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.