I took my general on May 20, 2019 as a first year graduate student. The Committee Chair Professor J. Kollár is responsible for algebraic geometry, one of my two special topics, and Professor S. Morel for model theory, the other. We also have Professor Y. Shlapentokh-Rothman for general analysis.
This transcript is wriiten on May 26, 2020. I can only remeber few of the contents but I will provide as much information as possible.
General algebra and analysis:
Q: Show that if a continuous function (on $\mathbb R$) and its Fourier transform both have compact support, then the funtion is zero.
I answered that the Fourier transform can be extended to an entire function on $\mathbb C$ and use the principle that the zero set of a nonzero holomorphic function is discrete to show that the Fourier transform is zero. I think I was asked to prove that principle as well, maybe not. Then I was told that this is not the expected answer and Prof. Shlapentokh-Rothman suggested another method which I cannot remember. I don't remember, either, if I followed his sketch successfully or not.
Q: Show that the polynomial ring of one variable over a Noetherian ring is Noetherian.
I answered with the classical method of considering the ideal of the original ring generated by the leading coefficients of elements in an ideal of the polynomial ring.
Q: What is a UFD? Show that the polynomial ring of one variable over a UFD is a UFD.
I answered well, except I used some other well-known facts about UFDs as granted which may not be assumed so in the general exam. We filled the possible gap anyway.
Q: What can you say about a finite dimensional linear representation (over an algebraically closed field) of an abelian group?
I answered that we first apply the structure theorem of finite abelian groups, which was objected by Prof. Kollár pointing out that we don't assume the group finite. He also said that I can assume the group is finitely generated to make life easier. I answered that we can apply Jordan decomposition to each element of the group and pass to smaller subspaces.
Q: What can you say about a finite dimensional linear representation (over a field of characteristic $p>0$) of a finite $p$-group?
I answered that an irreducble representation is trivial, using that finite $p$-groups have lots of normal subgroups and the explicit description of the group algebra of a cyclic group of order $p.$
Model theory:
Q: State and prove the completeness of the theory of algebraically closed fields of a fixed characteristic.
I made the correct statement and answered that I have two methods to prove it, one by model theory and one by algebraic geometry. I was asked to present both and did the latter one first. I explained why a closed formula (of $n$ variables) corresponds to a constructible subset of the $n$-dimensional affine space over the prime field. This uses that constructible subsets are determined by their closed points, that closed points have residue fields finite over the prime field, and most importantly the Chevalley theorem to eliminate the existential quantifiers. The completeness corresponds to $n=0$ and you easily conclude. I was then asked to prove the Chevalley theorem needed. I answered Noetherian induction and generic freeness. Then I presented the method by model theory, which is a standard application of (a weaker form) of the Łoś-Vaught test, since any two algebraically closed fields of the same uncountable cardinality are isomorphic. I think I did not apply it directly but essentially proved it in our special case, for which I used upward Löwenheim-Skolem.
I cannot remember exactly the rest of this part but I do remember I mentioned the ultraproduct of all $\mathbb Q_p$ as valued fields. I guess I was asked to spell out a favourite theorem that involves both model theory and algabraic geometry or number theory.
Algebraic geometry:
Q: Study smooth curves of genus 0 (over a field that is not necessarily algebraically closed).
I answered that such a curve is a quadratic plane curve and $\mathbb P^1$ if and only if it possesses a rational point or a line bundle of degree 1. I don't remember what was asked then, but I remembered I was pretty awkward in the rest of this question. Anyway, Prof. Kollár led me to the solutions he wanted with suggestions and hints.
Q: Compute the genus the normalization $C'$ of the projective curve $C$ given by $Y^2=\ldots$ over $\mathbb C$. (I cannot remember the explicit equation.)
I guess this is the question that you really should expect if you choose algebraic geometry as your topic. The standard method is to use Hurwitz formula. I didn't. I computed the Euler characteristic of the structure sheaf $\mathcal O_C$ of $C$ (which only depends on its degree), and pointed out that the difference between the Euler characteristics of $\mathcal O_C$ and of $\mathcal O_{C'}$ is the length of $f_*\mathcal O_{C'}/\mathcal O_C$ where $f$ is the normalization map. I pointed out that this sheaf is supported on finitely many closed points and identified all the stalks. I didn't compute the final result because it is pretty complicated in my exposition and I said we should leave this job to a computer. (I also explained why it reduces to a finite computation of matrices.) Then I was suggested the standard way and I followed. At some step I turned to explicit computation of the local rings at infinity to find the ramification index and was suggested I find the uniformizer by insepection on the affine equation. I could do this inspection formally, but I said I must do the computation to make sure that things are correct, etc., etc. We agreed that I was able do the computation, but much more elaborately.