I'm new to this website, so I'd apologize if what I'm posting isn't relevant. I'm currently a sophomore, but since I already took essentially all the upper-level math and physics classes in my school (small LAC), I'm considering to graduate after my third year. I'm currently at Math in Moscow and taking Riemann Surfaces, Commutative/Homological Algebras, Homology and Cohomology, and Complex Analysis. I understand that graduating a year early is always discouraged, but since I have nothing to do anymore in this school and don't want to pay expensive tuition (60 k!) for that, I'm seriously considering this option. Could you give me any advice regarding which PhD programs I should aim for based on the following information? If I have no chance for any well-respected PhD programs, should I rather apply to any Masters programs?

Undergrad Institution: Top 20 LAC

Major(s): Mathematics

Minor(s): X

GPA: about 3.75 (major)/3.7 (overall)

Type of Student: International, Male, Asian

GRE Revised General Test: not yet taken (but I believe my verbal score will be low, just like many int'l Asian students)

Q: (%)

V: (%)

W: (%)

GRE Subject Test in Mathematics:

M: 890

Program Applying: Pure Math

Research Experience: 1 REU (I have another REU in this coming summer, and I hope I can publish a paper from that work.)

Awards/Honors/Recognitions: getting 41 in Putnam (2013)

Pertinent Activities or Jobs: currently attending Math in Moscow

Any Miscellaneous Points that Might Help: I took several reading courses to study category theory, differential geometry, and statistical physics at graduate-level.

Any Other Info That Shows Up On Your App and Might Matter: Until the last semester I have taken abstract algebra, rep theory, experimental mathematics, advanced set theory and logic, and field/Galois theory as well as QM, statistical mechanics, and EM.