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bio website math.umn.edu/~garrett
location US
age 63
visits member for 3 years, 7 months
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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
awarded  Good Answer
Jan
24
comment Any formula for the exact number of primes below a given bound?
One might claim that the real point is the precise relation between primes and the zeros of zeta, as made clearer in Guinand's and Weil's extensions of Riemann's "Explicit formula", to be a sort of Fourier duality between the two. I find the relation more interesting than either questions about counting primes, or any sort of "raw" version of questions about the zeros...
Jan
22
answered Finite-dimensional space naturally isomorphic to its double dual?
Jan
20
awarded  Nice Answer
Jan
17
comment What is $B$ in $\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$
Zeta itself has zeros at negative even integers, and these are called "trivial", so your reference to "non-trivial" zeros would ... by convention, anyway... exclude the negative even integer zeros. But the product expansion is correct if you do include them, just that your sentence about "non-trivial" is (by convention) incompatible with the necessity of including them. Either put the Gamma factor on, and then say that only "non-trivial" zeros enter, or leave the Gamma factor off, but include negative even integer zeros, and don't say that "only non-trivial" zeros appear. That's all I meant.
Jan
17
comment What is $B$ in $\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$
If you're writing the product of just zeta, without the Gamma factor, rho must run through negative even integers (trivial zeros) also, so your parenthetical sentence can't be exactly what you mean.
Jan
17
comment How unitarize an irreducible representation of a finite group?
Averaging: given one inner product $\langle,\rangle$, define $(v,w)=\sum_{g\in G} \langle \psi(g)(v),\psi(g)(w)\rangle$.
Jan
14
comment Does Idele group of norm 1 preserved by the norm?
Yes, $|a|_w=|N_{L_w/K_v}(a)|_v$ for $a\in L_w$ and $w|v$. Thus, at the "bottom" (namely, $\mathbb Q$ for the number-field case, and a genus-zero field for the function-field case) the product rule holds by reducing to primes and using the fact that the "integers" are a principal ideal domain...
Jan
14
comment Does Idele group of norm 1 preserved by the norm?
The ("metric") norms on extensions of the "bottom" field's completions are obtained by composing the bottom field's (metric) norms with the local Galois norms.
Jan
13
comment Is the solution of a PDE always the convolution of the Green function?
Well, depending how you interpret things, I'll say the obvious: the without-boundary case allows $\mathbb R^n$ to operate as a group, and no boundary terms, either. The "domain" case wrecks the group structure, and loses the no-boundary property. (Most boundary conditions are not stable under translation...) Specifically, different sorts of boundary conditions must give different solution formulas...
Jan
13
comment Is the solution of a PDE always the convolution of the Green function?
Green's functions depend on boundary conditions.
Jan
13
comment Compute complex Gaussian integral
Again, there is no necessity of treating large $z$ in $\det(A+z(e_{ij}+e_{ji})$...
Jan
12
comment Compute complex Gaussian integral
To define a square root of a non-zero function on a region it is only required that the region be simply connected, which a convex region certainly is.
Jan
11
revised Prove that there exists infinitely many pairs of relatively prime integers $(a,b)$.
edited tags
Jan
11
revised Compute complex Gaussian integral
added 661 characters in body
Jan
11
comment Compute complex Gaussian integral
One need not be too precise about domains to prove holomorphy of both sides in each entry... the main thing is to be sure of connectedness, which follows from convexity of the cone of positive-definite real matrices. I can fill in further details tomorrow, ...
Jan
10
comment Is arrow notation for vectors “not mathematically mature”?
@Ruslan, I cannot speak for contemporary physics styles, but this convention seems archaic and text-book-y, reminiscent of styles from several decades past. Not that it's a bad thing, but that contemporary styles outside of textbooks may not follow that, if, for example, the questioner is wondering about contemporary stylistic tendencies.
Jan
10
answered Compute complex Gaussian integral
Jan
10
comment Compute complex Gaussian integral
The imaginary part must be skew-symmetric, as @NickThompson's (counter-) example shows. Then the identity principle in complex analysis gives the general result.
Jan
10
awarded  Nice Answer