Thinking about math research?
There is so much more math out there than what you learn about in classes, and research is usually where the most interesting math can be found! Getting involved in research is an excellent way to figure out what kind of math you like to do -- or whether you like doing research at all. Luckily there are summer programs you can apply to, called REUs (Research Experiences for Undergraduates), that will pay you to do research with other math students from around the country. Research is also a great way to see how the math you learn in different classes can connect. After all, when you're thinking about problems that nobody in the world currently knows the answer to, you get to try any technique you want to see what works!
To search for REUs, the two best sites to use are
https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044
http://www.ams.org/opportunities (make sure to check the appropriate boxes on the left to filter out programs that don't apply to you)
If you're a Colgate student, you can find non-REU summer research opportunities with professors early in the spring semester through the Portal. Keep an eye out for emails directing you to them.
Some things worth keeping in mind:
There are typically many, many more applicants to REUs than there are positions available. If you do not get accepted anywhere, it does not automatically mean you're not a good mathematician! It's very difficult not to take that sort of news personally, but it's true. It's not uncommon for a single program to get over 100 applications when they may only have 10 positions available.
Because of the selectivity of REUs, I strongly encourage applying broadly -- for example: as much as it might be nice to attend a program close to home, there will probably only be a small number of them (if any!).
REUs typically require one or two letters of support from professors. Make sure to leave plenty of time between your first application deadlines and when you ask for your potential letter-writers to agree to write a letter for you. By "plenty of time", I mean minimum one month, but preferably at least 6-8 weeks. This will give your letter writers plenty of time to tailor their letter specifically to you instead of providing something more generic.
Deadlines for REU applications are often in the spring semester, even as early as February 1. Websites for the REUs usually update with more specifics during the fall. Before the websites update, you can take a look at what prior years' projects involved, so that you can get an idea of what next year might look like.
Thinking about grad school?
If you're thinking about going to grad school for math, then you probably like math quite a bit. Grad school gives you the opportunity to not only learn tons of math with like-minded folks, but contribute to the collective mathematical knowledge of the world! With the help of an advisor, you will be able to push the boundaries of mathematics beyond what is known. You'll become an expert in your area, and people will turn to you for answers because of it. There will be opportunities to travel to a wide variety of places -- college towns, mid-to-large cities, and maybe even internationally. You'll likely have the chance to teach math, too, which helps you to firm up your own mathematical foundation in addition to helping others. Even if you aren't looking towards academia in your long-term goals, an advanced degree in math can boost your marketability and/or earnings potential. Since my background is in academia, most of this advice will assume that the reader is (potentially) interested in academia.
Before getting into the advice, something that is important to be aware of is that the academic job market is generally not great. It's a bummer, and getting an academic job in math is certainly possible, but it's true. As is often the case, there are just too many applicants for all of the available positions. On top of that, grad students are usually paid enough to get by, but not much more. It can be challenging to get through those years -- it's doable, but there's not much cushion, financially. There are tons of positives to joining academia, but it's important to know what you might be facing once you get there.
Where should you apply?
Here is a site where you can search for programs based on lots of different factors. You can search by degree program, research specialties available, geographic region, financial support availability, and more.
But which of these are "better" than another? This is a very difficult question to answer. Some people rely on the US News & World Report rankings (which has a list specifically for graduate programs in math), but their lists are notoriously noisy -- so much so that Columbia University has declined to actively take part in the ranking system entirely. The list provides a heuristic, but don't put too much weight on which department is ranked #n vs. #(n ± ε).
Instead of worrying too much about rankings, it's better to find the right fit for you. Some things that will help you narrow that down are:
Think about what you'd ultimately like to do with a graduate degree, and look for programs with a history of success in that area. For example: if you want to end up working at a liberal arts college like Colgate, take a look at which programs were commonly attended by your professors. Go to the websites for math departments at schools similar to Colgate and take a look at which programs those professors attended.
If you have at least a vague idea of what kind of research you'd like to focus on, look for departments with plenty of faculty in that area. A good way to get started would be to ask your professors who do work in that area for recommendations on departments that have thriving research groups in that area. For example: the University of Minnesota has a large (and excellent!) combinatorics research group, while some departments which can be considered "peers" overall may not have a combinatorics group at all.
