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美国留学 | 教授发来了积极回复,怎么保持联系?

最近陆续发了十几封套磁,有三个教授发来了比较积极的回复,我就回信表达了感激,和希望能在他指导下学习,我会尽快申请,如果他需要其他信息请联系我这样的回信。我觉得应该下offer前都一直和教授保持联系,表达自己已经准备好去他项目了比较好,但是有点无从下手。比如看了他paper之后可以提一些很浅显的问题吗?因为我是本科生申phd,感觉提出自己创新性的想法可能有点难,还有什么其他的和教授保持联系,留下好印象的方法吗?最终目标我想套出教授愿不愿意给我funding,和明确向招生委员会推荐我,希望得到大家的指导。
下附三个回信
1. Thank you for your interest.  You have a promising background.  I intend to accept 2-3 students this year.  Please let me encourage you to submit an application to be considered.

2. Thank you fro reaching out to me. I will be accepting one or two more graduate students in my group. Your CV does look excellent and would be very competitive amongst other applicants. Therefore I would strongly encourage you to apply for admissions into my department.

3. Thanks much for your interests in our program.  Also let me know your TOEFL and GRE scores if / when you have.  Check our departmental web to see if your scores meet the requirement.
In addition to background of geoscience and atmospheric sciences, our program also requires a good level of math and computing.  Mostly importantly, one shall be interested in modeling research.  Sounds you are well qualified in these regards.
(这个后来我把TG成绩告诉他就没有回信了)

另外我这月底还要刷一次T,近三周没空集中看论文,下个月再交流他的paper会不会太迟了,显得我不够积极?