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Background: I am a sophomore who has taken one analysis course, and one linear algebra course. I am taking abstract algebra, 2nd read analysis, and topology this semester. I have participated in a directed reading program last semester and learned some representation theory and matrix groups. Using Serre's representation theory for finite groups and Tapp's Matrix Groups for Undergraduates. However, I don't think I understand algebra as deep as analysis. Since somewhat algebra seems more abstract to me.

Since I found them both interesting, but interesting in a different way. I wonder if it is true that researching in analysis requires not as many pre-requisites as researching in algebra? Since I noticed that there are a lot of topics in algebra, but not as many as in analysis. I understand that whatever topic I choose to dwell in doesn't really matter much toward my graduate study since I could always change before that. So I wonder what is more beneficial to me? Given my background, would it be easier to do an undergraduate project in analysis than algebra?

Alvis Nordkovich
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    If you're looking for something accessible, and you want to do something original, right now, at this very moment, then neither analysis nor algebra is a good choice. I would look into combinatorics. – Matt Samuel Jan 08 '19 at 03:00
  • That said, undergraduate projects in math aren't usually original. Mine was, but I was extremely advanced, and had essentially already taken a full year's worth of graduate courses before I discovered something new. Of course other people also do this, but it's the exception rather than the rule. Harvard warns their graduate students that their dissertation might be only a minuscule advancement in their field, and that's normal and not an indication of failure. – Matt Samuel Jan 08 '19 at 03:06
  • Thank you! I totally understand that undergraduate research is more reading than really researching. I believe I concern is that is there much content in algebra than analysis? @MattSamuel – Alvis Nordkovich Jan 08 '19 at 03:14
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    There is a massive amount of content in both of them. Analysis is sort of a younger subject, but it's still hundreds of years old. In any subject except combinatorics undergraduates may only barely get into the topics of the early 20th century. – Matt Samuel Jan 08 '19 at 03:20

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