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Seeking Advice on Preparing for Mathematics Graduate Programs

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Dear TheGradCafe friends,


I am a college sophomore 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.






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I'm a little surprised that you are considering taking Linear Algebra and Abstract Algebra at the same time. I think the norm is to take Linear Algebra first to develop some "math maturity" before moving on to Abstract Algebra. However, you have studied some algebra on your own so you may be fine. You can try taking them both at the same time and drop one of the courses if the course load is too much.

Some schools only require Analysis and Linear Algebra, some schools may require more. This is something you can look up online for each school you are interested in. You can also talk to your advisor(s) and see what they suggest. Since you are interested in applied math they may suggest developing a strong background in analysis.

Undergraduate research is very important. Of course it is not as rigorous as graduate level research, but any research experience is a good research experience. Grad programs like to see when applicants have done undergrad research and it will make your application stand out. It shows that you know the process and that you will be capable of doing it in the future.

Putnam score will be impressive (you can list it under an "Awards" section in your CV) but I doubt that will be more important than other components of your application (such as research experience, letters of recommendation, etc.). I think your undergrad research and coursework will be more important.

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"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."

No offense, but this sounds pretty silly. Putnam problems are by and large toy problems. Now I do recall one problem that was related to Hensel's lemma, but if you want to study number theory, study number theory. The Putnam can still be a rewarding process, and if you do well, you can put it on your CV, but don't expect it to make much of a difference or to contribute to your study of number theory.

I would recommend you think about applying to CS programs, even if you start out with a Master's. There's much more crypto work in CS than in math. Of course you'd need to take some classes like OS, Computer Systems, discrete math, automata theory, algorithms, data structures, etc.

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