Why should we listen to you?
If you are considering basing your entire grad school application process on a single obscure blog post, then perhaps a life in research is not your best option. I highly encourage you to seek advice from a range of professors, administrators, researchers, and grad students, especially those who know you and can offer more personal advice. The lessons and suggestions below are merely anecdotes drawn from my own experiences applying to grad school during Fall 2010/Spring 2011.

For full disclosure, I applied to a hodgepodge of programs that would support research in theoretical neuroscience (which translated to Neuroscience, Biophysics, Physics, and Applied Math programs), as well as just about every fellowship for which I was eligible. With the generous guidance of about a dozen professors and grad students, I managed to snag a Churchill Scholarship to spend one year at the University of Cambridge and a Department of Energy Computational Sciences Graduate Fellowship (DOE CSGF) to support my PhD studies in the US thereafter (at the time of this writing, I have not decided exactly where). I was also fortunate enough to receive offers of admission from the majority of schools I applied to but deferred or declined them all to spend a year in the UK.

My goal in writing up this guide is not to help you design your college experience, craft the perfect resume, and con your way into grad school, but rather (1) to leverage the obscene amount of time I spent gathering information on grad schools and fellowships to save you time and (2) to help you make a reasonably well-informed decision.

The Process

1. Identify faculty, schools, and programs (in that order)
2. Contact faculty and students
3. Apply for fellowships
4. Apply for grad schools
5. Worry
6. Interview and/or visit
7. Worry
8. Decide

1. Identify faculty, schools, and programs (in that order)
Why in that order?
You are going to grad school to learn how to do research (or to escape a poor job market, but my recommendation stands); finding an appropriate research advisor is one of the most challenging and important parts of ensuring that you have a good experience. By focusing on schools or programs first, you may become distracted by far less important details, such as coursework, impressive web pages, and ambitious program names.

Are you saying schools and programs do not matter?
Not quite. A school defines where you will live and who you will interact with, and a program defines your first-year course requirements and may restrict who you can work with. My point is that working with a great advisor at a good school is, for most people, going to be a better experience than working with a mediocre advisor at a great university.

What criteria should I consider in looking for faculty?
Interesting research – Obviously, you need to be interested in their research.

Well-connected – Since your advisor will likely be your primary liaison for establishing collaborations during grad school and finding jobs afterwards, you might want to find someone who is well-connected and collaborates frequently in your research community. Frequent invites to speak at seminars and conferences, a diversity of co-authors on publications, and frequent citations of his/her research (in journals, not USA Today) are all good signs.

Advisement style – Advisors vary widely in how closely they interact with students. Some will assign their students projects and meet with them for a couple of hours multiple times a week. Others prefer to let their students struggle a bit to define their own research problem and may not require any meetings, serving only as a source of advice a couple of times per month or semester. Where your optimal advisor falls on this spectrum of involvement depends on your own personality and preferences. Emails to grad students, interviews, and visits (discussed below) will help you determine the advisement style of the faculty in whom you are interested.

Availability – Your advisor also needs to have the time and funding to take on students. At a minimum, they need to be willing to advise you, but depending on your preferences for interaction, you may want to consider how much time they actually have for their students. A quick glance at their website may reveal whether they are taking on students but if not, a brief email asking this question is absolutely acceptable. As for determining how much time they actually have, emails to grad students, interviews, and visits are again helpful and will be discussed below.

Great explainer – Another helpful quality in an advisor is the ability to clearly explain complicated ideas. Not all great researchers make great mentors, and having an advisor who is a great explainer can help you learn a lot more during your time in grad school. Looking up talks and papers by a professor, as well as interviews and visits, can help you infer how well they explain their work.

Personality – Finally, you need to get along with your advisor. Choosing an advisor because you thought they were “funny” or because they shared your interest in crocheting may sound absurd, but you will be spending between four and seven (or more) years working with them. Due to the nature of scientific research, you will inevitably run into many failures, dead ends, and frustrations, and overcoming such obstacles will be far easier if you have an advisor you enjoy working with. Again, not all great researchers make great mentors.

These criteria of course reflect my own particular preferences, and your own may differ. I strongly recommend drafting your own list before contacting professors or visiting schools, so that you have some idea of what questions you should be asking and what you should be looking for. Update your list as you learn more throughout the application and visiting process.

What criteria should I consider in looking for schools and programs?
Livability – If you grew up in Orange County and will need to be rushed to the hospital if a snowflake impinges upon your sensitive tanned skin, then perhaps the University of Chicago is not your best option. If you were raised climbing trees and having picnics in Oregon, then think carefully before applying to NYU. If you are a diehard rock climber, then perhaps you should pass on the University of Kansas. At a minimum, you should be sure that you can lead a moderately satisfying life in any city containing a grad school to which you apply.

Community – Your school and program more or less define the people you will collaborate, eat lunch, and socialize with for several years. The people in your program are usually the first that you get to know (since you will likely take classes with them for at least a year or two), though how close-knit these communities remain over the next few years varies greatly by school and discipline (mathematicians, for example, typically shun human contact and live in hermit caves after year 3). I have had one professor tell me that the single factor that contributed most to his positive experience in grad school was the enthusiasm and motivation of the students around him; it is much easier to deal with the trials and tribulations of PhD research when you are constantly reminded of how much fun science can be. In addition, your colleagues in grad school will likely become your collaborators afterwards, so surrounding yourself with people who do good work may serve you well in the future. Visiting your program and talking to students ahead of time is the best way to assess whether the community will be a good fit for you.

Multiple advisors – While you may enter grad school gung ho on working with a particular professor, your interests may shift, that professor’s interests may shift, funding may be an issue, or you simply may find that you two do not get along. To prevent yourself from needing to switch schools (or worse, work on something you are not interested in), look for schools and programs that have more than one faculty member you can see yourself working with. Also, if your interests span multiple disciplines, you may need multiple advisors with different areas of expertise anyways.

Funding – It is socially acceptable to ask faculty and students about the funding situation. (I ignored this question throughout my applications and visits until professors actually started asking me to ask them about funding.) In my experience, most programs fund students for their first year or two while they are finding an advisor (which may be done through fellowships, teaching assistantships, or both), and advisors fund their students thereafter. Though funding will likely not be an issue if you are in the sciences, grad students complaining about excessive teaching loads or that they could not work with their first choice of advisor due to funding problems are red flags. Ultimately, you can circumvent this issue if you are lucky enough to snag your own funding (see discussion of fellowships below).

Advisor flexibility – Make sure that your program gives you the flexibility to work with any of the multiple advisors in which you are interested, regardless of their department affiliations. In my experience, this was never an issue, though some more traditional departments may emphasize “training their own.”

Coursework – This criterion is listed last for a reason. While long lists of interesting courses make for impressive websites, it is your advisor, research, and colleagues that will be far more important in the long run. In addition, many (most?) grad students end up teaching themselves and each other skills as needed during research. Do not be distracted by the red herring of coursework. That said, make sure that you can tolerate the course requirements for any program you apply to.

Just as for faculty criteria, this is my own biased list, and you should draft your own as well.

How should I find faculty, schools, and programs?
Ask professors and/or grad students who know you well and/or are knowledgeable about your area of interest for suggestions. Browse review articles, textbooks, and papers and note faculty whose research interests you; follow up by browsing their websites. Diligently scanning the complete faculty profiles for schools and departments in which you might be interested also works, though exhaustive search is linear in problem size (in other words, its time-consuming). As for programs, do not feel as though you must apply to only programs with a particular name; it is more important that you find a program that will give you the flexibility and support to do the research you want to do with the people you want to do it with (again, I personally applied to programs with titles ranging from “Physics” to “Neuroscience” to “Applied Math”). Keep a detailed list of faculty, schools, and programs as you go. For perspective, the Google Doc I used for this purpose outputs a 53-page PDF. Gird thy loins!

2. Contact faculty and students

*Gasp* You can email professors?!
Yes, and you should. It will (1) help guide your decision of where to apply, (2) help you write a more-informed application, and (3) possibly help your chances of admission, since a professor may lobby the admissions committee on your behalf. All of this assumes that you avoid racial slurs, links to raunchy YouTube videos, and absurdly poor spelling and grammar in your emails.

How should I email professors and what should I ask them?
Keep your messages short. The likelihood that a professor will read your email varies inversely with its length. Avoid sending resumes or making lengthy introductions. Simply include a sentence or two mentioning your relevant background, that you are applying to grad school, and that you are interested in their research group. Appropriate questions include whether they have the time and funding to take on PhD students and specific questions about their research or potential projects. Be sure to read their webpage and a few of their papers (or at least the abstracts) before contacting them. Failing to do so and instead asking “So what do you do?” will likely give off the impression that you are incredibly lazy and may actually hurt your chances of admission. Also, if you email a couple of professors in a row, be careful not to mix up their names. I did this once and, not surprisingly, I did not receive a response.

Why would I want to email grad students?
First of all, they are a goldmine of useful information about programs, universities, and particular labs. Second, they are much more likely to respond to your messages than professors, as they typically have far more time and receive far fewer emails. Throughout the application process, I actually found grad students to be far more helpful than any other single resource. Many are eager to talk about their own research, the advisement styles of various professors, details of their programs, and just about any question you might come up with. I even had lengthy phone conversations with several especially helpful students. Of course, as I mentioned for professors, make sure to do your due diligence first. Read faculty websites, relevant papers, and program websites before contacting students, and avoid asking questions that are answered elsewhere.

3-4. Apply for fellowships and grad schools

What are fellowships?
Funding, mostly distributed by the government though also by several private organizations, that will follow you to whichever school you choose.

Why should I apply for fellowships?
Preparation – Fellowship deadlines are typically earlier than those for grad schools, and many of the essays, recommendation letters, and other application materials that you prepare can be reused. Inevitably, you will start putting together your grad school applications too late, and the forced preparation of fellowship applications will save you from missing deadlines.

