UPenn math PhD oral exam problems

今天下午刚考完PhD oral exam，有兴趣把题目写一下～


先简单介绍一下，这边的oral exam就是PhD qualify exam。oral exam的意思就是当面在黑板上做题。每个人首先选一个major topic和一个minor topic，然后选3个老师组成committee。考试时间依老师心情而定，从两个小时到6个小时不等。今天考的比较多，一个半小时就考完了，大概老师也累了吧～


Major Topic: Differential Geometry
(下面提到的度量除非特别说明，都假设是complete的)


1.Basic problems: find all totally geodesic submanifolds in S^n (Standard Sphere) and H^n (Hyperbolic spaces)


2. Some questions about minimal submanifolds:
Give some examples of minimal surfaces in R^3 (I said 2-planes, catenoids, helicoids). Show that catenoids are complete (w.r.t the metric).
If M^n is a minimal hypersurface in R^{n+1}, what can you say about the (sectional) curvature of M?
Can a minimal hypersurface of S^n be contained in an open hemisphere of S^n?
Let M^k be a minimal submanifold of R^n of codim more than 1, can M^k be compact? (I didn't solve that one)


3.Sphere packing
Assume M is a n-dim complete Riemann manifold with Ric>=-K(n-1), where K>0. Let r<R be 2 positive numbers. How many balls of radii r in M can be packed into a ball of radius R in M? Can you bound this number from above or below? Relate this problem to the problem of covering B_R by B_(2r). (How many B_(2r)'s are needed in the last problem?)


4.Let E be the tautological line bundle of RP^n, can (the total space) of E carry a metric with sec>0? sec>=0? What if I replace RP^n by CP^n (warning: then the tautological line bundle will become a COMPLEX line bundle)?


5.Challenging question: Let g be a metric on S^2, s.t. sec>=1, and S^2 has a closed geodesic of length exactly 2pi. Show that g is the round metric.(I didn't have time for this question)


Minor topic: Complex Algebraic Geometry
I didn't do a good job in this part.. In fact I got stuck at the first problem..
Let C be a complex algebraic curve of genus 5, can C be embeded into P^2? (I only solved this one..)
Let L be a line bundle on the above C, if the map associated with linear systems of L maps C into P^2, and the map is a non-degenerate morphism (i.e. everywhere defined , the image is not contained in any linear P^1 of P^2), what is the minimal degree of L?


Another problem: give an example of effective but not ample divisor (I gave the exceptional divisor of a blow-up of P^2).

lyleffective 发表于 2015-12-17 08:42
哦，那也好啊，不过要是考六个小时也真是可怕

我也要考math qualify，是4h 笔试，希望这次出 ...

看老师。oral exam其实是个很不正式的考试，难度完全由老师自己决定。笔试正规一些，不过4小时笔试也太长了＝ ＝。。宾大以前也有笔试，不过据说“越出越难，以至于部分题目连faculty都不会做了，就索性取消了”＝ ＝。。

喔 前排

当面做题，确实有压力啊，还是笔试好。。。

lyleffective 发表于 2015-12-17 05:24
当面做题，确实有压力啊，还是笔试好。。。

问的都是基础的概念问题和大定理的直接应用，而且做不出来老师会给提示。笔试反倒是可以出得很难呢。。

A.Einstein 发表于 2015-12-17 07:04
问的都是基础的概念问题和大定理的直接应用，而且做不出来老师会给提示。笔试反倒是可以出得很难呢。。

哦，那也好啊，不过要是考六个小时也真是可怕

我也要考math qualify，是4h 笔试，希望这次出题教授按常理出牌

A.Einstein 发表于 2015-12-17 11:38
看老师。oral exam其实是个很不正式的考试，难度完全由老师自己决定。笔试正规一些，不过4小时笔试也太长 ...

那是得有多难。。。上古神题吧。。。搞那么难干嘛