What are my odds of getting into these math masters?

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A senior math major is pursuing a master's degree to enhance competitiveness for top 20 PhD programs, citing concerns about preparedness and the math subject test requirement. They are applying to several schools ranked 30-50, including the University of Washington and Georgia Tech, due to their lack of a subject test requirement. With a strong GPA of 3.86 and significant coursework, research experience, and leadership roles, the applicant seeks feedback on their chances for admission. Another participant shares their own experience of applying to PhD programs and emphasizes the importance of building relationships with faculty and securing research opportunities. Overall, the discussion highlights strategies for improving application prospects in competitive academic environments.
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I am a senior math major and my goal is to get into a top 20 PhD program. Right now I do not feel competitive enough to get into those schools, so I am pursuing a masters degree first.

I'm applying to these schools:

-University of Washington

-Texas A&M - College Station

-University of California - Irvine

-UNC - Chapel Hill

-Georgia Tech

-Purdue

-University of Southern California

-Rensselaer Polytechnic Institute

I picked those schools because they are ranked in the top 30-50 and do not require the math subject test. I'm not prepared to take that test right now and do well on it. And coming from such a small program, I don't think I'm prepared anyways to succeed in a top 20 school.

Here are my stats:

School: Unranked, unknown state university

GPA: 3.86

Classes:

-20 college math classes total (from Calculus II and beyond)

-2 semesters of algebra, 2 semester of analysis, 2 semesters of geometry, 3 independent studies, complex, actuarial science, combinatorics, differential equations, graph theory.

Letters:

-1 would be a very strong one from a professor I've taken several classes with and did research with

-1 would be from a professor I've had several classes with including an independent study

-1 would be from my REU advisor.

Research:

-REU

-Two papers in the process of being published, one from REU, one from summer research with a professor

Talks:

-Gave talks at two conferences at separate universities

-Gave special sessions talk at the AMS/MAA joint meeting in San Diego

GRE scores: Unknown, but assume at least good, if not excellent

Extracurricular:

-Math club president

-Pi Mu Epsilon vice president

-Putnam team member for 2 years (scored a 0...)

-Honors program

-Eagle Scout (2009 Eagle Scout of the Year)

Those are my stats. How would you rate my chances if you could quantify it?
 
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I have similar stats, including a publication, and no GRE subject. My undergrad was from a small and new program. Last year I applied to PhD programs at Duke, UT Austin, Vanderbilt, and Arizona. I got turned down outright at Duke, and wait-listed everywhere else, but didn't end up getting in anywhere. I got into the masters program at my undergrad school and I'm applying to PhD programs again this year. I've been focussing on building relationships and improving my CV and cover letter, in addition to putting up a website with some applications I've written and pretty pictures of stuff I've been working on. I feel a lot better this year, and I won't be applying anywhere that I haven't been invited to by a faculty member.

See if you can still get into an REU this summer at one of the schools, that should help a lot- they expect you to be familiar with the research going on and have an idea who you want to work with. If you've communicated with a professor, that helps. If they're interested in working with you, you're in. Without any contacts, you're probably not.
 
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I just noticed that your question was specifically about masters programs, which I have less experience applying to (...despite the fact that I'm in one).
 
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