potential Equation
Exactly. Dirac's marvelous intuition in the late 1920s was nicely rigorized in one fashion by (B.Levi-) Sobolev's spaces by the 1930s, as promoted and amplified by Gelfand-et-al in the "Generalized Functions" 6 volumes.
Jul
18
awarded  modular-forms
Jul
17
answered How to construct the symmetric power function from a modular form?
Jul
17
revised How to construct the symmetric power function from a modular form?
edited tags
Jul
16
revised Make mathematical sense of the Dirac well Potential Equation
edited body
Jul
16
comment Make mathematical sense of the Dirac well Potential Equation
Ooops, yes, I'll change that sign on the epsilon. The general, basic things about Sobolev spaces are treated in most or many books on PDEs, especially linear ones. Folland's book (or Tata lectures), Brezis' book, are two that do this sort of thing. The question of multiplication is an immediate corollary, then, since $H^{+s}$ and $H^{-s}$ (in various contexts) are in duality, so "pair" to $L^1$, at least. The Sobolev imbedding business is treated in those sources, certainly. Googling "Sobolev space" should give lots of useful results, too.
Jul
16
answered Make mathematical sense of the Dirac well Potential Equation