This is an insanely long post, but I appreciate anyone who takes the time to read it and respond. I just feel like my situation is kind of unique (and I am very concerned about graduate school). If I should make a separate post for this, I will, but I thought this would be a good place to ask questions.
I am a 4th year undergraduate Mathematics student and Computer Science minor (I will graduate at the end of my 5th year) at a state school and I recently got really serious about graduate school.
I kind of went off track after my first semester due to a painful Calculus II teacher, taking a bunch of gen. eds, so I didn't take my intro to proofs class until last spring, which was when I was invited to, and joined, the honors program.
This school year, I decided to take the Honors Abstract Algebra sequence (intro/prerequisite course for grad students) as my first upper division course. It has been a huge challenge (my professor went to Harvard undergrad and Berkeley for his PhD in Algebraic Geometry, so he is a genius), but I have learned SO much. I plan to take the graduate level sequence next year, and I am very excited! I now want to pursue a PhD with a focus in Algebra, but I am concerned because this is the one course that I have gotten a B in. Essentially all of the schools I am looking at are top 50 schools, because I really don't want to ensure that it is worth it, since a PhD in pure maths is going to be extremely challenging, and I want to be a professor at a school that is at least as good as my current institution.
Basically I want to know, do I even have a chance?
I have taken the following courses
(Sophomore Level) Ordinary Differential Equations I - A Multivariable Calculus - A Linear Algebra I - A
(Junior Level) Honors: Intro To Abstract Math - A Intro To Scientific Computing (Computer Science Version) - B+ (So dumb. I forgot an online quiz which dropped my quiz average from a 90 to a 75, but I'm not horribly worried since this is applied math and it's listed as a CS course) Probability & Statistics - Expected A
(Senior Level) Ordinary Differential Equations I - A Honors: Abstract Algebra I - B Partial Differential Equations - Expected/Certain A Honors: Abstract Algebra II - If I pull off an A- on the final, I should get an A-
Unfortunately I was unable to spread out my courses nicely because I was kind of a late bloomer, so next year, I plan to take: Honors: Advanced Calculus I & II (Senior Level Analysis Sequence) Modern Algebra I & II (Graduate Level Sequence) Honors: Topology (Senior Level) Complex Variables (Senior Level Complex Analysis)
My main concern is the fact that I am going to telling admissions that I want to focus on Algebra, and this is the one course that I have gotten a B in! What is also very bothersome for me, is that even though I got a B, I am probably more prepared than some students who got an A in the same course. Do the people doing admissions consider this? My professor really works the hell out of us, but I feel that my math skills have grown exponentially in this sequence. Will they also consider the fact that this was my first real upper division math course? It is quite a jump for the average math major.
I hope to ask my Algebra professor for a graduate school recommendation, but I feel weird about it because I got a B the first semester, and he is also just VERY awkward. I essentially live in his office, and he knows I am hard working -- I type very detailed proofs for the homework (Artin), and my homework average both semesters was a 92-94. It's just his tests are so hard; no one ever finishes them, and there are very low distributions. There are also only 5 students in the course, so whoever scores the highest get's the A. Apparently he teaches Jordan Canonical form in the introductory Linear Algebra course, which I personally think would be kind of a shock for the average math or engineering student. He is also the one who initially suggested I take the grad sequence after I told him I plan to continue on to grad school. I try to talk to him and get to know him better and let him know my plans and he's just not the best with conversation :( Should I be this concerned about asking him for a recommendation?
I am so passionate about math. I love linear algebra, so this course has been amazing since Artin's book has a huge focus on Linear Algebra. I am confident that I can be successful in grad school because some days I spend up to 12 hours just studying math. I do it for fun too, I even have a vault of Linear Algebra test questions that I collect/create, for when I (hopefully) one day can be a linear algebra professor. This summer I plan not to work (maybe some tutoring) but prepare for the Analysis Sequence, review Algebra, and study for the Math GRE (I took it last month and am still waiting on results). Most summers, I study a bit to prepare for my math classes, but this summer, I really want to make an effort to do so every week. It's crucial that I keep my mind sharp and continue preparing for graduate school if I really want to do this.
So many forum threads I see, people always say stuff like "Oh if you got a B then you have not a chance at any top schools.", which freaks me out. I hope that my experience as a tutor (~2 years), SI Leader for Computer Science, and grader for the junior level Abstract Algebra course will help my application. I have 2 other professors that will certainly write me awesome recommendations. I also received 5 departmental math scholarships/awards since last April, which I guess can help my application? I know that my statement of purpose will also strengthen my application, since I am very passionate, I know myself, and my heart is definitely in it 100% (I think my personal essays helped me get so many scholarships). And lastly, I am trying to increase my GRE score as much as I can, and plan to apply to a diverse collection of over 10 schools, all around the US.
Okay, I write WAAAY too much. But I've just been so overwhelmed with everything. I just want to be successful and do what I love, and show people the beauty of mathematics!
Any advice on other ways to strengthen my application, or other tips for preparing for grad school are very appreciated!