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Profile evaluation | Fall 16 | PhD Math | GRE V160, Q170, A4.5 | Math GRE 740(68%) | TOEFL 116 | UG: 8.5/10 | PG: 3.5/4

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Profile evaluation for PhD Mathematics
I am basically trying to shift to Mathematics from Engineering.

UG College: BITS Pilani, Goa Campus (India, well-known)
Branch: Chemical Engineering
GPA: 8.53 / 10

PG College: Erasmus Mundus program in University of L'Aquila (Italy, not well-known)
Branch: Mathematical Modelling
GPA: 3.5 / 4

Field of Interest: Stochastic Analysis and Financial Mathematics
Degree Goal: PhD

GRE General: 330 (V160 + Q170) + AWA 4.5
GRE Subject (Mathematics): 740 (68 %ile) :(
TOEFL iBT: 116 (R30 + L30 + S27 + W29)

Academic Projects:
1) Master's thesis: On a pre-published numerical method in Mathematical Finance. 6 month research; decent, nothing great, no significant new results.
2) One short project on a classic problem in probability and statistics.
3) Very short summer project on a problem in Kaggle in Machine Learning.

Recommendations: Two strong + one weak (all University of L'Aquila)

Work experience (after UG, 1.5 years) 
Mu Sigma Business Solutions Pvt. Ltd. | Senior Business Analyst | Innovation and Development department 
I was involved in implementing Machine Learning algorithms for Big Data using R. I was also involved in training the other teams on Big Data and R, and created training content on them.

Oracle Financial Services Software Limited, Bangalore | Business Trainee | 5.5 months 
Development of a project-employee management tool.

Central Food Technological Research Institute, Mysore | Research Assistant | 2 months 
Characterization of Ionic and Mixed Reverse Micelles (Irrelevant) 

Publications: None

Misc Achievements: Merit based scholarship for Masters education

Tentative Universities
Super ambitious:
Princeton (ORFE)
Brown (Applied Math)
U Illinois Urbana Champaign

U Texas @ Austin
U Washington
UC San Diego
Purdue U
Ohio S
Georgia Tech
Carnegie Mellon U (Financial Math)

Boston U
UC Santa Barbara (Stat)
Michigan S (Stat)
Texas A&M U
U Southern California

Iowa S
North Carolina S
Louisiana S
Florida S (Financial Math)

1. I have no clue where I stand. I am not sure if the universities I have denoted 'Reach' and 'Safe' are really so for my profile.
2. I do not have a very strong math background on paper (all UG math courses missing). In light of this, should I send my not-so-good GRE Math score when I have the option not to?
3. Should I also apply to lower universities?
4. Please suggest any other university I should look at?

Thanks for any suggestions and apologies for the long post.
@Everyone: Sorry if this is the wrong forum for this post. I am a new user here. Thanks for understanding.

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