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22649
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 8 months
seen 5 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 Tate's thesis for Artin L-functions
... are highly non-trivial, such as Wiles' argument that, in effect, L-functions of (nice...) elliptic curves are automorphic, to prove Fermat. It was Langlands' epiphany in the 1960s that non-abelian Artin L-functions (Galois objects) should be automorphic.
2d
comment Tate's thesis for Artin L-functions
Yes, perhaps to further clarify, "work" might mean "give L-functions with analytic continuation and functional equations". This sense is guaranteed for many automorphic L-functions, especially the "standard" ones attached to cuspforms on $GL(n)$, etc. On the other hand, Artin L-functions (and Hasse-Weil zetas of varieties) and other L-functions coming from Galois repns do not have any obvious general analytic continuation (for example), ... so are proven to have such by proving that they are, in fact, automorphic. Classfield theory does the abelian case. Further cases ... [cont'd]
2d
comment Hilbert Space Tensor Product vs. Algebraic Tensor Product
Yes, in brief, $V\otimes W^*$ can be identified with all finite-rank linear maps $W\to V$ for any vector spaces over any field, by $(v\otimes \lambda)(w)=\lambda(w)\cdot v$. Over $\mathbb R$ or $\mathbb C$, the usual Hilbert space structure on the algebraic tensor product is literally the Hilbert-Schmidt norm.
2d
comment Hilbert Space Tensor Product vs. Algebraic Tensor Product
Couldn't find the MSE question, but... if the algebraic tensor product is given the natural Hilbert-space structure, that is the same as Hilbert-Schmidt norm, and that completion is essentially the collection of Hilbert-Schmidt operators from one to (the dual of) the other. Certainly not every Hilbert-Schmidt operator is finite-rank.
Dec
17
comment Graduate program applications that require questionnaires and other non-letter material
And I was amused to be told by some of these on-line sites that the street address I gave them (the actual math dept address) "did not parse", or something. But, yes, as @JoeSilverman suggests, there're many issues about "leadership" and also questions about class rank (which no faculty would have reasonable means to answer)... Questions made up by people presumably uninvolved with actual admissions, and whose notion of it comes from movies or novels. :)
Dec
17
comment Graduate program applications that require questionnaires and other non-letter material
@KConrad, indeed, but/and this centralized "management" of grad admissions is some peoples' full time job, and once it's all in place and they're hired for that job, it's hard to eliminate. Further, some of the computer-not-literate academic departments seem to positively like this central "management", so there is by-far insufficient unanimity to eliminate this fake central unit and its constructs (and senseless burdens on people). Sure, if a Vice President were inspired to do this, it'd be one thing, but for mere working faculty to suggest changes... don't get me started. :)
Dec
17
comment Graduate program applications that require questionnaires and other non-letter material
I have to apologize for Univ of MN's shenanigans. These are not generated by the math dept at all, but by a central bureaucratic unit... The application set-up is somehow aimed to be one-size-fits-all for nearly all graduate programs. I've complained, etc., many times over the years, but Central Administration's belief is that academic depts needs lots of oversight and control, or we'd admit just anyone, and grant degrees willy-nilly. Again, my apologies...
Dec
17
awarded  Nice Answer
Dec
16
comment Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
It certainly at-least-potentially makes sense, just as in case $G_v$ (for $v$ archimedean) might happen to be compact, the holomorphy condition is vacuous. I am not practiced with induction scenarios in which this is the correct "bottom" stage, unfortunately. A presumably bad answer to your original question would be that, sure, because $Sp_n$ contains a copy of $GL_n$ (depending on indexing...), any torus can be imbedded there (if there're really no further conditions...) I'd wager this is not what you wanted, but I don't know.
Dec
16
comment Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
Ah! Ok, zero-dimensional in that sense, and no "hermitian" condition. Then I guess I don't know what condition(s) must be satisfied for a legitimate morphism "of Shimura varieties". The "hermitian condition" is about morphisms $G\to H$ over $\mathbb Q$ so that on non-compact factors over $\mathbb R$ the induced map on (hermitian) symmetric spaces $G_v/K_v$ is holomorphic. In that case, there is a "physical" induced map of complex analytic varieties... But this is evidently not what you're asking about! :) Thanks for clarifying.
Dec
15
comment Zeros of the derivative of Riemann's $\xi$-function
I think you intended to write "7/10 of the zeros of the derivative"...?
Dec
15
comment Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
Is there any analogue of "hermitian condition" on the morphism of groups (to $Sp_n$)? If so, then there's an obstruction for the hermitian-type groups $E_6,E_7$, which admit no hermitian-type imbeddings to $Sp_n(\R)$, as Satake observed in the 1960s. Could you clarify?
Dec
14
comment Why do we teach calculus students the derivative as a limit?
Edward Nelson's "IST" version of non-standard analysis is vastly more user-friendly. Alain Robert's book on non-standard analysis takes that viewpoint, and is a marvel of lucidity.
Dec
11
comment Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$
@JohannesHahn, ah, you're right. So that's the relevant dichotomy, apparently! Good. Thanks!
Dec
11
comment Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$
Always: rational repn of the norm-one subgroup of the unique quadratic extension, together with Galois automorphism. This can be made as explicit as one's knowledge of a non-square.
Dec
3
comment What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
@Vincent, I think there's no simple general relationship, apart from some natural dualities (when the objects are known to exist). E.g., for $p$-adic groups, the compact induction (corresponding to tensor product) is not left adjoint to restriction, although it is left adjoint to something. I saw this in P. Cartier's Corvallis notes on repns of $p$-adic groups... where some of the difficulties are illustrated.
Dec
2
comment What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
Isn't your first construction a right adjoint, and the second a left adjoint, to suitable forgetful/restriction functors? (The analogues coincide for complex repns of finite groups.)
Nov
29
reviewed Close Powers of orthogonal matrices is closed
Nov
28
comment Automorphism group of a modular curve and its action on the set of cusps
Probably the question is intended ask about automorphisms of the "open" modular curve, and/or automorphisms of the compactification that stabilize the set of cusps...?
Nov
20
awarded  Good Answer