Grad schools care – Many grad programs require their current students to apply for fellowships, and if you can say that you have done so before even applying, you may impress them with your apparent competence.

Academic freedom – There is a slight chance that you might actually win a fellowship and if so, you will then have a reliable source of funding that follows you to whatever university and research group that you choose. This means that funding is no longer an obstacle to your working with a particular group. It means that your research will not be tied to a particular grant so that you have far more freedom in selecting a thesis topic. It also frees you from taking on teaching assistantships for funding, thus giving you more time to focus on your research. You may still decide that you want to teach or your program may still require a semester or two of teaching in order for you to graduate, but you now have flexibility. Lastly, if you are wait-listed or rejected from a school that you are still keen on attending, a fellowship may encourage the admissions committee to reconsider, as you are now essentially free labor.

What are the major fellowships?
In increasing order of benefits, there are the Ford, NSF, NDSEG, DOE CSGF, and Hertz fellowships. Each involves some combination of tuition, living stipend, research/travel grants, research internships, and conferences, lasts for 3-5 years, and may only be used at schools in the US. I will abstain from listing the benefits of each program, as they may change year to year. Check their websites for up-to-date info. In my case, I missed the deadline for the Ford, applied for the NSF, NDSEG, CSGF, and Hertz, received the NSF and CSGF, and made it to the final round of interviews for the Hertz (more on this experience below), so I will offer comments on each of the NSF, NDSEG, CSGF, and Hertz.

What about international fellowships?
There are several fellowships available to Americans interested in heading to the UK for grad school, including the Rhodes (1-3 years, Oxford), Marshall (2-3 years, any school in the UK), Churchill (1 year, Cambridge), and Gates (1-4 years, Cambridge). Fulbright Scholarships (1 year) are more flexible and eligible for study in countries all over the world. International fellowships typically have earlier deadlines (September-October) and require nomination by your undergraduate institution. Note that many of these programs allow you to pursue a master’s (or two) and then return to the US for a PhD program and so provide an excellent opportunity to sample another academic culture and travel without committing to a PhD overseas. I will not comment further on any of the international fellowships here, except to mention that, beginning this September, I will be spending one year at Cambridge on a Churchill Scholarship and will write more about this experience as it unfolds.

What can you tell me about applying for the NSF?
The NSF is by far the largest of the fellowship programs. In 2011 for example (the year I applied), they gave out about 2000 fellowships. The NSF also has the earliest application deadline (mid-November) and so is likely the first application you will submit. Three other unique features of the NSF are that (1) the application requires a project proposal (in addition to a personal statement and summary of previous research), (2) the selection committee strongly emphasizes “broader impacts” on society, and (3) once the results are announced, you will receive “ratings sheets” which offer brief feedback on your application from three reviewers. Given that in all my years of applying for schools, scholarships, jobs, and library cards, I have never received any more feedback than a simple rejection or acceptance letter, I found this third feature of the NSF alone worth the time spent applying. As for the project proposal, I found it to be the single most time-consuming piece I had to prepare while applying for fellowships and grad schools. Writing (and re-writing) an NSF-style project proposal is, however, great practice for applying to grad school and, if you are aiming for academia, applying for the grants that will feed both your children and grad students. In preparing your proposal, I highly recommend (1) starting at least two months before the deadline and (2) eliciting feedback from professors, grad students, and other researchers. Starting early will give you plenty of time to write, revise, seek feedback, and repeat. Eliciting feedback is essential because, unless you are a child prodigy, this is the first real grant application you have ever written and you will be amazed at how much your proposal will evolve with a few rounds of feedback. I should also mention that the NSF will not hold you to carry out the exact project that you propose. The proposal is really meant to test your knowledge of your field and ability to express your thoughts clearly. For this reason, it is probably better to write a proposal about something concrete that you know well (such as an extension of your undergraduate research) rather than the ambitious yet vague project you would really love to work on. Finally, while grad schools and most fellowships will focus almost entirely on your research potential, the NSF also emphasizes your “broader impacts” on society, which include the direct impact of your proposed project, public outreach efforts, participation in mentoring programs, and even promotion of international collaborations (a complete description is available from the NSF). Do not overlook these. The ratings sheets that I and other applicants with whom I have spoken all mention the presence or absence of broader impacts in all three essays, and I am fairly certain that if you submitted a surefire proposal to unify quantum mechanics and general relativity, explain consciousness, and solve three Millennium problems in one brilliant swoop, but left out references to broader impacts, you would be swiftly rejected by the NSF (if it makes you feel better, you would still get the Hertz). Now, if your research is in mathematics or the pure sciences and is unlikely to change your grandma’s life in the next three years, then you are probably terrified at the notion of being forced to write about broader impacts. Fear not! As I alluded to above, broader impacts include public outreach efforts, mentoring, and other activities not directly related to your research, and the NSF will not penalize you for not attempting to cure cancer. Given the novelty of writing about your “broader impacts” and drafting a full-blown NSF project proposal, it can be quite helpful to begin by looking at a few sample essays and ratings sheets, and I have provided such links in the section on additional resources below.

What can you tell me about applying for the NDSEG?
The most unique feature of the NDSEG is that your application must pique the interest of a Department of Defense research office. This does not mean that you necessarily need to be building the Death Star or mind control devices. In fact, every NDSEG fellow that I contacted mentioned that they made no special efforts to reference military applications of their work when they applied. Also, while the NDSEG site mentions that you can browse the websites of the various research offices to see what kinds of projects they find interesting, I would advise against it for two reasons. One, no NDSEG fellow I contacted mentioned any perceived benefit in doing so. Two, government websites are notoriously difficult to navigate and out of date. It seems the best strategy is to apply just as you would for any other fellowship and hope for the best.

What can you tell me about applying for the CSGF?
The CSGF stands for “Computational Sciences Graduate Fellowship”, but “Computational Sciences” is broadly defined as any research leveraging computers and/or mathematics, thus you need not have a server farm in your basement to qualify. In addition to the very generous financial benefits, the CSGF is unique in requiring a summer internship at a national lab. This feature can either be viewed as a necessary obstacle to receiving your funding or as a great opportunity to try out a non-thesis-related research project at one of 17 stellar labs across the United States.

What can you tell me about applying for the Hertz?
Not only is the Hertz the most competitive (about a dozen superstars pick one up each year), lucrative (5 years of stipend and tuition), and prestigious fellowship in the US, it is also by far the most fun to apply for, as it includes two rounds of intense interviews.

After the initial applications in October, about a quarter of applicants are invited to interview. If you are among this lucky bunch, you will receive an email reminiscent of a James Bond movie, requesting that you don a suit, meet at a hotel, and:

At five minutes before your interview time, please go to a house phone in the lobby, dial the operator and ask for Dr. [your interviewer's name here]‘s room. He will pick up and let you know what suite they are in and if they are ready for your interview. The interview will take place in a completely separate living room area of the suite. The interview is a formal technical interview, lasting 45 to 60 minutes. It is patterned after the PhD oral exam and you may be asked to perform calculations, discuss your previous research work, and to demonstrate the breadth and depth of your technical knowledge. Please bring paper and pen in the event you are asked to perform calculations during your interview.

After receiving this message, I was preparing to be doing push-ups and integrals at the behest of a military officer. In reality, although the interviewers asked impeccably sharp questions, the conversation was quite casual. (I was, admittedly, slightly disappointed that I did not get to do push-ups and integrals.) My question topics ranged from my personal motivations (e.g. “How did you become interested in neuroscience?”, “Where do you see yourself in ten years?”) to my previous research (e.g. “Tell us what you did in this project.”) to creative puzzles (e.g. “If you had a friend on the other side of the moon and were banned from launching anything into space, how would you communicate with him?”). In all cases, the interviewers (there are typically two) continually peppered me with sub-questions, ranging from clarifications on specifics (e.g. “And about how large is a neuron?”) to speculations on broader issues (e.g. “And what implications do you think that has for the field of neuroscience?”). Note that you will almost certainly be stumped by one or more of the questions asked and that this does not necessarily disqualify you. I stumbled on at least two questions in the first round and still managed to limp my way to the second round. (The second round interview is very similar.) Personally, I applied for the Hertz mainly because the interviews sounded like fun and I was not disappointed. If you have a similarly sadistic sense of pleasure, then this is the fellowship for you!

UPDATE (11/25/2011)

Several people lucky enough to score a first date with the Hertz Foundation have requested additional interview tips. Here are a few additional thoughts:

Know your research well, not just the technical details but also the broader significance. Be able to explain why your projects were important and exactly what role you played in each one. One question I was unprepared for was: “What is the most creative idea you have had in your research?” Another was: “What is the most interesting new tool available to scientists in your field?” You should be able to come up with better answers to these than “I vectorized my Matlab code” and “Google Plus”.

Most importantly, be prepared to be humbled. The interviewers will probably be nice, but they will also probably push you to and beyond your intellectual limits. That is their way of testing you, so do not feel ashamed if you cannot answer some of their questions. What you should not do is remain silent for long periods of time, mumble, or bullshit your way through an answer. Instead, vocalize your thought process, and be honest and precise about any uncertainties you have. The interviewers want to see how you think, and they cannot do so if you sit there daydreaming.

Finally, remain confident. Treat the interview like a game, and I am sure you will have lots of fun. I truly enjoyed my interviews and would do it again in a second.

Sounds great! Can I receive all of these fellowships?
Up until 2010, it was possible to accept multiple fellowships and use them one after another. Beginning in 2011, however, the US government decided to spread its funding and no single student can accept more than one scholarship from a federal source (which includes the Ford, NSF, NDSEG, and CSGF). Since the Hertz is awarded by a private organization, it might be possible to accept it along with a federal fellowship, but this is between you and your funding sources. In any case, if you win a Hertz, you should have cured cancer and disproved P=NP by your fifth year, so a second fellowship should not be necessary.

