Here is some general advice that I give to undergraduate math students.  For now, this is mostly for students that are interested in doing research, and is really skewed more towards pure math.  This advice represents only my views, and not those of NC State or the mathematics department.  (This page is evolving.  I hope to include more resources as time goes on.)

Advice on advice: I want to preface this page with an important disclaimer.  Below are strategies and views that I find helpful, but may not necessarily lead to success.  I think it's important to seek out advice from a variety of mathematicians, such as faculty teaching your classes, academic advisor, research advisor, REU mentor, etc.  It is good to ask your peers, but take things with a grain of salt, since they have not necessarily served on admissions committees, read/written letters of recommendation for graduate school, etc.  Also, sometimes you are in a situation that is somewhat unique to you and the generic advice does not apply.  

Advice on being a strong applicant for graduate school:

In my view, the most important indicators for application success are: 1) good grades, 2) taking lots of math courses (especially graduate level), 3) strong letters from faculty, 4) research (especially if it's published).  

There are lots of other things that can help with your success.  I'm going to list a few that come to mind which might help your application to varying degrees: 

• Posters and presentations in conferences, seminars, etc

• Participation in conferences

• Directed reading programs, independent study, or undergraduate thesis

• Academic or research awards (e.g. Goldwater)

• Outreach and mentorship activities 

• Teaching experience (TA, grader, tutor, etc)

• Math competitions (e.g. Putnam) 

• Other math activities (math clubs, math circles, etc)

It's not reasonable to do all of these, but a lot of stronger candidates have a handful of these activities.  

Advice on letters of reference: 

For graduate schools, one generally needs 3 letters of recommendation, but these are important for lots of other things (e.g. fellowships, REU's, etc).   There are several things that can make a letter effective, so it is worth investing in thinking strategically about who will be writing your letters in the future.  (This advice is very academia/math specific.)

• Strength.  While you likely know that you want the letter to speak highly of you, you may not be aware that it is reasonable to ask someone if they would write you a *strong* letter.  (It's better for them to tell you no or that it won't be strong enough for the programs you are aiming for than to have them write you a weak letter.)

• Content.  If a letter simply says that you got good homework and exam scores, this won't be as strong as a letter that can speak to your specific strengths (e.g. "she is extremely creative", "they have the best intuition in analysis I have seen in my career", etc.)  For the same reason, it helps to have letters from people that can speak to your more advanced work: research, graduate courses, independent reading, etc.

• Who.  At least for pure math, make sure to get letters from professional mathematicians.  Don't get letters from graduate students, since they are likely to be discounted.  The more fancy and senior the person is, the more likely they are to carry weight.  If you do an REU or something like this, having a letter from someone outside of NC State can help calibrate you against students from a range of institutions.

Of course, you may not be able to find 3 people that fit all of these boxes, but try to plan the best you can accordingly.

Advice on undergraduate research: 

Many graduate schools now really emphasize having research experience that is in some way connected to a professional mathematician.  This can come in lots of forms: research with a faculty member in your department (possibly through their graduate students), summer REU's (Research Experience for Undergraduate programs), Polymath Jr.  

For applying to REU's, I recommend starting with this website (which also has a valuable advice section).  

For doing research, there's a few ways to approach this.  

• At NC State, Jordan Snyder is the Director of Undergraduate Research.  You can reach out to him to help discuss potential research opportunities.  If you're in the Honors Program, then I also have some recommendations for faculty working with honors students.

• Talk to a faculty member you've taken a class with that you had a good relationship with and enjoyed the material.  They might have a project for you or can direct you to someone who can.   Often you will have to start with reading some background material before you can get into research.

• Look through faculty webpages to see who is doing some research that looks like a topic that's interesting to you.  Then reach out to them.  It can be helpful if you've taken a look at what they're doing and tried to digest it a little bit, but don't worry if you don't understand everything that goes into what they're doing.

• If you're being turned down by faculty, then try to get involved in a Directed Reading Program.  This can help you get into some more advanced topics that can help get you ready for research.  

• You can get course credit for research (e.g. MA 499 at NC State) or sometimes funding 

Some things to apply for: 

There are a number of fellowships, research funding, etc.  In general, you should apply for things if you are interested.  There is a tendency to think that you might not be good enough, but often times that's just self-doubt.  You can always talk to an advisor to see if they think you are a good fit and/or candidate.  A couple of the most prestigious are:

• Goldwater Scholarship. (For sophomores and juniors.)  This provides a scholarship for undergraduate students interested in research in STEM fields.

• National Science Foundation Graduate Research Fellowship Program. (Usually the year you apply for grad school.)  This provides significant funding to supplement your graduate program.  It often pays more than the usual stipend in a graduate program, and gives more time for research in graduate school by reducing some teaching responsibilities.  It is very prestigious.