(Mathematics) Grad School at UCI

Introduction

Graduate school can take some getting used to, no matter where you go. As an undergraduate, grades mattered most, and the focus was primarily in the classroom. As a graduate student, the focus changes: while classes are still important, research is now the ultimate priority.

With this change of focus, a change in tactics is necessary. The methods you used to succeed as an undergraduate may not be completely applicable to graduate school. On this page, I hope to give an idea of how to adapt to the new goals and expectations of graduate school. I'll be writing primarily for the audience of students entering graduate school in mathematics at UCI, but the tips should be more broadly applicable. It is my goal to collect and convey the lessons I've learned as a graduate student, as well as the collective wisdom of the graduate students I've spoken with over the years.

Disclaimer:

This advice is just my point of view. There are many points of view on how to best proceed through graduate school, and each has its strengths and weaknesses. Please consider this advice to be a starting point in formulating your own point of view, rather than a definitive guide (which it is not). I hope that it proves useful to you. If you have ideas or contributions for this guide, I'd greatly appreciate them; please email me at:

pmacklin @ math . uci . edu

Before You Arrive at UCI

At UCI, you are required to pass comprehensive exams in algebra and real analysis. When you arrive, you will take advisory exams in each of these subjects. The good news is that if you pass one or both of these, you are exempted from the corresponding comprehensive exams. That means that you have an excellent opportunity to get a running start on your graduate career. Furthermore, the advisory exams are generally easier than the corresponding comprehensive exams, so there's extra incentive to pass these exams! A little work at the beginning will save you a lot of work later on.

Choose which subject you feel stronger on of the two, and study that subject exclusively in the summer before you arrive at UCI. Download any available old advisory/comprehensive exams here. Be sure you can work all problems thoroughly and rigorously. Also, work through any old exams you may have from your undergraduate courses. If at all possible, you should try to work the exams (even after you have solved them once) in a timed format.

By focusing your attention on one of the two exams only, you increase your odds of passing that exam; if you divide your time evenly to study for both exams, you very well may nearly pass both of them, which doesn't get you anywhere.

If you feel equally strong on algebra and analysis, I'd recommend focusing on analysis. The analysis comprehensive exam has a reputation for being harder than the algebra exam, and so it is most desirable to pass the analysis advisory exam.

Your First Year

In your first year, your primary goal should be passing the comprehensive exams. Teaching will also be new; it is important that you don't lose sight of your long-term goals and priorities and spend too much of your time on teaching. While you do want to do a good job, remember that you're here to earn your degree, and that means that you need to pass your exams.

You should also be careful to not focus too heavily on research at this point, as failing the exams will prevent you from advancing in the program. However, you should attend the mathematics graduate student colloquium (MGSC), a series of talks organized by and for graduate students on ongoing research within the department. It's an excellent way to get a feeling of what you'll be doing later while meeting your fellow graduate students and building community.

Case 1: You Passed 2 Advisory Exams

Congratulations! Skip to year 2!!!

Case 2: You Passed 1 Advisory Exam

Congratulations! In a sense, you're done with half of your first year here. You'll need to take a yearlong course sequence on the advisory exam you didn't pass. Make that class your main priority during this academic year. Take the homeworks seriously, as they should be viewed as studying for the comprehensive and qualifier exams.

Also, you'll need to choose an upper-level sequence to start preparing for qualifier exams; you can choose among real analysis, complex analysis, and algebra. I'd recommend choosing the subject in which you passed the advisory exam.

In April and May, focus your energy on studying for your remaining comprehensive exam. If you do this, you should be able to pass it in June, and then you can spend your summer focusing on your qualifying exam. Don't take the qualifier exam in June; you are only allowed to take each qualifier twice, and it will detract from your studies of the comprehensive exam. After you pass your comprehensive exam in June, spend the summer studying for the qualifier and take it in September.

Case 3: You Didn't Pass an Advisory Exam

Don't worry. This is not a terrible thing; you still have great chances of succeeding; you'll just have to work a little harder during your first year. You'll be taking yearlong sequences in algebra and real analysis. Make these courses your top priority for the year. Take the homework seriously; you should view it as studying for the comprehensive exams.

In April, choose which subject you feel stronger on, and start to focus on studying for that exam. Experience has shown that the best approach is to focus your attention nearly 100% on that exam. This way, you can pass the comprehensive exam on your strong subject in June and spend the summer preparing for the other comprehensive exam. Splitting your attention in April and May between the two subjects (when you also have teaching and other duties to distract you) increases the odds that you'll fail both exams and have to spend the summer split on studying for two exams.

Your Second Year

In this year, you'll primarily be focused on your qualifying exams, but you should also start thinking about research. I haven't already been doing it, you should be attending the mathematics graduate student colloquium (MGSC), a series of talks organized by and for graduate students on ongoing research within the department. By now, you should have some inkling of what you do and don't like. Start attending specialized seminars and talks. Start learning LaTeX so that it will be easier to write papers later. But again, don't lose focus: you can't research later if you don't pass your qualifying exams now.

Case 1: You Passed 1 Advisory Exam

Hopefully by now, you passed your one remaining comprehensive exam and your first qualifier. This year, take the yearlong sequence on your remaining qualifying exam. In April and May, focus primarily on studying for the exam, and when June comes, go for it!

