Seeking Your Advice on Preparing for Mathematics Graduate Programs S.E advisers,

I am a rising college junior (five-years tract) in US with a major in mathematics and an aspiring mathematician in the fields of theoretical computing and cryptography.  I apologize for this sudden interruption but I wrote this post to seek your advice on preparing for the mathematics graduate programs; I am currently attracted to the applied mathematics.  Through introduction to the computational biology, I found my true passion and love toward the mathematics, which led me to switch my major from the microbiology to mathematics on the last (Spring) semester.  Since then, I have been taking basic calculus courses and also self-studied the preparatory materials (such as proof-writing) for the abstract mathematics.  I am currently self-studying the abstract algebra (Artin, Dummit/Foote), abstract linear algebra (Hoffman/Kunze), and vector calculus (Lang).  I will be taking the following courses on the Fall:  Vector Calculus-computational, Linear Algebra with introduction to proofs, and either Abstract Algebra I or Mathematical Statistics I.  At the start of August, I will be starting to conduct the supervised undergraduate research on the theoretical computing (abstract and math-heavy) and computer security (more on the programming than mathematics), which I am very excited.  

I am curious what are the essential factors for admission into mathematics graduate programs.  Do undergraduate research (such as mine) will be an important factor too?  I know the biology programs strongly require the research experience but I frequently heard that the mathematics programs focus less on the undergraduate research since most undergraduate-level research in mathematics are viewed as less matured than the graduate-level research.  Does Putnam score also play an important factor?  I am interested in preparing for the Putnam Competition as it has many interesting problems on algebra and number theory (topics of my interest) and I have been preparing vigorously with the problem-solving-strategy books as I never participated in any mathematical competition before.  About my current course schedule, should I take the Abstract Algebra I with other courses?  At my university, the AAI hs the pre-requisite of linear algebra, but I was granted a special permission for enrollment since my research advisers strongly recommend me to take the abstract algebra for the theoretical computing and computer security.  I am currently studying the abstract algebra and linear algebra to prepare for the courses and upcoming undergraduate research.  

Thank you very much for your time, and I apologize for this long post.  I look forward to your reply.

Sincerely,

PK
No, undergraduate research is not a very important factor for graduate admissions in mathematics.  It's nice if you have it, and if by any chance you publish a solo author paper in a respectable journal that's fantastic, but that kind of fantastic stuff is done by people who would assured of admissions to top programs anyway.  Since every letter ever written about a math student's undergraduate research is positive, admissions committees just don't get enough information to use in tough admissions choices.  


  I am a rising college junior (five-years tract) in US with a major in mathematics and an aspiring mathematician in the fields of theoretical computing and cryptography. I apologize for this sudden interruption but I wrote this post to seek your advice on preparing for the mathematics graduate programs; I am currently attracted to the applied mathematics. Through introduction to the computational biology, I found my true passion and love toward the mathematics, 


You have named at least four different fields in one introductory paragraph.  This is a bit disconcerting to a potential admitter.  We might worry that you have not managed to settle down on any one thing for long enough to study it in any depth.  Maybe that worry is totally groundless, but you should do what you can to allay it.


  Does Putnam score also play an important factor? I am interested in preparing for the Putnam Competition as it has many interesting problems on algebra and number theory (topics of my interest) and I have been preparing vigorously with the problem-solving-strategy books as I never participated in any mathematical competition before.


Similar answer to the last one but more extreme.  If you score in the Top XX, that looks good, but it would be very rare for that to be the breakthrough good thing about your application.  Moreover, whereas one must admit that undergraduate research in mathematics is becoming increasingly popular and viewed increasingly seriously over the years, the same cannot be said about the Putnam: if you never take it, no one cares.  If you got a zero every single time, no one cares.  (I took it in my freshman year only, honestly don't remember my score but suspect it might have been in the single digits, and I got into every program I applied to, including the top three.)  I was never excited about contest math so I need to ensure that my own proclivities don't creep into my answer, but quite soberly: studying for the Putnam (or any university math contest) might be fun and it might be educational, but it is not much of an investment in your future.  If you like algebra and number theory, great (OMG! me too) -- study algebra and number theory.  Working Putnam problems would teach you more of that than watching Game of Thrones, but it's still quite indirect.


  About my current course schedule, should I take the Abstract Algebra I with other courses? At my university, the AAI hs the pre-requisite of linear algebra, but I was granted a special permission for enrollment since my research advisers strongly recommend me to take the abstract algebra for the theoretical computing and computer security. I am currently studying the abstract algebra and linear algebra to prepare for the courses and upcoming undergraduate research. 


This type of question needs to be answered by someone who is familiar with you and with your department's program.  We're not.  It's exactly the sort of question you should ask your faculty advisor.  If you don't have a faculty advisor yet, no problem: this is an excellent opportunity to get one.

One final comment: based on your writing, I suspect you are not a native speaker of English.  Good English language and writing skills are important for an American PhD program in mathematics: a low enough GRE verbal score can kill an application.  (Conversely, in my program, in order to be competitive for the top drawer internal fellowships you have to have a high "academic index", which is some weighted numerical measure of your GRE quantitative, your GRE verbal and your GPA.  So it's serious.)  Shoring up shaky English skills is more of a priority than studying for the Putnam exam, if you ask me.