I posted on gradcafe yesterday and got some insight regarding admissions to PhD programs for statistics, but I was hoping you guys could help me out/let me know if I'm being realistic with respect to mathematics grad schools. My primary concern is no original research in mathematics, and would like your opinions on how much weight top programs put on this in admissions.

Undergrad Institution: Imperial College London (Mathematics 4 year MSci integrated masters, graduated June 2018)

GPA: ~3.9 to 4.0 equivalent

Type of Student: International Male

Masters Institution: Oxbridge (Masters in Mathematics, currently attending, will apply to schools next year with grades from this in hand)

GRE General Test:

Q: 170

V: 167

W: 4.5

GRE Subject Test in Mathematics:

M: 92%

Programs Applying: PhD in Mathematics and Statistics

Research Experience: Two summers with different professors in the stats department, one of which led to a paper being published. Expository final year thesis in probability.

Awards/Honors/Recognitions: Deans list.

Pertinent Activities or Jobs: Internship as a quantitative trader. Will be working as a quantitative researcher analyst from July 2019 (hopefully only for a year then entering grad school).

Letters of Recommendation: 1st from the stats professor where the paper was published, 2nd from an analysis professor (supervisor of thesis), 3rd from an algebraic geometer. Should be strong.

Relevant Grades: I thought I'd give a description as I will be an international applicant. I didn't officially take the courses written in italics (its rare if not impossible to be given permission to take extra courses for credit, certainly not for this many) but my letter writers verify that I studied them to a high standard independently either under themselves or their colleagues. "A" grade equivalent in everything but a couple courses taken in the first year.

Analysis -- Analysis I, Real Analysis, Complex Analysis, Measure and Integration. (Functional Analysis, Analytic Methods in PDEs, Fourier Series and Theory of Distributions, Stochastic Calculus)

Algebra -- Algebra I, II, III, IV sequence (starts from undergrad vector spaces/linear algebra, groups and rings and builds to homological algebra and graduate groups/rings). Then Galois Theory, Lie Algebras, Commutative Algebra. (Infinite Groups, Group Representation Theory, Modular Representation Theory)

Geometry/Topology -- Metric Spaces and Topology, Algebraic Topology, Algebraic Geometry, Differential Topology. (Manifolds, Riemannian Geometry, Complex Manifolds)

Probability/Statistics -- Probability and Statistics I, II, (measure theoretic) Probability and Markov Processes. Then Applied Probability, Time Series, Generalised Linear Models. (Statistical Theory, Statistical Inference)

Applied stuffs -- Methods I, II, DEs, Multivariable Calc/Fourier/PDEs, Applied Analysis. (Function Spaces and Applications, Advanced Topics in PDEs)

I have not yet decided what courses I will take to examinations this summer, but it will likely be a mix of analysis, stats and geometry right now.

Applying to where: Only really looking at top programs in either field tbh. Especially given that I may go back into industry at some point, branding is pretty important, and supposing I do want to come back to the UK it'd be very useful to have an internationally renowned name behind me.

Statistics - CMU, Berkeley, Stanford, Harvard, Chicago, Columbia

Mathematics - Berkeley, Stanford, Harvard, Columbia, Chicago, Princeton, MIT.

I'm not really sure if my profile is suited more to statistics or maths programs honestly - while I have research in statistics, and my job is also statistics heavy, my coursework and extra reading is dominated by maths.

Does anyone know what the calibre of incoming students at these programs is roughly like? Having checked the syllabi and done some timed past papers I know I could pass Harvard/Stanford quals and Berkeley prelims right now, and judging by the level of depth in some transcripts I'm almost ready to take Berkeley/Princeton style oral quals with concentration in say representation theory and algebraic geometry. But my concern is, if given the plethora of talent that these schools can select from, is basically everyone in my position or better with regards to mathematical knowledge upon entering the program? Because if so, I'm quite worried about my lack of original research now.

Thanks if you read, any thoughts would be greatly appreciated!