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Let $G$ be a (smooth, connected for convenience) algebraic group acting on a variety $X$ (over a reasonable base scheme). The theory of equivariant perverse sheaves is a functor $\operatorname{Perv}_G(-) \colon \mathrm{Var}_G \longrightarrow 2-\mathbf...

Working in $ZFC$, the statement "$0^\sharp$ exists" is often liberally taken to be one of many known equivalent statements.
However, working in $Z_2$ or $ZFC^-$ (with collection, well-ordering), most of these equivalent forms require knowledge of forbidden uncountable cardinals; we have to be a little bit more precise. We can luckily define the notion of an E.M. set, remarkability...

Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$.
The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be associated a defect group $D_B$ (unique up to conjugacy). The defect of $B$ is $d$ where $|D_B|=p^d$....

I am just an ordinary student, and I have never had the chance to ask this question to mathematicians. Perhaps this is the first opportunity I have to ask it here.
When we refer to "Collatz numbers", we mean all natural numbers that eventually reach $1$ under the Collatz process. By odd-step Collatz numbers, we consider only the odd numbers that appear at each step of the sequence, skipping all even numbers. For...