What are applications like in general?
Applications typically ask for a mixture of essays (e.g. personal statement, summary of previous research), recommendation letters (typically three, though as many as five), general GRE scores, transcripts (sometimes official, sometimes unofficial), and personal info (address and such). Some may ask for a subject GRE score in your area of focus. Grad schools have application fees (typically $60-$120), but fellowships do not.

How should I prepare to apply for fellowships and grad schools?
Contact your recommendation letter writers and start preparing your personal statement early (I started during the summer and found it to be just enough time to hit all of my deadlines in the fall). If you can, consider taking your general and subject GRE even earlier (during your junior year) so that (1) you can retake them if you are not satisfied with your scores and (2) you are not preparing for GREs, classes, and applications simultaneously (personally, I found applications to be almost a full-time gig). Make organized spreadsheets of every item that is due; when you are juggling half a dozen deadlines in a single week, you will inevitably forget to send a transcript (or even an entire application) if you are not organized. Be sure to gently remind your recommenders of approaching deadlines; they will forget (though rumor has it that faculty members are so notorious for late letters that review committees will wait a week or more for letters to trickle in anyways). In general, do not worry about contacting the various organizations to make sure that your materials have arrived. They will contact you if something important is missing.

How should I prepare for the GREs?
If you are a native English speaker, all you need to do for the general GRE is to take a couple of practice tests ahead of time to understand the format. ETS (the inept company that administers the GREs) offers test prep software that allows you to simulate the riveting experience of using a computer in the early 1980s… and taking the painfully primitive test in an official GRE test center. There are also plenty of practice books available from companies like the Princeton Review that offer additional practice tests. The test is similar in nature to the SATs and your score is probably not particularly important, so you should not spend too much time preparing for or worrying about the test. If you are not a native English speaker, I hear the test is considerably more difficult. Sorry.

The subject tests, on the other hand, are far more challenging and do require practice. The only subject tests I have had experience with are physics and math. As for math, it is rumored that universities do not particularly care how well you did on the subject test, though they require it anyways. Physics programs, on the other hand, worship the physics GRE, and rumor has it that you need a 900+ score for top programs to even look at your application. Fortunately, there are four old exams floating around the interwebs as well as a fantastic website discussing various approaches to solving the individual problems. I recommend taking each of the (timed) tests on your own and reviewing the web solutions afterwards to fill in the gaps in your learning (I took each twice over a period of 6 weeks and did fine on the exam). Ignore prep books for the physics test; they are a waste of money, as the practice exams and web solutions offer plenty of preparation.

Who should I ask to write my recommendation letters?
Ideally, a Nobel Prize winner who swears that you are the spawn of Isaac Newton and Mother Teresa and will revolutionize your field before quals. Realistically, at least two professors who have overseen your research and an academic advisor who can speak about your coursework, participation in academic programs, and general background and personality. Having recommendations from a professor outside of your university (e.g. from a summer research program) or one who plays poker and shares oscilloscopes with faculty on the review committee (yes, scientists are humans too) are considered pluses. However, the party line is that you are better off with a good letter from an unknown professor who knows you well than a mediocre letter from a Nobel Prize winner who met you once at a seminar.

How should I write my personal statement?
As far as I can tell, there at least three things that grad schools want to hear about: (1) your interest and motivation for doing research, (2) your experience in and preparation for doing research, and (3) your short and long-term goals. Grad schools commit a lot of time and money to accepted grad students, and a PhD student who burns out after two years and heads to Hollywood to pursue a life in film is generally considered a poor investment. For this reason, grad schools want to know that you have reasons for being interested in research and, ideally, a history of pursuing it. In addition to demonstrating that you have faced the frustrations of research, have a realistic picture of what to expect, and yet are still naive enough to want to pursue a career in it, a history of research indicates that you will be able to hit the ground running in grad school and will require less initial training than a born-again English major who decides upon graduation that they want to investigate the origins of our universe. Finally, grad schools want to know that you have invested some thought in your future and thus have some idea of why you want to go to grad school and what you expect to get out of it. You should mention professors you might be interested in working with as well as general project ideas that you might pursue. If you have contacted those professors (and you should), you might mention that, as well as project ideas that stemmed from your exchange. You need not specify fine details, such as the exact molarity of the buffer you will use in step 2 of your first experiment or the variance for the Gaussian noise model you will use in your modeling project. In fact, if you are too emphatic about one particular project, this might even be detrimental to your case, as grad schools may assume that you are inflexible and might be disappointed or even drop out if that particular project does not work out (for example, because the advisor has too many students already). Unfortunately, if you claim to have interests in every field from C. elegans genetics to atomic microscopy to quantum field theory, grad schools will assume you are naive and have not thought hard enough about your future. In summary, you must navigate between Scylla and Charybdis and convince grad schools that you are creative and goal-oriented yet realistic and flexible. Most importantly, start writing early and get feedback from friends, grad students, and (if you are very lucky) professors. Seeking multiple rounds of feedback on each of my essays was probably the single most helpful thing I did while applying.

Fellowships, on the other hand, are slightly different. They seem to be slightly less interested in your interests and motivations and more interested in your previous and proposed research. Several fellowships, including the NSF, require project proposals, and in these cases, feel free to go into far more detail than in your grad school applications.

Will I be hanged, drawn, and quartered for breaching the page or word limits for essays?
I doubt any one actually counts words in your essay, although an extra page may raise eyebrows. Streamlining your essays for readability over poetic flourish is a good idea, but do not stress over a few extra words.

How many schools should I apply to?
As many as you see yourself potentially striving at. I know students who applied to as few as 3 and as many as 15 (and pre-meds are known to commonly apply to 20 or more schools), though the average is probably between 5 and 10. I personally applied to 13 schools, a number at which I arrived by applying to every school which I could plausibly see as my top choice after visiting. Reasons to limit the number of schools to which you apply include applications fees (typically $60-$120/school), time spent applying (though this decreases rapidly since you can recycle essays), and difficulty scheduling visits in the spring (if you are fortunate enough to be accepted). Reasons to apply to more schools include increasing the probability that you get in somewhere, increasing the number of visits you get to do in the spring, and allowing for a “borderline” school to surprise you during a visit. Your own optimum school number will depend on how many schools fit your interests, your pre-application confidence in your admission likelihood and school preferences, your financial status, and your time for and interest in spring visits.

Where can I find more additional resources on applying for fellowships and grad schools?

• advice from other previous applications, including:
• fellowship and grad school application from Stanford CS grad student and NSF fellow Philip Guo
• NSF advice and sample essays from Boston University physics student and NSF fellow Alex Lang
• my own application essays, including:
• NSF Personal Statement
• NSF Previous Research Experience
• NSF Proposed Plan of Research
• NSF ratings sheet
• sample 2-page grad school personal statement
• sample 1-page grad school personal statement
• additional tips from the NSF (though I found these to generalize to other applications) on:
• review criteria
• tips for applying

5. Worry
Now that you have submitted all of your applications, it is time to sit in front of your email inbox, hitting refresh on your browser 24 hours a day for the next couple of months.

In all seriousness, relax. If you have taken the time to read this far in an obscure blog post on applying to grad school, you have likely worked reasonably hard throughout your college career and on your applications, and your hard work will soon pay off. You can also take solace in that you will get to spend the next several years (at least) doing something you love while your sucker college classmates get jobs they hate, slowly lose friends and gain weight, and decay into a passion-less existence. You can also track user-submitted admissions decisions at GradCafe, but I recommend against it. Admissions decisions and interview offers are not sent out all at once, so finding out that others have been accepted, rejected, or offered an interview before you have heard back is not predictive of your own situation. Therefore, you will likely do little more than fuel your own delusions and anxiety.

6. Interview and/or visit

Wait, I have to interview?
This varies by discipline. Some programs require in-person interviews, others may conduct phone or Skype interviews, and still others may admit/reject students based on their applications alone. In the latter two cases, accepted students may be invited to visit before making a decision. For some reason, interviews and invited visits seem to be more common in the life sciences than in math and physics. This may either reflect the cultures of the fields (biologists are usually slightly more gregarious than mathematicians) or the amount of available funding (as a general rule, the more applied departments usually have more money).

Should I be nervous?
Not at all. Interviews and visits are the best part of applying to grad schools! Be prepared for several weeks of:

• one-on-one time with faculty to discuss their current research, potential projects for yourself, or just about anything else
• plenty of time to talk to current grad students
• lab tours
• the opportunity to meet other students from all over the country and world with similar interests
• …and (surprise!) free travel, meals, booze, and entertainment!

That’s right – spring visits are basically extended science parties. For both interviews and visits, programs typically plan a mix of one-on-one meetings with faculty (30 minutes to one hour), info sessions, lab tours, faculty talks, student poster sessions, lunches, dinners, and receptions with students and/or faculty, and entertainment, including everything from parties to hikes to aquarium visits. “Interviews” are as much recruitment as they are actual interviews, so prepare to courted. Also, in my experience, any reasonable expenses will be reimbursed, from baggage fees to taxis to lunch to coffee. I even heard one rumor of a group of students who ran up a $2000 wine bill during dinner and managed to get comped, though I recommend against trying this yourself.

Does that mean I am guaranteed admission if I am invited to interview?
Not quite, though the odds are in your favor. Most programs claim not to have a particular quota in mind, but rumor has it that typically somewhere between 50% and 75% of interviewees are accepted. Note that more students are accepted than desired to enroll since programs anticipate that not all students will accept their offers. Note also that a post-interview rejection does not necessarily mean that you were a disappointment in person. Interviews are as much about judging your own fit and interest in the program, and faculty may have sensed that your interests were better served elsewhere.

How should I prepare?
Publish three Nature papers, lose 20 pounds, and memorize all of Shakespeare’s sonnets.