If you aren't sure about what sort of research you'd like to do, that's completely okay! In that case, I recommend looking for departments that seem well-rounded in terms of what research areas they offer. If you start dabbling in one area and end up not liking it, you'll have plenty of alternatives.
Other recommendations:
Plan ahead! It is never too early to think about where you may want to apply and what you will need to do to prepare your application. Applications are typically due late in the fall semester or very early spring semester, so it might not be a bad idea to prepare a lot of your material during the previous summer. That way, you won't have to worry about doing it on top of your usual coursework.
How many applications should you send? There is no single correct answer, although conventional advice is 4-8 by some people, and 5-10 by others. I personally lean towards 10, if not more. I recommend applying to between one and three departments that you think would be a real stretch, because you truly never entirely know what will happen. I also recommend applying to two or three programs where you feel confident that you comfortably satisfy all of their requirements or expectations. Don't be mistaken, though: there is no such thing as a "safety school" when it comes to grad school. But applying to programs in which you feel you more than satisfy their expectations can at least increase your chances of being accepted somewhere. The bulk of your applications should aim towards programs somewhere in between these two. Regardless of how many you applications you send in, always ensure that they are as strong as possible. Applying to 100 programs isn't going to help you if your application material is sloppy, generic, impersonal, etc. Moreover, I strongly recommend only applying to programs in locations that you can see yourself living in happily -- or at least tolerably -- for roughly 5-6 years. Keep an open mind about what this means to you, but, for example, if you know for certain that long, cold, snowy winters will make you miserable, then you may not want to bother putting your energy and money into applying to programs far up north.
When writing any statements (such as personal statements), ask for feedback from friends, your advisor, the writing center, etc. Keeping their suggested revisions in mind can help to strengthen your overall application! Be willing to write several revisions, or even throw away early drafts entirely in order to produce new statements. This is another reason to plan ahead!
Like REUs, most programs will require some letters of support from people who can speak to your work and/or your potential as a student, or as a summer researcher. Think about who would be best for ask for a letter of support. Make sure to leave plenty of time between your first application deadlines and when you ask for your potential letter-writers to agree to write a letter for you. By "plenty of time", I mean minimum one month, but preferably at least 6-8 weeks. This will give your letter writers plenty of time to tailor their letter specifically to you instead of providing something more generic.
Some graduate programs will require GRE scores to be sent in. The GRE is an awful predictor of long-term grad school success, but it still gets used sometimes (possibly against the department's wishes -- it could be something they're required to ask for at the behest of administration). So, unfortunately, it might be a hoop you'll need to jump through in order to have a complete application for certain programs. It also costs money to take the exam and have scores sent to departments, so, once more, plan ahead.
Good grades in your math classes are important, but they're not the only important thing! Search committees will also look at research experience, involvement/leadership positions in math-related clubs, whether your potential research interests align with what the department can offer, and so forth.
I strongly discourage accepting an offer to a graduate program if there is no funding offered to cover your tuition and/or if there is no stipend offered. Funding most frequently comes in the form of a teaching assistantship or research assistantship.
Like with REUs, there are typically many, many more applicants than there are positions available. Like with REUs, a realistic scenario for a large math department may be 100 applicants for 10 available positions. Of these, there could be 20 applicants who are clearly perfect fits that the department would be crazy to deny -- but half of them still get rejected! So, it bears repeating: if you do not get accepted anywhere, it does not automatically mean you're not a good mathematician!
If you're on the fence about whether to pursue a Master's or a Ph.D. right away, you've got options!
There are some departments that only offer Master's degrees, and you can use that time to help determine whether you want to continue onto a Ph.D. These programs can be excellent stepping stones to more competitive doctorate programs, since you'll be further prepared beyond your undergraduate work.
The vast majority of programs that offer Ph.D.s also offer Master's degrees. It's very common for grad students to complete the work for a Master's along the way towards their Ph.D. -- so if the Ph.D. ends up not working out, they still have the Master's that they earned.
For a program that offers both Ph.D. and Master's degrees, there are usually fewer Master's students than Ph.D. positions. Moreover, Ph.D. students are almost always offered funding support through teaching or research assistantships, while this may not be as universally true for students in the same department's Master's program. Because of this and the reason above, applying for a Ph.D. position is sometimes considered the "safer" route.
I'm happy to chat in more depth about anything related to REUs and grad school, so if you have more questions that aren't answered here, feel free to send me an email!