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Hello, as I finished my junior year at my new undergrad university (came from average community college), I would like to start preparing and assessing what "level" I should most look into. Everyone has their dream school, and mine is UC Berkeley. Granted, my hopes of being admitted there are slim at best. Nonetheless, I would be remiss if I didn't try and apply. That said, I have some other programs I might have some shot at, even if it's a long shot. I would appreciate an honest look at where I stand, and then maybe some examples of institutions I should consider "reach", "within my reach" and also "safety." 

Type of Student: Domestic White Male
Applying to: Pure Mathematics PhD

Undergrad Institution: University of Washington (Tacoma Campus)
Undergraduate Major: Pure Mathematics
Undergraduate GPA(As of now, going into my senior year): 3.71 Overall, 3.58 Math Major GPA

Work Experience: I work as a tutor for a well-known tutoring company and applied for a tutoring position through the university.

GRE General Test: Not taken yet (July 14 test date)
GRE Mathematics: Not taken (took a practice test, got a 440! Yikes! I am studying though, mostly needed a calculus refresher (integration))

Undergraduate Relevant courses:

Community College(Freshman/Sophomore):

Calc I,II (AP class), Calc III (sequences, series, convergence, basic vector functions) 3.0, Calc 4 (multivariable/vector calculus) 2.7, Differential Equations (2.0, but retaking this fall. I didn't do well in the class at community college but now that I am at university for my junior/senior year I thought I'd retake it), Linear Algebra 2.7

University(Junior/Senior)

Prob. and Stat. for Engineers and Scientists 3.0, Math Reasoning (Proof writing class/formal logic) 4.0, Math Seminar (Capstone prep/ how to conduct math research) 4.0, Complex Analysis 4.0, Abstract Algebra I (Group theory with intro to rings/fields) 4.0, Abstract Algebra II(Ring/field theory up to field extensions) 4.0, Readings in Math (Independent study; Advanced Linear Algebra [non-matrix based]) 4.0.

To Be Taken This Coming Year:

Differential Equations, Geometry (Axiomatic geometry with emphasis on non-Euclidean geometry), Coding Theory (error correcting codes), Real Analysis I, Senior Seminar(Capstone presentation class), Topology, "Math, Power, and Society" (for graduation requirement), and hopefully another independent study on Galois theory.

Capstone:

Part of our graduation requirements require a capstone project. My advisor and I have decided to try and publish our capstone in an undergraduate journal by the end of summer. We will be working throughout the summer on the group structure of elliptic curves and its connection to RSA encryption. This will be my only research experience, and only at an undergrad level.

Other Considerations:

At my university (small secondary campus 1 hour drive from main state university campus) I am fortunate to have a great relationship with my professors. We are (for the most part) on first name bases and I feel very comfortable for them to be able to attest to my abilities and potential for graduate studies. Furthermore, my "ace-in-the-hole" (if you could call it that) is one faculty member is rather well known and has a good reputation. She is known mostly for her communications of mathematics, and is the President-Elect of the MAA (She will become official president soon). She has directed me in my linear algebra independent study and has had me as a student for my proof writing class. She is fully able to attest to my abilities, and hopefully she will be well known enough to give aid me in applying to my dream schools. Also, I have two other professors who I believe really know me and my abilities, and can recommend me enthusiastically.

I am also a member of the MAA, Spectra the LGBTQ mathematician association, I was inducted to our school's Pi Mu Epsilon honor society chapter, and received department awards for "Best in Class: junior" and "math ambassador"

Planning on Applying to:

UC Berkeley(dream school), Princeton, Penn State, University of Washington, University of Virginia, University of Wisconsin-Madison, Oregon State University, University of Hawai'i at Mānoa, and anything you might recommend.

I am really hoping for schools to learn a lot of algebraic number theory, especially in reference to Elliptic curves or group/field theory (I loved Artin's book on Galois theory).

I would appreciate an honest assessment of my chances. Granted, I am still going into my Senior year in undergrad, but I wanted to start prepping now. I will of course dream big, but I would appreciate honest advice on schools I have a shot at, and then safety schools.

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As recommended go check out the mathematicsgre forum, there's a lot more people there involved in math. 

But I will say that I know many people with applications that are quite a bit stronger than yours (people at schools like Berkeley, Stanford, and Harvard) with higher gpas, 5+ graduate courses (some even reaching double digits!), multiple high-quality REUs, GRE scores in the 90+ percentiles who weren't able to crack programs like Princeton. Your recommendations seems like a bit of a wildcard but I can imagine a situation where if your profs call you brilliant in the rec letters you may get in. 

If you have some sort of story about overcoming adversity with your low CC GPA and turning it around when you transferred, manage to get a pretty high score on the subject test I don't think Berkeley is an impossibility, but still a reasonable reach.

You should take all my advice with a grain of salt though as math is not my area.

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