But if you do not have time for that, remember that interviews are far more fun and casual than you likely imagine. Be ready to discuss your previous research (partially to test your understanding of your work and partially to direct the conversation towards topics of your interest). Be ready also to describe your current interests, with an eye towards potential projects. For those faculty with whom you will be interviewing (schedules are typically sent the week before you arrive) or with whom you are interested in speaking with at receptions and other events, read their websites and recent abstracts, and prepare questions about their research, intersections with your own interests, and possible projects. Thoroughly reading every paper they have published since their 5th-grade book report on “The Phantom Toolbooth” is however unnecessary, as most faculty are quite happy to give you an overview of their work in person. Essentially, you should read just enough to satisfy your own interests and prepare a few questions. Preparing questions will not only help you gain information relevant to your own decision, it also displays your interest, creativity, and preparedness and will likely help your chances of admission.

What should I ask faculty during my interview/visit?
See the advisor criteria I posted above for some ideas. Besides asking about current and potential projects, consider explicitly asking about advisement style and the funding situation.

What should I ask current grad students during my interview/visit?
While you can also discuss research with them, grad students are your go-to resource for questions on local living and entertainment, coursework, and what it is actually like to work with particular professors. In addition, I found the following three questions to be very helpful: (1) what is the worst thing about your school/program?, (2) what was the biggest surprise?, and (3) where else did you consider going and why did you choose to come here? See the school and program criteria I posted above for some other ideas.

What if I do not get to meet with a faculty member who I am really interested in speaking with?
Many faculty may be traveling or otherwise unavailable during interview/visit weekend, while others are simply over-requested by prospective students and cannot meet with everyone. If someone you requested is missing from your interview schedule, try asking the program administrator if you can replace a less desirable interview or fit in some one-on-one time elsewhere. For example, you might meet with a professor at a reception, over a meal, or during a lab tour. You can also speak with current grad students advised by that professor or other prospective students who interviewed with that professor. If you still feel neglected at the end of your visit, feel free to email a professor to ask questions, set up a phone discussion, or plan to meet at an upcoming conference.

What do I do if I was accepted but not offered a visit and would like to make one?
Let the program director know that you are interested in visiting and, in some cases, they may offer to help pay for a visit (the only school to which I applied that did not offer visits was willing to do this). Even if they do not have the funds to do so, they will likely be willing to set up meetings with professors and students if you can cover travel costs. If you are in this situation with regards to a school you are very interested in, I highly recommend investing in a visit. Choosing a grad school is a major commitment and you do not want to condemn yourself to several years working with an advisor you do not get along with on a project you do not care about in a city you despise simply because you were too cheap to splurge on a visit.

7. Decide
First of all, do not spend too much time during your visits trying to weigh your options and make a decision. Given the limited time of your visits, it is better to avoid this distraction and immerse yourself in the local research and social environment. Also, do not be surprised if you leave every visit wanting to call your friends and family to let them know you have a new top choice. As I mentioned, visits are disguised science parties and you will be swooned by almost every one. In the weeks afterwards, signal and noise will begin to separate, and you will slowly whittle down your choices. If you are fortunate, you will soon realize that every school is awful except for one perfect school that caters to your every need and desire. If you are less fortunate, you will realize that there are two or more schools at which you could be quite happy and productive. If this is the case, review the criteria on faculty and schools that you drafted in the fall (or if you were too lazy to do so, consider my own listed above) and consider your decision in light of these criteria. If you are still stumped, talk with friends, family, academic advisors, and research advisor about your decision. Other prospective students whom you met on your visits are also great for this purpose, as they are in the midst of a similar decision. Encourage and reflect on the feedback you receive, but also monitor whether your explanations of your indecision consistently belie an underlying bias toward one particular school. Often, I find that, long before I feel confident about a decision, my discussions with others betray that I have already made up my mind. If you have particular questions that remain unanswered, contact professors, grad students, or program directors, as most are quite happy to discuss your position in decision space, given their own vested interest in the outcome. Hopefully, you converge on a confident decision by April 15, the sacred agreed-upon admission decision deadline, as declared in the US Constitution.*

Acknowledgements
Thank you to Paul David, Alex Lang, David Sheen, Henry Yuen, Chris Rollins, Nick Steinmetz, Samir Menon, Peiran Gao, and Matt Goldstein for several rounds of generous feedback on my various application essays. Thank you to Nick Steinmetz, Samir Menon, and Peiran Gao for also seeding my initial list of faculty and schools with their own suggestions. Finally, thanks to Bartlett Mel, Andrew Childs, Paolo Zanardi, Stephan Haas, Kwabena Boahen, and Gene Bickers for their mentorship and guidance, as well as putting up with dozens of at-the-buzzer recommendation letter requests. I will spend the next several years trying to earn the generosity that each of you has offered to me.

Postscript
For students currently planning to apply or currently applying to grad schools and/or fellowships, feel free to post additional questions in the comments. For those who have run the gauntlet and survived, feel free to post additional advice (or refutations of my own) in the comments as well. I will periodically update this guide with any intelligent conversation that emerges.

*Unverified.

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What I Did Right
1. Not Selling Out Until I Found My Passion
I entered college without a clue of what I wanted to do with my life and entertained doing everything from making films to being a chef to being a sherpa. Instead of giving up on finding my passion, pursuing a career in medicine, law, or finance, and relegating my enjoyment of life to weekends and semiannual vacations, I spent my first couple of years at USC “shopping” for a passion by sampling from just about every program and opportunity USC had to offer. I explored international commerce in Hong Kong with the business school, tried to bring clean water to villages in Honduras and India with Engineers Without Borders and the USC Stevens’ Global Impact program, immersed myself in Chinese language and business through a summer internship in Shanghai via the USC Global Fellows program, led an initiative to bring the Flexcar car rental service to USC (now acquired by Zipcar), and backpacked western Europe. My major shifted officially from history to business to computer engineering to chemical engineering and unofficially to probably another dozen disciplines.

It was not until my third year at USC that I found something that stuck. That year, I took an introductory course in electricity and magnetism with Professor Paolo Zanardi that undoubtedly changed my life. Though I had taken physics classes in the past, it was Professor Zanardi’s style of teaching that introduced me to a new way of asking and answering questions about the world that resonated with my natural curiosities and loves of mathematics and problem-solving. I unashamedly abused his open office hours by drilling him with questions for hours on end multiple times per week and his ad-hoc whiteboard lessons and suggestions for further reading fueled late nights curled up in bed with a textbook, furiously solving problem after problem. In addition to being an excellent teacher, Professor Zanardi served as a living example to me that science was a viable and rewarding career path. Though I had fallen in love with mathematics in first grade, I had never known a scientist until college and had subsequently adopted the utterly false stereotype that scientists are no more than automatons who mindlessly carry out the “scientific method” at lab benches. It was meeting Professor Zanardi and other scientists that conveyed me to just how creative and exciting an endeavor science truly was. Professor Zanardi also offered me my first significant research experience the following summer in Italy, which made it clear – science was indeed the career for me.

After my decision to focus on science, life became, in many ways, far easier. I stopped worrying about classes or homework because I wanted to do the assignments anyways. I no longer felt an inclination to “build my resume” because the activities required to do so were exactly what I would have chosen to do in my free time. In summary, my internal motivations became aligned with external incentives and there was no longer any struggle to define what I should be doing – I just did.

An important caveat to this story – I was, in many ways, completely miserable during my first two years at USC. Devoid of any clear goals or direction, I felt foolish for not knowing exactly what I wanted to do. How could I not summon an answer to such an obvious question as: what is your passion? I spent hours and days writing and reflecting, trying to “discover” the answer by looking inward, and learned two important lessons. One, identifying your passion is not necessarily easy. Two, looking inward to “find yourself” is not necessarily an effective way to find your passion. Instead, I found that serial sampling of each of your latent interests and allowing yourself to get lost in each and every activity can be far more effective.¹

Given my experience, I offer the following suggestions.²

Suggestions for students: your first goal in life should be to identify at least one thing you truly enjoy. (Note that you may, or even should, find multiple such things.) Then, and only then, might your goal shift to receiving a particular type of training or degree. If you find at least one such thing before college, great; pursue it. If you do not, use college to carry out massive parallel experiments in each of your possible interests. Actually, even if you come into college with clear goals, consider sampling from secondary interests anyways. In each case, lose yourself in these activities. While reflection is important, constant reflection can be a barrier to actually experiencing anything resembling passion, so reflect with caution. And have patience. Do not settle for a career in something you do not truly enjoy. Dabble widely until you find something you are truly passionate about. Ignore stereotypes about possible career paths and the pressure to chose something based on your parents’ preferences or potential for fat paychecks. When you pursue something you are really passionate about, happiness and success will come naturally.

Suggestions for educators: encourage students to spend time testing out their spectrum of unexplored interests. Offer and advertise optional programs that introduce students to other disciplines without forcing them to change majors. Most importantly, encourage academic advisors to ask students what they really enjoy doing and hope to accomplish, instead of merely focusing on helping them to fulfill degree requirements and resolve scheduling conflicts. Be careful in making decisions and defining goals for students; learning to make your own decisions and form your own goals is an essential part of life that should also be an integral part of education.

2. Finding Great Mentors
Mentors played an essential role both in helping me find my passion and in pursuing it. Paolo Zanardi stoked my interest in physics, fueled my self-study outside of the classroom, showed me that science was a viable career option, and gave me my first significant research opportunity. It is safe to say that I am a scientist because of him. Bartlett Mel taught me how to combine mathematics and biology to “see the neural forest for the trees” and intuit how brains do what they do, as well as how to effectively bridge the communication gap between theoretical and experimental neuroscientists. Andrew Childs taught me how to become a more independent researcher and gave me my first opportunities to give a research talk and write a journal publication. Gene Bickers and Stephan Haas introduced me to the quirky world of academia, answered long lines of physics questions, and helped me navigate many professional and personal problems. Kwabena Boahen and Ted Berger allowed a naive physicist-mathematician to charade as a biologist in their labs and, hopefully, absorb some knowledge about the brain.

