11,810 reputation
22758
bio website math.umn.edu/~garrett
location US
age 63
visits member for 4 years, 5 months
seen 2 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.


Aug
25
awarded  Nice Answer
Aug
25
comment Primer on Eisenstein series
... [cont'd] is the trick (one way or another) of identifying (self-adjoint) compact operators in a "spectral decomposition", to use the discreteness of their spectrum. Y. Colin de Verdiere used the (Lax-Phillips/Faddeev-Pavlov) discrete decomposition of spaces of pseudo-cuspforms to prove meromorphic continuation of genuine Eisenstein series, by observing that after an elementary modification, they satisfy a differential equation whose resolvent is compact... For example, at math.umn.edu/~garrett/m/v/cdv_eis.pdf there is an explication of the latter.
Aug
25
comment Primer on Eisenstein series
The "rigidity" is the Mostow-Margulis-etal theorems that assert that most co-finite-volume discrete subgroups of semi-simple real Lie groups are "arithmetic". Further, the "congruence subgroup problem"'s positive resolution for essentially all higher-rank groups is that mostly these discrete groups are "of congruence type", so p-adic and adelic ideas are relevant. Langlands was trying to preserve some generalities that turned out not to exist, to some degree. The "discretization"... [cont'd]
Aug
23
comment Has Frucht's theorem been successfully used in inverse Galois theory?
I'm not following that first "Logically"...
Aug
19
comment Springer GTM Reprints in China?
@DimaPasechnik, I'm not sure that the copyright thang is really true, although I've heard it myself. By this point, I think it's that <large international corporation> wants to persuade people to not disrupt it's practices... not illegal practices, but, um, profitable-unless-disrupted. I've had a textbook of mine (foolishly giving away the copyright in the 90s...) translated and printed in Chinese in China, and, amazingly, that version only costs maybe a buck or two, while the U.S. version cost nearly 100 bux. "Go figger..."
Aug
19
comment Springer GTM Reprints in China?
Such things have existed since at least the 1980s: the pricing of books in India and China "had to" make them vastly cheaper, given currency exchange and local economic conditions), or Springer (et al) could not sell any at all. No, they did not sell anything at a loss, or... why bother? The point is that the "Western" market could then (and still now) bear much bigger mark-ups (over cost of production), and has precedents for believing that it's ok, or ... something.
Aug
17
awarded  Nice Answer
Aug
16
comment Gauss-Wantzel theorem, Fermat primes and solvability of S_n
But what are the chances that two small integers would differ by exactly $1$? :)
Aug
15
comment Question about expression of a sum of two Hecke eigenvalues
It seems to me that the sum in each term is a problem.
Aug
15
comment Question about expression of a sum of two Hecke eigenvalues
For prime powers $p^\ell$, the corresponding eigenvalue is of the form (a trace, thinking of Shintani-Cassleman-Shalika... formulas) $(a^{\ell+1}-b^{\ell+1})/(a-b)$, where the product $ab$ is normalized in some way, without much loss of generality to $ab=1$ (just depending on the "weight"). So whatever identities follow from this... Does this respond to the question?
Aug
15
comment Question about expression of a sum of two Hecke eigenvalues
I would think that for $n$ composite it is not generally possible to do this.
Aug
15
comment Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?
One thing that distinguishes functions of a single complex variable from those of several is that the region in which a power series converges has no simple canonical form in more than one variable, and certainly the region need not be a ball or polydisk, with the usual example $\sum_n (zw)^n$ in two variables. If one tries to address this issue, one will indeed run across Reinhardt domains and such, but that's not really necessary for basic convergence properties: a polydisk will suffice, I gather.
Aug
12
comment Group of real analytic isometries of $g$-fold product of the Poincare upper half plane
The conclusion is certainly correct, and has been known at least since Siegel's early work. Possibly his "advanced analytic number theory" does this (it's just Schwarz' lemma). I heard it as old-standard fact in 1976 in a course given by Shimura...
Aug
12
reviewed Approve Finding optimal linear transformation for intersection of convex polytopes
Aug
10
comment Eisenstein part of the theta series of lattices in same genus
Siegel-Weil formula gives a general assertion that "in principle" should include such a thing, but probably requires a bit of work to "unpack" into more immediate combinatorial terms...
Aug
4
awarded  reference-request
Aug
3
comment Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)
It would be helpful, I think, at least to me, if you filled in some of the details. E.g., is everything happening with periodic functions? Integral on a circle? In what sense "equality"? As distribution? Pointwise? Etc... With mild hypotheses, Fourier series in two variables, on circle $\times$ circle work very well, in the context of Sobolev spaces, which amounts to polynomial-growth constraints on "coefficients"... assuming one can tolerate Fourier expansions which are "merely" distributions, and so on. Clarify, please?
Aug
2
awarded  Nice Answer
Jul
29
comment Zeta-Determinant for shifted Laplacians on the circle
Repaired the computational errors since the question got bumped upward anyway...
Jul
29
revised Zeta-Determinant for shifted Laplacians on the circle
deleted 3 characters in body