**Undergrad:**T10 UK university for maths

**Masters:**Part III maths at Cambridge

**Major(s):**Mathematics for both (technically "Pure Mathematics" at Cambridge)

**Minor(s):**N/A

**GPA:**High(?) First Class in my undergrad

**Type of Student:**International White Male

**GRE Revised General Test:**Not taking

**GRE Subject Test in Mathematics:**Ditto (would have to fly over to France or Bulgaria to take)

**Program Applying:**PhD Pure Math, looking to specialise in functional analysis/operator theory. Very ambivalent towards PDEs, I'm only really looking at FA/operators for now.

**Research Experience:**Not really research per se but did a project on Leray-Hopf solutions to the Navier-Stokes equations as part of my undergrad (not original, more of a survey). I will be doing a Part III essay somewhere in functional analysis, tending towards geometry of Banach spaces considering the specialisms of my desired supervisor.

**Awards/Honors/Recognitions:**None since high school (one of the downsides of small fish big pond)

**Pertinent Activities or Jobs:**I wrote much of ProofWiki's coverage of functional analysis and measure theory. Competed in a small inter-university integration competition this year and plan on helping coordinate next year's.

**Any Miscellaneous Points that Might Help:**My coursework is probably the strongest thing about my application. (if it even matters much) Grad-level courses would be three courses in functional analysis going up to C* algebras & Borel functional calculus and another course "Unbounded Operators and Semigroups", manifolds, analytic theory of PDEs, non-linear analysis, modular forms, measure-theoretic probability. Otherwise decent mix of undergrad courses, courses in groups/rings, linear algebra, real/complex/multivariable analysis, point-set topology, basic non-Euclidean geometry, number theory, measure theory, set theory, logic and a first course in undergrad statistics. My last year was slightly disappointing (had issues with insomnia around exams) with a few 2:1s but still did fairly well. (my overall grade is mid 80s)

Currently my list of North American schools is:

UIUC

Purdue

Vanderbilt

Dartmouth (throwing this in because it doesn't have an application fee mainly)

Alberta

Kent State

Nebraska-Lincoln

Maybe throwing in Oregon too

This'd be in addition to 4 or 5 in the UK to get a total of like 11-13 schools. My first reference letter will be from my undergraduate tutor who has a pretty high opinion of me, and who I met with quite a lot during my second year. Very recognisable name internationally but in an area that's not mine. Second letter writer was the supervisor of my third year project and was very impressed with my work. While he's in PDEs and harmonic analysis more so than functional, he has a low collaborative distance with quite a few people at the universities I'm applying to, so he might be a known name to some. (though I'm not sure if any will know him well) Third letter writer is a bit murkier, I will basically have the option between my first year undergrad tutor (who I met a few times and wrote one of my letters for Cambridge) and my tutor in Cambridge who I'll have only met a few times by time of applying. Neither are in relevant areas but the latter is a bit more well known than the former.

Questions:

(1) Are there any big names that I'm missing? Texas A&M is the no brainer but unfortunately they're requiring the subject GRE. I asked if they'd consider an application without it (nothing to lose just asking) but haven't got a response.

(2) I'm worried my letters will seem a bit weak, especially going out with a whimper when they see the third. Is that justified here?

(3) Am I shooting a bit too high? Should I look into more safeties? I think if I fail to get a (funded) PhD offer anywhere this year, I will probably just go off into industry, I don't really want to be reapplying year after year, so I really want to take my best shot this year.

Thanks a bunch!