Reading books and websites or attending classes is no substitute for working with a great mentor. Great mentors have followed a similar route to your own and can offer recommendations that take into account your strengths and weaknesses and anticipated obstacles on the road ahead. Finding appropriate mentors can be challenging and time-consuming because (1) they must possess knowledge relevant to your goals and (2) you must get along with them, but the trial & error necessary to find them is worth every bit of time and energy.

Suggestions to students: seek out multiple appropriate mentors. The great value of being on a college campus is not the ability to take classes (you can do that online); it is the interaction with people who are currently or have in the past pursued goals similar to your own and can offer relevant advice.

Suggestions to educators: establish programs that help professors learn and share best practices on mentoring. Additionally, establish formal programs for students across the university to help them identify and learn from appropriate mentors. Emphasize mentorship as an essential piece of undergraduate education.

3. Building the Communities I Wanted that Did Not Exist
While I spent my first semester of college reveling in the sheer number of different communities available to me, I soon realized that there were still a couple missing.

First, I wanted to live among a community of inquisitive, clean, and passionate people, so that my day would be infused with interesting conversations (and not marred by the overflowing sinkfuls of dirty dishes, typical of a college dwelling). Initial attempts, including my misguided joining of a fraternity, failed miserably. Eventually, I decided to build my own living environment. I found a 7-bedroom house and hand-picked friends and friends of friends to fill it. That was easily one of the best decisions I have ever made. The three years since have been chock-full o’ stimulating conversations³ and have convinced me of the benefits of communal living.

Second, I wanted to get off-campus, out of the city, and into nature more often. Despite USC’s prime position as a basecamp for hikes and camping trips exploring the geological chaos surrounding Los Angeles, I could not find a single student hiking group. So I started one. What began as a small group of friends organizing under the banner “USC Trekkers” soon ballooned into a 300-member Facebook group, and I have spent the past three years hiking just about every other weekend. Though I rarely use that Facebook group for organizing hikes these days, it seeded a now quite strong community of hikers at USC. I hike and camp now more often than ever with a revolving community of friends, and a fantastic official USC student group, SC Outfitters, has since spawned and gained popularity.

Suggestions to students: if you are searching for a community that does not exist, build it. It certainly takes effort to organize and develop a new community, but the results and experience are well worth the time and energy. If the communities you are searching for already do exist, great; join and improve them. However, due to the sheer volume of students organizations at a large university, it is easy to drift apathetically from organization to organization, feeling that each one is tolerable but “not for you” and finding yourself at graduation realizing that you did not experience college as you wanted to. Do not let this happen.

Suggestions to educators: make it easy for students to find the guidance and resources necessary to start new communities (USC does this quite well actually). In particular, offer avenues for students to easily organize living communities around mutual interests.⁴

4. Learning Outside the Classroom
As my family and friends know, I read textbooks like novels and have since high school. Doing so has helped me in several ways. One, it guided my selection of courses by indicating areas of interest or weakness. Without reading on my own, I would have been at the mercy of degree requirements and minimally informative course descriptions in allocating my time and energy at college. Two, it allowed me to get much more out of my classes, in part by knowing how to ask the right questions. I have found that learning requires at least two passes through a body of knowledge. During the first pass, one gains a sense for the general concepts and relationships between them but spends a great deal of time confused and unsure of what questions to ask to alleviate this confusion. During the second pass, one now has a sense of the general story and can focus on the details as well as asking the right questions to clarify confusion, recognizing the essential assumptions and tools necessary for the production of knowledge, and solidifying links between the important concepts.

Of course, textbooks are neither the only nor necessarily the most appropriate route to learning outside the classroom. In fact, I would argue that what I did outside of the classroom and beyond textbooks was most important to my education (textbooks and courses merely enabled me to do some of these things). For example, leading an initiative to bring Flexcar to USC taught me that proactively solving your own problems often helps others in the process, an 8-week internship in Shanghai developed my Chinese language skills far more than 3 semesters of coursework, working to bring clean water access to a village in India with students from engineering, international relations, and health promotions taught me the value of interdisciplinary teams, starting a weekend hiking group taught me how to organize and motivate people, helping to build an open-source web platform for scientific collaboration (CoLab) taught me how to monitor the zeitgeist of a community and channel it into a useful tool, and perhaps most importantly, doing research at and outside of USC conveyed to me the joy of scientific problem-solving and taught me how to pursue original research and communicate it to others. Looking back at my time in college, it is these non-coursework opportunities that I value most, not the classes. As I mentioned above when discussing mentors, the value of college is not the coursework; it is the professors, students, and opportunities that one gains access to.

Suggestions to students: do not restrict your learning to your coursework. Read outside of classes to inform how you choose courses and projects in the future. Preview course material so that you can ask the right questions and get the most out of your time with a professor. Perhaps most importantly, go beyond courses and textbooks; seek out opportunities and start projects to help you explore your interests and solve problems important to you.

Suggestions to educators: your primary goal should be to instill a curiosity and love of learning and problem-solving in your students. With that in place, they will carve their own paths. Also, de-emphasize the lecture. Encourage students to view video lectures or read textbooks outside of class and emphasize Q&As, discussion, and collaborative problem-solving in the classroom in order to maximize the value of student-professor interaction time. Integrate into courses projects that allow students to solve real-world problems of interest or importance to them. Offer plenty of non-major-specific, optional programs that allow highly motivated students to gain real-world experience solving problems of interest to them. Offer resources for students to easily propose and implement new projects and programs. Be flexible in allowing students to take less courses, forego homework assignments, or take a semester off to pursue such opportunities.

Where I Screwed Up
1. Teach
I have found that I do not truly understand something until I can (and do) teach it. When teaching, you are forced to understand every nuance of the relationships between concepts, the big picture as well as the fine details, and which assumptions are required for certain arguments and why, and you must be able to anticipate and answer every question that a naive but inquisitive student might ask. Combining this with my suggestion above that learning requires two passes, my educational mantra for achieving deep understanding has become: learn twice, teach once. Ideally, I would love to see teaching experience integrated tightly into education so that each generation of students is encouraged to teach and mentor the generation younger than them. For example, middle schoolers might help first graders learn to read, high schoolers might help middle schoolers learn algebra, college students might help high schools learn neuroscience, and so on. Beyond gaining additional familiarity with some body of knowledge, teaching also offers valuable experience in presentation, including how to combine an understanding of someone else’s background and the material to be presented into a coherent and satisfactory explanation.

Despite my love, appreciation, and proselytization of teaching, I failed to gain any consistent, formal teaching experience in college whatsoever. I did not mentor local high school students,⁵ tutor other college students in introductory material that I knew quite well, or even study or do homework in groups, which would have exposed me to spontaneous opportunities to teach. My rationale for foregoing the first two opportunities was that I did not have enough time to learn all the things I wanted to learn and do research and teach. My rationale for not studying in groups was that I did not want to water down my learning with socialization, piggyback on the problem-solving abilities of classmates, or devote the additional time required to organize such study sessions. In retrospect, I may have been mistaken in both of these choices. As I mentioned, teaching is an excellent opportunity for learning, as well as a rewarding experience in its own right. In the future, I will find ways to inject more teaching opportunities into my life.

Suggestions for students: teach! Teach to solidify your learning experience. Teach to learn to present. And teach because its fun and rewarding to witness and be responsible for the spark of understanding in the eye of another.

Suggestions for educators: integrate teaching experience into every level of education. Have middle schoolers teach primary schoolers, high schoolers teach middle schoolers, and college students teach high school students. Recognize that peer-to-peer teaching is not only valuable for conveying ideas to the taught student, but also for solidifying the understanding of the teaching student and for providing opportunities for spontaneous mentoring on additional academic and non-academic issues. Offering small, optional tutoring problems is not enough; make teaching required and easily accessible.

2. Have a Long-Term Project
I changed research groups in college more often than I changed running shoes (approximately 10 times to 3 times). I dabbled in groups working on everything from cognitive science and neuroscience to nanoengineering and neuromorphic engineering to quantum information and computational physics.⁶ In doing so, I gained an appreciation for the spectrum of how science is done as well as confidence that I have chosen the field that is most exciting and important to me. However, I also missed out on the opportunity to nurse a research project from vague proposal to implementation to publication. Most of my projects were done in collaboration with grad students or post-docs and focused on a sub-problem of someone else’s project. Only once did I feel that I truly owned a project and even then, the original project proposal was made by a professor (we collaboratively worked out the details).

In retrospect, my serial research sampling seems like a necessary part of converging on what I wanted to do with my time and energy in the future (I did not even consider science as a career until late in my sophomore year and had a bit of catching up to do). However, ideally I would have converged on a general topic before college, carefully chosen a research group based on advisor compatibility and research focus, and nursed my own project from vague proposal to publication throughout my undergraduate career.

Suggestions for students: to the extent possible, come into college with an idea of what you want to accomplish. Start a project or organization as early as possible to tackle a problem of interest and importance to you. Explore every aspect of that project and own it.

Suggestions for educators: encourage students at an early age (well before college) to begin considering what they enjoy doing and what is important to them. Emphasize reflection, independent decision-making, and goal-setting, so that students are more likely to enter college confident of what they want to accomplish. In college, emphasize these same themes in academic advisement and work with students to develop a set of goals that include an independent, long-term project of personal interest and importance.

In Summary…
…I found my passion, learned from great mentors, built the communities I wanted but did not yet exist, and did not restrict my learning to the classroom, but I also missed out on opportunities to teach and pursue a long-term project. All things considered, I have changed a lot over the last few years and while that is not sufficient for indicating progress, it is at least necessary.

Feel free to learn from my perceived successes and failures… or make your own mistakes. I confess the latter is probably more fun.

