I am currently a freshman at a t50 institution, domestic male, not URM, expecting a degree in honors mathematics, honors statistics, and computer science (still not decided if I want to follow through on the computer science major, but will get a minor). Currently have a 4.0 GPA, expect 3.8+ GPA by graduation. I would like to go to a pure math PhD program that is strong in analytic and algebraic number theory, combinatorics, algebra, and maybe probability and harmonic analysis.

**Coursework**:

I have taken/am taking topology, analysis 1, 2, linear algebra 1, 2, abstract algebra, grad intro algebra, grad measure theory/integration, and a reading course in field theory/algebraic number theory.

I expect to finish the commutative algebra 1, 2 sequence, complex analysis, probability 1, 2 sequence, functional analysis, algebraic topology, diff geometry/topology, class field theory, graph theory, and several topics courses. I have space in my schedule for 15-20 grad courses in math/theoretical cs total by the time I graduate.

**Research**:

I have had a very computational research experience. It was fun, but will likely not be mentioned on apps and did not result in publication. I will have an REU this summer in additive number theory, and I will have a research project during the year in algebraic number theory next year. My mentors for both projects say they expect we should be able to produce publishable results. I expect to do 2 more REU's (if I can get into more) the following summers and continue research in number theory. Also, my faculty mentor here says that my REU project looks very unique and would probably look a lot better than most other programs I applied to. I think I can reasonably publish 3-4 papers at the minimum as a coauthor by the time I graduate.

**Exams**:

I scored a 19 on the Putnam. This was my first attempt at competition math, and I believe I should have done better. I think that I may be able to get a 30-40 on the 2024 Putnam, and I plan to study consistently this time around. Is a 30-40 something that would help my application for a HYPSM etc school?

I plan to study for the GRE this summer and next summer. What should I be aiming for on the GRE quantitative and GRE subject score? Is a 85 percentile solid for both? 90?

**Other Things**:

I think I will be an officer of the math club, and I will likely be a tutor. I am pretty sure I will win a few awards related to math from my school. I also have an interesting and compelling story to tell in my personal statements/in interviews about my introduction to mathematics.

I also think I will get a very strong letter of recommendation from my faculty mentor here. I am going to try to get a letter of recommendation from one of the famous faculty at my school in my current areas of interest, and I think I have made a good impression on a couple. I will also try to get a recommendation from an REU mentor.

From the admissions profile threads, it is not clear to me that a profile like this would get accepted to a t10 school. Would I need more focus in my application, better scores, better gpa? Are my areas of interest too competitive? Idk what to think. Is it appropriate to reach out to admissions committees at top universities to ask their opinion since I am not currently in the process of applying, and how would I go about doing this? Thank you so much if you got to this point :)