--------------------------------------------------------------------------- Eaman Eftekhary's generals (Jan 17, 2001) Topics: Algebraic Geometry, Algebraic Topology Committe: J.Kollar (chair), W. Browder, R. Gunning (lasted about 1:40) ---------------------------------------------------------------------------- Gunning began with asking some questions in analysis: G: geometric property of holomorphic functions? regions conformally equivalent to the interior of unit disk (Riemann mapping)? talk about extension of a hol. map from a triangle to disk to a neighborhood of triangle. different types of singularities? (Big Picard?) [and some other questions that I've forgotten] G: different notions of convergence for functions over [0,1], L^p spaces? how are L^p spaces related? prove it. does the convergence in L^1 imply pointwise convergence ? what if the sequence is bounded ? [dominated convergence]. how do you prove that L^p is complete? [again, I don't remember the rest of it!] Then Kollar started with algebraic geometry: K: compute the genus of y^3 = x^6-1. I started with char(k)=2,3 and said that I should use Frobenius map to reduce to another equation, Kollar said what's the inverse of this map,... then I moved into computations, blowed up the curve at its singularity a few times till I got a nonsingular curve, counted the # of pts mapped to the singular pt of the initial curve, composed this map with (x;y;z) --> (y;z) to P^1, computed the ramification divisor and finally got the genus of the normalization using Hurwitz formula, but I didn't know that this is actually the geometric genus of the initial curve and Kollar was not happy with it. Then he asked me to define arithmetic genus, and I did. he asked me how I prove the genus formula, I said through Hurwitz formula. He didn't like that way, and asked whether I can do it differently. I wrote the sequence 0--> L(-C)--> O_X --> O_C -->0 (X=P^2) and computed the arithmetic genus using [C]=[dH]. Then he asked me to state the explicit computation giving h^i(P^n,O(m)) and Serre duality. He mentioned that I didn't use nonsingularity of C and hence the genus formula is true for arbitrary curve, giving arithmatic genus. K: talk about the extension determined by the function fields of C and P^1 determined by the map you gave. I said it's finite of degree 6 and showed that it's separable when char(k) is not 2 or 3, in case char(k)=3, I mentioned that the function is reducable,.... Then they asked Browder if he is willing to ask some algebraic topology questions! B: what are Stiefel-Whitney classes good for? (I showed that P^n does not immerse in R^{2n-2} for n a power of 2) ---------------------------------------------------------------------------- comments: as everybody will probably note, the exam was not as fast and smooth as it appears here, I made some mistakes, but they kindly tried to let me find them, by myself. doing math, when you're doing with 6 expert eyes following you, is harder than doing it for yourself. ---------------------------------------------------------------------------