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A good science writer must combine the dry precision of a mathematician with the relaxed storytelling of grandpa. Error too much on the side of the mathematician and you produce an unmotivated collection of facts that is about as fun to read as a 1960s computer punch card. Error too much on the side of grandpa and you leave your reader stealthily checking his watch and wandering when you will return from yet another sidetrack on the merits of the seat cushion textiles used in pre-1973 Chevys and get back to your main point.

In his Wisdom of the Hive, Seeley struck this balance perfectly, balancing a brilliant overview of all that is known on the foraging behaviors and allocation of workers in honeybee colonies with informative discussions on the methods and motivations behind the seminal experiments.

In “Honeybee Democracy” however, Seeley errors a bit too much on the side of grandpa. The first eight chapters of this book are bloated with anthropomorphic speculation on the inner lives of bees, gushing commentary on the brilliance and diligence of Seeley’s colleagues, and various anecdotes unrelated to his experiments. The admittedly interesting results and experiments on how honeybees select their new hive locations could easily have been summarized in a magazine article but instead are spread thinly over 200 pages of stealthy watch-checking and anxious squirming.

That said, the last two sections shine. Chapter 9 covers a lucid analogy between honeybee home selection and the neuroscience of primate decision-making. The underlying message is clear: replace bees with neurons and the mathematical principles behind the two systems are tantalizingly similar. Though I mentioned it in my review on Wisdom of the Hive, I echo: neuroscientists and entomologists would do wise to start throwing parties together. The computational problems faced by social insects and neural networks are often very similar and if Nature is as clever as we credit her for, then she has likely recycled her best evolutionary solutions.

Chapter 10 concludes the book with an insightful overview of lessons on effective group decision-making that Seeley has borrowed from his bee friends. While I usually find these extrapolations to human behavior cringeworthy (for the last time Deepak Chopra, special relativity and quantum mechanics do not imply that all viewpoints are equally valid and all of the Earth’s creatures are connected by a magical consciousness field), Seeley’s suggestions are well-motivated by his studies of bees and genuinely helpful for human groups. He advises that groups [1] be composed of individuals with mutual respect and shared interests (to unify goals and enable discussion), [2] led by a leader who acts as mediator rather than driver of discussion (to avoid Bush administration-like kowtowing), [3] initially seek diverse proposals independently generated by group members (to ensure that all potentially useful ideas are laid on the table), [4] aggregate group knowledge through debate (to enable each group member to make an informed and ideally independent decision), and [5] to anonymously survey the group opinion often (to effectively identify contentious decisions and accelerate convergence once a clear winning proposal begins to emerge). I found the most interesting feature of honeybee home selection to be that bees “advertising” a new home site do not directly recruit the support of their fellow bees; instead they recruit their independent assessment. That is, recruited bees play the role of skeptic, examine the candidate home site for themselves, and perform an assessment that is independent of the initial enthusiasm conveyed by the original advertising bee. Seeley is (rightfully) emphatic in his discussion of lessons [3]-[5] that a certain level of independence among the members of a group is essential to effective decision-making.

Seeley also includes a very brief but fascinating review on the concept of “signal ritualization” in the context of bee behavior (I first encountered this concept in the work of theoretical biologists Maturana and Varela). The idea is that evolution may sometimes seize upon an incidental action and modify it to produce an intentional signal over time. The example Seeley offers is the “buzz-run signal.” In order to prepare for flight, a bee must rub its wings together. Thus, wing buzzing is a natural indicator of impending bee flight. Yet bees have even learned to buzz their wings without flying in order to encourage other bees to prepare for a group takeoff. In other words, buzzing has been “ritualized” from an incidental predictor of flight in the buzzing bee to a signal encouraging flight in nearby bees.

Two questions I have for any entomologists that happen to stumble across this review. One, in light of Seeley’s suggestion that honeybee colonies have responses resembling metabolism and immune responses, I am curious whether colonies also exhibit behaviors analogous to aging and learning? Two, Seeley mentions that the number of dance circuits in a waggle run reflects the quality of the advertised home site, but have any studies probed whether rate and duration of waggle runs serve as separate channels of information?

In conclusion, if you are considering reading this book, I suggest replacing the first eight chapters with Wisdom of the Hive and then reading the last two chapters of “Honeybee Democracy” for their fascinating connections to neuroscience and human group decision-making.

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My Goodreads Rating: 3 of 5 stars

Clearly Braun is not familiar with the recipe for modern pop science texts. Where are the extrapolations from statistically insignificant correlations to bold sermons launching the next consumer craze? Why have they been replaced with tempered, conservative statements accurately reflecting the uncertainty of the scientific process and our current state of knowledge?

Genre-bending accuracy aside, Buzz is a handy user manual for the human body and the two drugs you almost certainly abuse it with – caffeine and alcohol. Braun employs an entertaining, Magic School Bus-style strategy of conveying the science from the point of view of our molecular stars as they journey through your poor unsuspecting body. If you maintain a healthy information diet (or frequently [ab]use either substance), you are unlikely to find many stunning surprises in the discussion of behavioral consequences (Egads! Alcohol disrupts learning and proper sleep and caffeine improves cognitive speed on mundane tasks and is a mild diuretic?!), but the basic science behind their commercial production and effects on the human body offer a few fascinating tidbits:

1) Alcohols are actually a quite large family of molecules. The one you are most well-acquainted with and commonly refer to as “alcohol” is ethanol. However its not the only member of the family capable of getting you drunk. Methanol, just a carbon atom away from ethanol, can also induce intoxication. The reason you do not see methanol on the shelf at your liquor store, however, is that a methanol hangover comes with a slightly less appealing side effect than a mere hangover – permanent blindness. Methanol is broken down into formaldehyde by an enzyme that is found in your retina… and formaldehyde is not something you want your eyeball getting cozy with.

2) That most of the table wine you find weights in at 12% alcohol content is no coincidence; its a necessary condition of the production process. Ethanol is typically produced by the gasping breaths of suffocating yeast cells, and in a 12% ethanol bath, ethanol can no longer diffuse across the yeast cell wall, inducing the drowning cell to shut down.

3) Caffeine, contrary to popular belief, is not exactly brain fuel. It works by blocking the activity of adenosine, an inhibitory neurotransmitter that seems to build up in the body throughout the day. Thus caffeine works by “turnING off the brake” rather than “hitting the accelerator.” This is important because it makes it nigh impossible to overdose on caffeine. On the other hand, this means that if you are a lifeless drone devoid of passion, caffeine cannot rescue you.

One question I leave for researchers of caffeine is: does there exist a biochemical means by which caffeine might make us think or remember that we are/were much smarter under its guidance than we really are/were? Many claim to be granted creative superpowers by caffeine but current research has not been able to support these claims. Perhaps caffeine only increases our beliefs about our cognitive abilities and not our abilities per se.

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Book Review
Never have I read a book that communicates the process and logic of scientific discovery so well. Like erotic literature for the scientist, Wisdom of the Hive not only conveys what entomologists know about bee colonies but the graphic details of they found out. Seeley prefaces every discussion of experimental data with the precise thought process that led him or colleagues to perform the experiments as well as a clear overview of all methods used. He follows each discussion with scandalously honest assessments of what can and cannot be concluded from the results. He even has the grit to discuss competing hypotheses (i.e. views he does not hold), past and present misconceptions in his field (i.e. times he and others were wrong), unresolved problems (i.e. stuff he hasn’t figured out), and suggestions for future experiments to resolve these problems (i.e. ideas he has not yet had time to pursue and that could be taken up by others). Perhaps most importantly, Seeley has the discipline to not blow his scientific load early and lead discussions with experimental results and conclusions. Instead, he carefully walked me through the historical results and thought process that lead to a particular question, considered possible routes to resolve this question, and only then revealed that: “Oh by the way, I’ve performed this experiment and here are the results and how I interpret them.” In other words, Seeley never answered a question I didn’t have; he takes careful steps to ensure that I was practically begging for the answer when he presented it. The only danger in going into such detail is that Seeley has to spend the first four chapters and eighty pages introducing the reader to bee physiology, experimental methods in entomology, and the broad topics covered in the ensuing discussions of experiments. I was sipping from these initial pages like a forager bee from a 2.5 mol/L sugar solution feeder after a week-long thunderstorm, but those not sharing my enthusiasm should take note – the book really shines from Chapter 5 onward.

Despite the focus on experiments, Seeley also paints a coherent theoretical picture over all by emphasizing abstract principles of information flow within a hive. Thus, despite the dozens of experiments mentioned and the dazzling complexity of the beehive, I feel confident that I could take up a summer internship in a beehive and never break decorum. He also includes a summary at the end of each chapter to highlight the most important experimental results and open questions. Every field needs a Seeley – someone to provide a comprehensive and even-handed review of methods, past experimental efforts, current agreed upon and disputed hypotheses and models, open questions, and suggestions for future research directions and experiments.

This masterful work can be read as a comprehensive review of information flow in bee colonies, a how-to guide for designing and carrying out experiments, or a near-perfect example of scientific writing for a general audience.

What I Learned
Despite several endeavors into the complexity and chaos literature, I’ve never encountered a better treatise on how global organization emerges from local interactions. Bee colonies elegantly optimize the allocation of labor and collection of resources to satisfy current and projected needs even though colony resource levels and needs are neither known to any single bee nor readily available in a centralized signal. Instead, individual forager bees integrate information about their colony’s needs with the profitability of resources they have discovered, and if the resource is judged important by that bee, the bees performs a “waggle dance” to recruit other bees to join him in foraging from his discovered source. The details of the waggle dance indicate the location of the resource while the duration of the dance is a measure of how important the bee thinks his discovery is to the colony. Since other bees sample dances unbiased, longer dances result in more bees recruited. No Department of Labor or managerial staff – just individual, information-processing, dancing bees. Foragers can also regulate their personal foraging vigor to increase or decrease resource collection as well. (Why not go all out all the time? Its not energetically efficient to do so, and energy seems to be a constraining resource in bee colonies. There is no bee McDonalds or manufacturer of bee Oreos.) The emerging picture is this: if you want to design a complex and powerful organization in which individual members possess as little information on the actions and goals of the organization as possible, the US government a bee colony would be an excellent model.

