Zhen Lin
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394/400 score
 Feb22 comment The computability of Kleene's $T$-predicate In effect, Kleene's $T$-predicate is the arithmetical version of a universal Turing machine. That should tell you how to construct it. Feb21 comment American undergraduate applying overseas The examination is at the end of the year, spread over the course of about 3 weeks. Feb21 comment American undergraduate applying overseas Most PhD students in Cambridge take Part III or equivalent, but this year there's one who started straight after undergraduate studies in America. There is some cultural difference – for example, we don't do quals, and there are no compulsory classes. Feb21 comment How is the set of all programs countable? @TobiasKienzler A program is not just a lookup table, so it is very bad form to identify it with the graph of the function it computes. Feb21 comment Example where Čech and derived functor cohomologies don't agree. Answered on MO. Feb21 comment Examples of categories where morphisms are not functions If you soften your "and" to "or" then that's easy: take the category of sets and relations. But it turns out to be isomorphic to a different category whose objects are structured sets and whose morphisms are functions. There is also a locally small category which is demonstrably not equivalent to any concrete category. See also here. Feb20 comment Example where Čech and derived functor cohomologies don't agree. More seriously, one can prove without any hypotheses at all that Čech cohomology (after taking the direct limit over all open covers) always computes the correct $H^1$, so any counterexample has to be in $H^2$ or higher. Feb20 comment Example where Čech and derived functor cohomologies don't agree. A scheme is in particular a topological space, and their sheaf cohomology only depends on the topology. In fact, they are very simple as topological spaces. :p Feb20 comment adjunction relation My definition just a functorial way of saying, take a coproduct of such-and-such-many copies of so-and-so. (It is a left adjoint, by the way.) Feb20 comment What is the notation for set of prime factors of a number? Essentially, $\operatorname{Spec} \mathbb{Z}/(n)$. Feb20 comment Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction Actually, isn't $(X \times_S S') \times_{S'} T \cong X \times_S T$ the pullback pasting lemma? This isomorphism has a somewhat different flavour than, say, $\ker (f \times g) \cong \ker (f) \times \ker (g)$. Feb20 comment Why do we accept Kuratowski's definition of ordered pairs? Well, if you make some assertions about truncating the universe at some small rank or something silly like that, it might be important what kind of pairing function you use... (There's a comment in Hodges's Model theory about someone using definitions which made Lemma 4.3.1 false.) Feb20 comment Why do we accept Kuratowski's definition of ordered pairs? You should say what $s$ does to sets that are not finite ordinals. Feb19 comment Freyd's Geometric Finiteness : An Example Computation No, continuous does not mean preserving arbitrary joins. It would be more precise to say that it preserves "jointly epimorphic families", in the sense that if $\mathfrak{U} = \{ U_i \to V \}$ is an open cover of $V$, then the image of $\mathfrak{U}$ under $F$ is jointly epimorphic. However it is true that a left exact functor $F : \mathcal{O}(\mathbb{R}) \to \mathcal{B}$ must have image in the subterminals of $\mathcal{B}$, so in this case it is just a matter of preserving joins. Feb18 comment Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal. I am referring to the notion of a graded ring. But perhaps your supervisor has a better solution; I'm not a professional algebraic geometer. Feb18 comment Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal. You can use a kind of degree argument where $X_1$ has degree $2$, $X_2$ has degree $3$, and $X_3$ has degree $6$. (Also, are you one of my students?) Feb18 comment An example of decomposing a projective variety This one is somewhat difficult. You have probably seen the solution by now, but if not, I'll just quickly say that one component is the line $V(X_2, X_3)$, and the other is the twisted cubic. (Also, the variety is supposed to be considered in $\mathbb{P}^3$.) Feb18 comment To what extent is a scheme morphism determined by its topological map? In the same vein, it is not quite true that a scheme morphism $X \to Y$ is determined by its action on points even if $X$ and $Y$ are varieties over some algebraically closed field $k$. You also have to know that the morphism commutes with the structural morphisms down to $\operatorname{Spec} k$. Feb18 comment Why isn't this a valid argument to the “proof” of the Axiom of Countable Choice? No, there is certainly no problem with creating finite choice functions. Everyone has explained this. The problem is in assembling them together to make an infinite choice function. Feb18 comment Why isn't this a valid argument to the “proof” of the Axiom of Countable Choice? @AloizioMacedo More to the point, mathematics is not temporal. A variable is not undefined at one time and then defined at another, so that you can jump over infinitely many steps and have them all simultaneously defined. Even if you could perform infinitely many tasks and jump ahead to a time when they have all been completed, that does not mean that things will be in a consistent state at that point. For example, consider the process where I turn on a lamp at $t = 0$, turn it off at $t = \frac{1}{2}$, turn it on again at $t = \frac{3}{4}$, etc.; is the lamp on or off at $t = 1$?