My project. I'm nowhere near done but a) Better to not write all of it at the end b) that way I don't forget how to prove things I know how to prove c) catch errors in my logic
I took a course on the history of postmodern art (mostly US-centric) which was fascinating in my last qtr of undergrad. Wished I'd engaged with the dept more then.
Is there a nice way to integrate some $2$-form over the ellipsoid $x^2+y^2/4+z^2/9=1$? I parameterized it using spherical coordinates, but I got a long and ugly expression.
Hmm. I am pretty sure I have done this before but I am having trouble figuring out why $\int_{\gamma} \omega = 0$ for any loop $\gamma$ implies $\omega$ is exact ($\omega$ here is a $1$-form)
@AkivaWeinberger I think I recall the definition of an ultrafilter and what a principal one is. Just not what a filter actually is (nor why we should care about them)
@TobiasKildetoft In one of our first conversations you mentioned that projectivity over a group algebra gets more interesting in weird characteristic. Do you know of any references of results in that direction? It suddenly became relevant for my undergraduate project.
@TobiasKildetoft Yesh, that's the case I've been working with (characteristic not dividing the order of the group.) By 'question at all', you mean there are cases where there are no nontrivial projectives?
@TobiasKildetoft Indeed. So, to rephrase my question: for a finite group and a field of characteristic dividing the order of the group, under what conditions is a module over the group algebra projective? (Asking for references in that vague direction, not concrete answers.)
@AndrewThompson yeah, it is probably hard to get into if you are not already familiar with that way of looking at things (though as I recall, it does have some results on this precise question)
@AndrewThompson by which you mean an equivalence of the projectively stable categories of modules?
Thanks. The endgoal is to say something about Hochschild homology of group algebras.
I do find something in Auslander-Reiten-Smalø about relative projectivity for group algebras of finite reptype, however I am unable to see how that's immediately helpful.
Hmm, I think there is something about the dimension of projectives always being a power of the characteristic. But this may be just for simple projective and possibly just in the special case I am familiar with of finite group of Lie type in the defining characteristic
See attempt below
I am interested in effective computations in finding approximate spectral decompositions in some suitable format.
Let $A: H \rightarrow H$ be a Hermitian operator on an $n-$dimensional Hilbert space $H$ with the spectrum $\{\lambda_1, ... \lambda_m\}, m \leq n$. Then, $A$ can ...
See attempt below
I am interested in effective computations in finding approximate spectral decompositions in some suitable format.
Let $A: H \rightarrow H$ be a Hermitian operator on an $n-$dimensional Hilbert space $H$ with the spectrum $\{\lambda_1, ... \lambda_m\}, m \leq n$. Then, $A$ can ...
![i.imgur.com/oGXR3We.png?1[/img]](http://i.imgur.com/oGXR3We.png?1[/img]){kind=link}
![i.imgur.com/6tHDTsf.png?1[/img]](http://i.imgur.com/6tHDTsf.png?1[/img]){kind=link}
