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Profile review -- seeking advice on what programs I could realistically get into as an applied math major

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I feel that I am not good enough and I have doubts about getting into any PhD programs, especially given that I have not taken PDEs yet. I did apply to applied math REUs this summer, and I was very lucky to get into a very selective program and 2 other REUs at good schools, but I feel like I am not good enough. 

Here is my current profile:

  • Current institution: top 25 undergrad but not known for math
  • Domestic female applicant (I'm a US citizen)
  • Current GPA: 3.97
  • Classes taken:
    • Watered-down multi calculus class (red flag since I need to teach myself vector calculus so I can take PDEs) 
    • Real analysis
    • Real analysis II
    • Linear algebra
    • Math of data science
    • Numerical analysis 
    • ODEs
    • Graduate math class in subfield of numerical linear algebra
    • Calculus-based mathematical probability, mathematical statistics sequence (non proof based)
    • Cloud computing class
    • Statistical inference intro
    • Reading group on theoretical / statistical machine learning
    • Some programming classes in R and Python, some non-rigorous stats courses
  • Additional notes: Planning to take during the semester I apply: PDEs (undergrad), Numerical analysis (graduate), nonlinear optimization (undergrad), regression analysis (undergrad); planning to take in final semester: possibly measure theory (this is only offered as a graduate course)
  • Research experiences: 2 NSF math REUs (in applied / computational math), about 1 year of machine learning research assistant employment
  • Interests: optimization, statistical machine learning, applied deep learning for scientific computing, signal processing, numerical linear algebra, especially w/ biomedical applications 
  • Other: I have extensive PyTorch experience. 

Historically, students with similar profiles compared to me from my school have gotten into top programs, but they're smarter than me / they have something I lack (like physics background, research is more of a fit, their recommenders have ties to those programs, etc.). I used to be an English major. 

**Here are some schools I am interested in for applied math:** 

  • Brown
  • Rice
  • JHU
  • CU Boulder
  • NCSU
  • UMD
  • USC

**Here are some schools I am interested in for statistics:** 

  • Duke
  • Rice
  • NCSU
  • Northwestern
  • Columbia
  • Cornell
  • UPenn (Wharton)
  • Yale
  • University of Washington
  • CMU (probably not worth applying here, probs gonna be rejected)

I might apply to Harvard, JHU, UPenn, and Columbia for biostatistics.

Please, someone slap me in the face with cold hard reality. Which of these schools are unrealistic for me to apply to? What are some lower-tier schools I ought to consider? Thank you 

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I don't know anything about applied math, but your list for stats is reasonable but a little ambitious if those are your only schools.  (Not sure why you're hesitant about CMU - Duke, Columbia, Penn, Washington are all at least on the same level).  You don't have a lot of safer match options besides Rice, so I'd add more in the 15-35 range (thinking schools like UIUC, UT Austin, Ohio State, etc)  Northwestern is a bit of an outlier in your list- it's not very well-known and is really only a place I'd consider if you are dying to live in Chicago for personal reasons - multiple people on this forum have had terrible experiences with the department.

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I know that logically I should apply to lower-tier schools but I would rather go to a lower-ranked school for applied math than a lower-ranked school for statistics. So I'll be applying to some applied math programs that are lower-tier. 

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