How do foragers determine their colonies’ needs? Again through local mechanisms – the search time for a processor bee to unload their delivery (in the cases of nectar, pollen, and water) and personal level of protein (in the case of pollen). Short unload time for nectar? Clearly not enough nectar is being collected. Dance a waggle dance to recruit more foragers. Long unload time for nectar? Clearly the processors need to ante up. Dance a tremble dance to recruit more processors. Sustained success of nectar foraging? Clearly the black locust trees are in full bloom. Perform a shaking signal to recruit more foragers. Surplus of protein in the diet for a pollen forager? Clearly the colony has plenty of pollen. Cease pollen foraging and go check out the waggle dance floor to see what the colony really needs.

These mechanisms also introduce the distinction between cues and signals. A signal is produced explicitly to communicate information, while a cue is a byproduct that may act to communicate information. Search times and protein in the diet are both cues while waggle and tremble dances are signals. Why would bees use cues? One reason is that they are easier to evolve. A cue requires only the evolution of a recognition mechanism for an exiting observable instead of the co-evolution of signal production and recognition. There are also cases in which signals would be difficult and expensive, such as employing a bee to survey the colony’s entire resource stores and broadcasting his findings. Why then do bees also use signals? For some information, there does not exist a cue. A returning forager loaded with nectar may be adorned with the scent of flowers which provide some information about his collection source, but the direction of these flowers is not encoded in him in any way. Thus, to recruit more foragers to a profitable source, an explicit signal (the waggle dance) is required.

Colonies also exhibit the influence of resource requirement variability on collection mechanisms and the differences when that variability is supply-driven vs. demand-driven. Because nectar and pollen availability are highly variable, bee colonies do not send all foragers to optimal collection sites but instead distribute them among non-optimal sites as well. This provides for the continual monitoring of resource sites and robustness against rapid shifts in supply. Also, since the variabilities in need for nectar and pollen are supply-driven, bees maintain stores of these resources in their hives. The variability in need for water, on the other hand, is demand-driven, and bees do not store water but merely collect it when needed.

Colonies are also capable of integrating external and internal signals to make decisions. High nectar availability (external) and nearly full combs (internal)? Clearly the colony is running out of space for honey storage. Build more combs. (By the way, how do processors detect comb fullness? Though results were not conclusive at the time of this book’s writing, probably long search times for empty comb.)

Colonies also employ combinations of mechanisms acting on various timescales to regulate their function. Pollen foraging is regulated both by the collection rate per forager (short) and the total number of pollen foragers (long). Why two mechanisms? The former is faster to adjust but provides less dynamic range, while the latter is slower to adjust but provides more dynamic range. The result is a rapid and robust combination of mechanisms allowing colonies to match pollen collection rate to pollen demand and supply.

The above also highlights evolution’s ingenious reuse of biological design principles: the use of search times in nectar, water, and pollen collection to indicate balance between colony demand and supply and the use of dances for communication (waggle and tremble) of resource needs and locations.

In closing, the above language I use is not accidental but is meant to suggest analogy with another system whose investigators might benefit from considering the design principles of bee colonies and the experimental techniques and theoretical concepts of its researchers. That system is the human brain. For those who listen carefully, discussions of global organization implemented by local interactions, the dual use of cues and signals, the essential role of variability, the integration of external and internal signals, the interaction of mechanisms acting on various timescales, the distributed storage of information, the use of excitatory and inhibitory feedback, and the elegant reuse of mechanisms should sound eerily familiar.

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1. The Synergy Between Microfinance and Open Science
At least in the US, the most typical flow of funding for science follows the pattern: taxpayer -> government -> scientists. FundScience, SciFlies, and EurekaFund ask, “Why not cut out the middle man?” Their idea is to enable citizens to fund scientific projects directly. While any one citizen probably can’t afford to fund anything but mathematics (coffee is cheap), the collective donations of many science groupies can easily add up to support more resource-intensive projects.

I really like this idea because it beefs up the incentive for scientists to adopt open science practices. Why? Consider which projects are most likely to be funded by microfinance. If I’m a citizen about to throw several hundred dollars into a scientific project, I want to be able to see the science. A published paper every few months (or year) is not enough. I want to see the process, I want live updates, and I want to feel like my donation is moving science forward. In other words, citizens will be more likely to fund open science projects than traditional proprietary projects.

Microfinance needs open science because it needs a way to attract citizens and get them excited about the ongoing science of a particular lab. Open science needs microfinance in order to create clearer incentives for scientists to share their science.

2. Reproducibility as the Standard for Open Science
Science is supposed to provide a systematic way for us bumbling fools to avoid deceiving ourselves. One way it does so is by enforcing that our theories be based on results that are reproducible. Yet consider the last paper you read. Where was the raw data from which plots were produced? Where was the simulation code? Where were the exact experimental protocols? Could you really reproduce the results of that paper without this information?

Science should not require trust in another’s scientific infallibility. If you publish an interesting new discovery, I should have the opportunity to convince myself of your discovery by reproducing it. Science that is not reproducible is not science; its marketing.*

The standard of reproducibility provides an answer to the question: “Just how open should science be?” If we truly mean to do good science and avoid deceiving ourselves, we need to release every bit of data, code, protocol, and communication that would allow a colleague to reliably reproduce our results.

If you’re interested, you can read, listen, or watch more on this idea from computational scientist and policy wonk Victoria Stodden.

3. Come for the Closed, Stay for the Open
There’s a problem with websites whose main benefits come from a large community of users – they’re only useful once plenty of people sign up and early adopters will be bored in the meantime. Successful websites should be useful to single users or small groups, even if all their friends & colleagues haven’t signed up yet.

For web apps promoting open science, this means that the successful sites will be those that prove useful to individual researchers or research groups, regardless of whether or not their colleagues also use the site. For CoLab (a website enabling online scientific collaboration that Casey Stark and I built and demoed at OSS 2010), this means creating a rich set of tools that is useful for managing the workflow of individual scientists or groups.

Doing so is essential to convincing those that are on the fence about open science to give it a try. The goal is to draw scientists in with slick project management tools for their closed group activities, expose them to the lively discussion and new collaborations being formed over the open projects on the site, and gradually convince them that openness makes science more efficient and fun.

(Thanks to Jason Hoyt at Mendeley for pointing this out.)

4. New Vision for CoLab – Enable Scienctific Debate Around Any Piece of Scientific Content
CoLab was inspired by PolyMath, Quantiki, and a few other experiments in open science from the theoretical physics & mathematics communities and was built by a pair of physics and math majors. Not surprisingly, the site is currently optimized for collaborating over projects that focus on discussion and equations. But Casey and I are aiming to make it stupid easy for all scientists to collaborate openly online, not just physicists and mathematicians. After a series of long discussions with Jean-Claude Bradley, Lee Worden, and other experimentalists who want to share more than equations, I think we’ve got a better idea of how to do so.

Our new vision for CoLab is to enable scientific debate around any piece of scientific content. We want to make it stupid easy to center a discussion around protocols, data, plots, published papers, papers in progress, simulations, code, or any other component of scientific research. As an experimentalist, I should be able to import a lab protocol, raw data, or manipulable plots based on a live feed from that raw data and discuss it online with collaborators across the globe. As a computational scientist, I should be able to import code or live simulations and troubleshoot online with anyone in the world who might be able to help. As a member of a journal club, I should be able to import a published paper and collaboratively highlight and annotate in-line with colleagues, from those in the lab next door to those in another country. As a researcher ready to publish, I should be able to host a working version of my paper online, collaboratively edit with any of my colleagues, and submit a link directly to a journal, without being forced to download the paper and make finishing touches offline. In short, as a scientist, I should be able to easily and openly discuss any piece of my science with my entire scientific community.

That’s no small task, but its what science needs and what we will continue to build.

*Update (August 4, 2010): After a fruitful discussion with Michael Nielsen (@michael_nielsen) and Seb Paquet (@sebpaquet) on Twitter, I should clarify that certain fields, such as astronomy, have fundamental barriers to reproducibility. As much as they might love to, physicists cannot summon supernovas on command. Thus, in observation-based fields, we should stress that data analysis be reproducible but not necessarily data collection. The key point is that information exchange between researchers should not be a barrier to reproducibility.

**Update (August 7, 2010): As pointed out by Greg Wilson in the comments below and Lisa Green of Creative Commons over lunch today, there have been plenty of open science conferences over the past decade. This sentence should really read: “…first major conference devoted to open science that this baby scientist & web dev noob had ever seen.”

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Why the Physicists Can
There is a cultural myth that progress in theoretical physics is made by emaciated hermit-geniuses who go off into the woods for months, scribble equations and day-dream in solitude, and return with profound insights into Nature’s inner workings. Although this is an exaggeration, there is some truth to it. Theoretical physicists can go off into the woods and do their work (albeit usually in the company of others) and can make progress while spending a great deal of time in solitude (although most don’t). Though theorists and experimentalists do work closely with one another, theorists are capable of running off without experimentalists for months or years on end and still making scientific progress.

Albert Einstein famously popularized the Gedankenexperiment – the thought experiment meant to elucidate scientific truth based solely on previous knowledge, logic, scientific intuition, and imagination. Though Einstein did not work alone (contrary to popular belief, he had many collaborators), many of his ideas were inspired by thought experiments. For instance, his inspiration for special relativity was based on his mental simulation of chasing a beam of light. More recently, string theorists have argued that, while decades or more ahead of realizable experiments, their approaches to describing the fundamental laws of nature represent a new kind of science that places great confidence in both the ingenuity of the human mind and the beauty and symmetry of Nature (no reference here – this is just what I’ve heard on the physicist circuit).

