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[数学应数] UPenn math PhD oral exam problems

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A.Einstein 发表于 2015-12-16 11:37:12 | 显示全部楼层 |阅读模式

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今天下午刚考完PhD oral exam,有兴趣把题目写一下~
. 1point 3acres 璁哄潧

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). From 1point 3acres bbs
. more info on 1point3acres.com

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


2. Some questions about minimal submanifolds:. 鍥磋鎴戜滑@1point 3 acres
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).. visit 1point3acres.com for more.
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)

. visit 1point3acres.com for more.
Minor topic: Complex Algebraic Geometry
鏉ユ簮涓浜.涓夊垎鍦拌鍧. 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? . From 1point 3acres bbs
.1point3acres缃

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

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lyleffective 发表于 2015-12-17 05:24:39 | 显示全部楼层
当面做题,确实有压力啊,还是笔试好。。。
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 楼主| A.Einstein 发表于 2015-12-17 07:04:03 | 显示全部楼层
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。。
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lyleffective 发表于 2015-12-17 08:42:24 | 显示全部楼层
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
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 楼主| A.Einstein 发表于 2015-12-17 11:38:00 | 显示全部楼层
lyleffective 发表于 2015-12-17 08:42. 1point 3acres 璁哄潧
哦,那也好啊,不过要是考六个小时也真是可怕. 鐣欏鐢宠璁哄潧-涓浜╀笁鍒嗗湴

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”= =。。
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lyleffective 发表于 2015-12-17 13:30:50 | 显示全部楼层
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
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