This year, you should also start looking for an advisor. Browse personal websites, talk to other graduate students, and attend talks to find advisors you're interested in. Talk with grad students to determine which faculty have work styles that match yours. And once you've narrowed down your search, approach these professors in their office hours or per email and see if they're taking students. They'll probably give you some papers to read, and you can start to gauge your interest.

Case 2: You Passed 0 or 2 Advisory Exams

Out of real analysis, complex analysis, and algebra, choose your favorite two. You'll be taking yearlong sequences in these subjects. Make these courses your top priority for the year. Take the homework seriously; you should view it as studying for the qualifiers.

In April, choose which subject you feel stronger on, and start to focus on studying for that exam. Experience has shown that the best approach is to focus your attention nearly 100% on that exam. This way, you can pass the qualifying exam on your strong subject in June and spend the summer preparing for the other qualifying exam. Splitting your attention in April and May between the two subjects (when you also have teaching and other duties to distract you) increases the odds that you'll fail both exams and have to spend the summer split on studying for two exams. This is the same strategy as for the comprehensive exams.

Your Third Year and Forward

Now, it's time to focus on research. You should have an advisor selected, and you should be working full time. You and your advisor should have a specific project in mind by this point, and you should start doing the background reading and/or programming required to understand the topic and prepare for your oral exam. You should be attending seminars regularly, including the mathematics graduate student colloquium (MGSC).

Starting working on papers and get ready to advance to candidacy and write your dissertation!

Tips and Resources for Studying for Exams

Studying for an Exam

Studying for an exam requires a long-term commitment and plan. Here are some tips I have picked up through the years after making plenty of mistakes of my own.

1. Form a study group: With more people working together, you'll see new angles to and shortcuts on problems that you probably would not have thought of on your own. Even if you don't learn anything new (unlikely), explaining things to others will help you solidify your own understanding.

2. Start early: I started studying for my complex qualifying exam 6 weeks before the exam. Real analysis was similar. Starting early will give you an edge and will also take the pressure off you as you prepare and as you take the exam.

3. Make a regular schedule: Make sure you work at least a little every day and make steady progress. I found that a good strategy was to set a goal of solving two exams per week.

4. Don't make notecards: Making notecards of all the major theorems is very time consuming; it can easily eat up 1-3 weeks of your study time. It is more important to practice solving actual problems, and you'll learn the theorems as you go. Furthermore, by learning the theorems by application, the reasons for the hypotheses will make more sense and be easier to remember.

   I have also found that starting your studies by writing out notecards is extremely tedious, and this tediousness can cause you to lose momentum and get depressed. On the flip side, if you wait to make notecards until the last 1-2 weeks (if you choose to make them at all), then you can focus on the 2-4 theorems you're having the most trouble on, rather than every theorem. Much more efficient.

5. Take timed tests: Make sure that you can work both accurately and quickly. The best approach I found was to first do as much of each exam in 2-2.5 hours as I could without references, and then spend the next several days correcting and formalizing my answers.

6. Revise your solutions: Don't just take the exams and move on; revise your solutions so that you can improve your arguments and make them more efficient. This in-depth thinking will help you understand the subject matter more completely, and this will pay off when you see exam problems that are different than the ones you have worked on.

   So for example, take your first exam on Sunday, and revise your solutions on Monday and Tuesday. Take your second exam on Wednesday and revise it on Thursday and Friday. Meet with your group on Friday to discuss your solutions, and spend Saturday revising your solutions based upon your group meeting. This will help you to polish your arguments.

7. Review your solutions: In the last several days before the exam, review your solutions so that all the techniques you have mastered are fresh in mind. Take this time to review any theorems that you don't feel 100% solid on.

8. Sleep: Because you will have spent six weeks or more studying for your exam, you won't need to cram in the one-to-two days before the exam date. Instead, try to get eight hours of sleep a night for a few days before your exam. This way, even if anxiety keeps you awake the night before your exam, you'll still be (relatively) well-rested and mentally alert for your exam.

   On a related note, if you're used to staying up until 4am and sleeping in until noon, you should start shifting your schedule so that your body and mind are accustomed to working at 11am, which is likely when your exam will take place.

Resources for the Algebra Comprehensive Exam

Please feel free to browse through my study materials here.

Resources for the Real Analysis Qualifying Exam

My old study materials are here.

Resources for the Complex Analysis Qualifying Exam

I submitted copies of some of the 12 most recent complex analysis exams so that everyone can download and solve them who wants to. They will likely be posted here. I'm also posting them here.

1. Fall 2002
2. January 1997
3. January 2000
4. January 2003
5. June 2001
6. June 2005
7. September 1997
8. September 1999
9. September 2000
10. September 2001
11. September 2004
12. Spring 2002

You are also more than welcome to dig through my study materials here.

Webpage Resources

I have created a script that will automatically generate a starter webpage for you, much like this sample.

To download this script and see instructions on how to run it, please visit the make_webpage link.

The generated website will have links to good HTML and CSS tutorials, and it will give you an example of a fully standards-compliant webpage.

Tips for Research

Resources for LaTeX