17,460 reputation
22961
bio website math.umn.edu/~garrett
location US
age 62
visits member for 3 years, 1 month
seen 16 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.


2d
comment What are some 'conceptualizations' that work in mathematics but are not strictly true?
@Max, not necessarily... even $L^p$ spaces of functions don't have pointwise values everywhere, etc.
Jul
23
comment What are some 'conceptualizations' that work in mathematics but are not strictly true?
I think it works very well indefinitely, if only one suitably enlarges one's notion of "function", e.g., dropping a too strict demand for "pointwise values" and such.
Jul
20
comment How can I describe Lie bracket for formal product of elements of Lie algebras
As in @darijgrinberg's comment, there is some danger in attempting to describe the universal enveloping algebra as "formal products". Sure, there's the Poincare-Birkhoff-Witt theorem, but that oughtn't be relevant too often. Use the mapping-property characterization.
Jul
16
comment Euler product of Dirichlet series
Just a small stylistic point: it's best to use language that gives the impression of making a request, rather than giving a command. I edited the language a little in this light.
Jul
16
revised Euler product of Dirichlet series
added 58 characters in body
Jul
12
comment $k[x_1,\dots,x_n]/\frak{a}$ is an $k$-algebra of finite type?
Yes, and, also, since $k[x_1,\ldots,x_n]$ is Noetherian, every ideal $\mathfrak a$ is finitely-generated, so that ideal of "relations" is also "finite".
Jul
12
reviewed Close Probability of their's constituting a triangle?
Jul
12
reviewed Close Find expression for sum of series
Jul
11
comment Can't understand a step in the advanced calculus book by thomas P. Dence
Whatever else is going on, this sort of maybe-it's-a-rabbit-out-of-hat style argument is idiotic and unexplanatory. Very popular, though, among the crowd that think's math is about pranking other people. The questioner is more-than-right to take some umbrage at this.
Jul
9
comment Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$
The number of partitions of $\ell$?
Jul
9
awarded  Autobiographer
Jul
2
comment Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?
The local condition that $f$ is $C^1$ does not exclude some pathologies, such as a path in $\mathbb R^2$ that behaves well except for curving around and coming back to run into itself... except omitting that last point. Thus, locally at that point, the image looks like a "T". But $f$ is smooth and has injective $df$ everywhere... but this does not exclude these tricky pathologies.
Jul
2
comment Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?
It is becoming less clear to me what hypotheses you do have, then. Was the situation meant to evoke counter-examples to simple-sounding versions of true theorems? Invariance of domain applies to (local) homeomorphisms. The "df" argument applies to $C^1$ maps when $df$ is reasonably behaved.
Jul
2
comment Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?
The differential $df$ of $f$ maps the tangent space of $U$ (at a given point) to the tangent space of the target. This $df$ is a linear map of those vector spaces, and, if injective, is ... again... an injective linear map of vector spaces. This gives the inequality of dimensions.
Jul
1
answered Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?
Jun
26
reviewed Approve suggested edit on Prove that $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$
Jun
26
reviewed Leave Closed I need help with a double sum
Jun
26
reviewed Close remainder, quotient problem
Jun
26
reviewed Close When writing a proof, why do we want to assume a different but equivalent condition given in the proposition?
Jun
26
reviewed Close How to prove this claim using Mathematical Induction?