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## All Questions

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### Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.

What is the point of $\pi$-systems and $\mathcal{D}$ / Dynkin / $\lambda$-systems? I am an analyst in the process of consolidating my measure theory knowledge before movi …
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### Proof that the factors of sigma(p^e) have two forms.

I found the proof by Kronecker that the expression X = p^e + p^{e-1} + ... + p^2 +p + 1 is irreducible. Four people translated the German, two in Detroit and two in Kiel, Germ …
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### Are there “reasonable” criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisim …
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### Convexity of a constrained optimization problem

Hi, this is a continuation of a previous question I asked about the convexity of an optimization problem I am working with. Consider the function \begin{multline} B_i(a_0,\mathbf …
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### Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been t …
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### Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgeb …
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### Conformal map of a doubly connected region to an annulus

Hi. I am a Mechanical Engineering student. I'm not good at complex variable theory and having problem with finding conformal mapping of a doubly connected region to an annulus (or …
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### Existence of Fermi coordinates on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ …
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### Frobenius upper shriek/flat of a dualizing complex

Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called bein …
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### Repairing the Lie operad in characterstic 2?

Recall: We present an operad (with $S_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the followin …
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### Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan

Where can I find a comprehensive treatment of this important result at the level of a very advanced undergraduate/beginning graduate student? What works develop the relevant materi …
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### When is a coarse moduli space also a fine moduli space?

Given a moduli problem, it appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space(is this strictly true?). In any case, assuming …
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### representation theory and finite order automorphisms

Let $\sigma$ a finite-order automorphism of a finite-dimensional complex simple Lie algebra $g$. Denote the order of sigma by $k$ and fix a $k^{th}$ root of unity $\omega$. It is w …
For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite nona …