Hello everyone too! I have a similar question so I posted it here. I am new so if I should post it anywhere else please notify me and forgive me.
I have sort of the same problem here. I am applying for Ph.D in Pure Math with an intend of doing research in algebra.
I am from a small greek university majoring in math. I was top of my class with a current overall GPA of 9.66/10. I have many algebraic courses covering introduction to Galois Theory, PIDs, UFD's, modules (projective, injective, cofree, essential extensions), tensor product, hom functors, simple and semisimple rings and aalgebras, categorical notions and group theory (free - solvable - simple groups Sylow theorems, Nielsen-Schreier Theorem). I have also had a basic course in algebraic curves which stirred my interest towards algebraic geometry ( i find it ineresting but have not dealt with algebraic varieties and so on). Some of the topics above I did in oral presentations as projects.
I have had also two semesters in functional analysis (Hahn Banach theorem, open mapping, closed graph, Hilbert base, hilbert dimension,weak - weak* topologies,compact linear operators) the second being a seminar course,one introductory in set theory (cardinals, ordinals, AC), one intro in measure theory (till integrals, the last part very quickly and not deeply), one in real analysis (argela-ascoli, weierstrass-stone,semicontinuity), one in topology of metric spaces ( compact, donnected etc..), one in discrete math (graphs, Euler-Bernoulli numbers, combinatorics, generating functions, difference equations). Also: 1 course in differential geometry (curves, surfaces, Egregium Theorema, developable surfaces, curvature lines, omfalic points etc -not Gauss Bonnet though), 1 in non-Euclidean Geometry (Hyperbolic mainly through a model-axiomatic approach, elementary methods used, not advanced differential calculus involved), 1 in differentiable manifolds ( smooth functions, total differential, tangent space, tangent bundles, implicit theorem and inverse mapping etc). What else? I think 1 elementary number theory, the standard calculus-linear algebra sequence, 1 in ODEs (not systems though).
Attending currently:1 in general topology, will attend complex analysis (residue theorem) next semester.
No research. Have done oral presentations in real analysis, algebraic topics. Also enrollment in graduate courses is not allowed for undergrads in my uni, but I attend unoficially graduate course "Algebra I" (commutative algebra) and will attend unoficially the graduate courses "Algebra II" and "algebraic topology" next semester.
I think I will have very good recommendations (however, overall I think me recommenders are not very well- known outside Greece)
I scored 730 (69%) GRE math subject. The General I will take next week.
My question is the same as Alexyar's (though he seems to have a much stronger background doing research an' all
) What universities should I apply to? Do I have a decent chance in some top program or am I beingg silly now?