For our purposes though, the important point is not to what degree theoretical physics can temporarily decouple from experiment; its that progress can be made by theorists at all without constantly holding hands with experimentalists.

Why is this possible? What is special about physics that allows this to occur?

First of all, this wasn’t always possible in physics. Go back to the toddler years of science when Galileo, Newton, and friends were paving the way for modern science, ask them whether they consider themselves “theorists” or “experimentalists,” and you are bound to get blank stares. The distinction between theory and experiment wasn’t made until centuries later. Galileo proposed in precise mathematical terms our modern concept of inertia and built pendulums and telescopes. Newton laid down his famous three laws and played with prisms and mirrors. In the early days of physics, theory could not stray far from experiment.

What allowed theory to periodically decouple from experiment was the establishment of a sufficient theoretical framework to begin with.

The Gedankenexperiment requires solidly tested laws from which to launch intuitive explorations. Galileo and Newton had no such thing. They had to stick very close to Nature and experiments because at that time, we knew very little about Nature. Einstein, on the other hand, had a little more going for him. He had Newtonian mechanics and Maxwell’s electromagnetic theory upon which to base his dreams about chasing beams of light. He didn’t have to actually try to chase a beam of light (which would have been a bit difficult) because he had a solid theoretical framework within to mentally simulate at least parts of the experience. In other words, a solidly tested theoretical framework can allow us to replace many basic physical experiments with thought experiments. Fast forward to today and physicists have established a much richer basis of well-tested laws, an environment that supports entire intellectual castles of theory, well-protected from the toils of experiment.

What I want to emphasize is that without an established base of theory, there can be no decoupling of theory and experiment. The modern state of theoretical physics has spoiled many scientists who tend to borrow their ideas of what theory work should look like in other fields from physics. But physics is quite different from other fields. Physics is very old, often deals with relatively simple phenomena, and has centuries worth of solidly tested theories. In other words, borrowing assumptions from modern physics is a grave mistake. For many fields, including neuroscience, it would be far better to borrow ideas about the coupling of theory and experiment from early physics.

Why the Neuroscientists Can’t
Modern neuroscience has very little theory to build upon. Sure, they’ve got the neuron doctrine, that the brain is made of individual cells, but that’s not much more than a special case of a more general law in biology. Beyond that, even the widely accepted notion that spiking neurons are the sole transmitter of information in the brain is a bit shaky. We are quickly gathering plenty of anatomical data, correlations between brain activity and behavior, and other interesting nuggets of phenomena, but broad theories to help us understand this sea of data are either non-existent or highly speculative.

Worse, its not even clear whether such theories will exist or what they will look like. While physicists simplified their game by focusing on “fundamental laws”, neuroscientists face the menacing challenge of historical accidents and messy hacks built by millions of years of evolution. Many (including myself) are banking on the existence of some basic laws that govern brain structure and dynamics, but these laws may look very different from the somewhat more “ahistorical” laws of physics.

What the Neuroscientists Can Do
So if you’re a bright-eyed, bushy-tailed, naive young physicist/mathematician who dreams of building theories of the brain, what do you do?

1. Avoid brain theories built by rogue engineers, physicists, and mathematicians who have never met a biologist.

To clarify my earlier comments, its not that brain theories don’t yet exist; its that good brain theories don’t yet exist. There are plenty of electrical engineers debuting their latest computer architecture models of the brain, computer scientists proposing their shiny new graphical models of learning, mathematicians arguing that synchronized feed-forward neural networks clearly solve the binding problem, and other rogue scientists who base their theories of the brain on their intuitions about how the brain should work rather than data about how the brain does work*. Plato & Descartes may have had to base their theories of how the mind works on pure introspection and phenomenology, but ever since Ramon y Cajal starting poking around neural tissue, we’ve had real live data to guide our intuitions. Decades from now, we will look back on the state of theory in modern neuroscience and wonder how so much nonsense was published. No field in science today is more polluted with bunk theories and outrageous publications than neuroscience.  If you want to understand actual brains, don’t fill your own with this drivel.

2. Form tight collaborations with experimentalists (and maybe even do a few experiments yourself).

There is room for theory in neuroscience – theory tightly coupled to ongoing experiments. Find people with patch clamps and MRI machines. Understand what they do and how they do it. Propose new experiments and try to help explain the results of old ones. The interesting and achievable projects in theoretical neuroscience of today are not the grand challenges such as explaining the hard problem of consciousness; they are explaining the tiny anomalies in experimental data that make you pause for a moment and scratch your head. Why does the distribution of synaptic strengths in rat visual cortex follow a lognormal distribution? Why does it consistently seem that roughly 90% of neurons are inactive? Why does ferret visual cortex activity rate and variance seem to rise throughout early development, peak, and then decline in maturation? These are the types of questions we need to tackle first before we can explain consciousness, thought, love, and all of that fun stuff. As much as I look forward to the possibility of understanding the brain well enough to support genuine Gedankenexperiment-style theory work, we’re not there yet. Now is the time for theoretical neuroscientists to imitate the physicists of Renaissance Europe, not those of Princeton and Waterloo. In other words, either get your hands dirty with experiments or make friends with someone who does.

Either way, don’t stray far from the data.

*For the sake of not making too many enemies, I’ll avoid references here, but you know how to use Google.

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Explorations in Mathematica
As you may remember, the last we heard from our heroes, they were in search of a version of Levinson’s theorem (relating the number of bound states to the winding of the phase of scattered states) that could be applied to graphs.  Given that neither Andrew nor I were quite sure how this relationship between bound states and the phase shift would play out, we decided to do some experiments.  First, we cooked up equations for the simplest graphs we could think of – a single weighted edge and then (drumroll) a single weighted edge with a self-loop on one vertex.  It quickly became apparent that if we wanted to take a look at meatier graphs, we were going to need some help.  So next, I hacked together a series of programs in Mathematica that could take graphs, plug a tail on some arbitrary node, calculate the transcendental equation whose solutions correspond to the existence of bound states, calculate the phase shift of scattering states, and then plot the transcendental equations alongside the phase shift, all with manipulable graph parameters.  In this way, I could jiggle the weights on the graph, watch bound states come into and go out of existence, and simultaneously monitor the phase shift.  I’ve posted my current Mathematica notebook here, so you can make sense of the above jibba-jabba and do some explorations of your own.  (WARNING: The documentation is limited to my stream of consciousness as I code.  Also, I’ll probably kill the link when I need the server space, so you might have to email me for updated the notebook if the link is dead.)

This was a really fun approach to research that I hadn’t taken before.

1. Recognize a hand-wavy possible connection between two quantities.
2. Investigate (by hand) a few simple cases and try to get a flavor for the relationship.
3. Investigate (by computer) much more interesting cases and try to pin point the details of the relationship.
4. Prove the relationship rigorously.

It was also a welcome opportunity to polish off the increasingly dusty programming portions of my brain and to expand my Mathematica repertoire.  (Mathematica can be an incredibly powerful aid to theory work if you take the time to learn it.  Its visualization tools are especially nifty.)

Bound State Zoology
Those interested in taking a peak at my Mathematica notebook will need a quick intro to bound state zoology.  It turns out that there are at least three distinct species of bound states.

1. Confined bound states – these guys live only on the graph and have zero amplitude on the tail
2. (Standard) bound states – these guys “leak” onto the tail; that is, they have an exponentially decaying amplitude on the tail
3. Half-bound states – these bound/scattering chimera have a constant amplitude on the tail and exhibit some characteristics of bound states and some of scattering states

From our investigations, it looks like the first two types contribute one winding of the phase and the third type contributes (go figure) half a winding.

An Unexpected Hiatus or “The First Crime in Waterloo Since Wellington Spanked Napoleon”
Mid-mathematical adventure, we hit a snag.  Given the abundance of Mennonites, smiles, and unlocked doors in Waterloo, I simply assumed that Canada was crime-free and kept my backpack in an unlocked locker while working out at the university gym.  Little did I realize just how far the local citizenry would go to snag copies of my Mathematica notebooks and, alas, my laptop and wallet were stolen.  Unfortunately, I had not made backups of my latest research.  Every setback is of course an opportunity to learn and improve and this was no exception.

The lesson: sync your research to multiple computers and ideally a trustworthy server (Dropbox is my tool of choice for this)

The opportunity to improve: as any programmer knows, code you wrote a week ago always looks painfully clunky.  In rebuilding my Mathematica work from scratch, I was able to integrate plenty of tricks and lessons I’d learned along the way.  The result – a way sexier notebook.

Complex Analysis Boot Camp
In the last week, Andrew and I converged on the general flavor of the version of Levinson’s theorem that we think we can prove on graphs.  Thus, away goes the computer and out comes the physicist’s favorite tools – pen and paper.  In our first stabs at proving the theorem, Andrew and I took completely different approaches.  My caveman approach was to try to adapt a simple trick involving the spectral resolution of the identity that Marcel Wellner used to prove a version of Levinson’s theorem for continuous potentials back in the 1960s.  Andrew’s far more sophisticated and elegant approach was to apply the topological reasoning of complex analysis to our problem.  Since my caveman club began to look a little too primitive mid-way through my proof attempt, I decided to take this opportunity to learn a little complex analysis, one of the (many) major holes in my undergrad math education.

And sweet Feynman has it been a fun last couple days!  I picked up Tristan Needham’s Visual Complex Analysis from the UW library and this book has reminded me why I fell in love with math as a wee lad.  The book’s pedagogical approach is to teach math the way mathematicians actually think about it – visually.  Needham’s book is chock full of nifty pictures of Riemann spheres, conformal mappings, branches, and more.   (I’ll post a review sometime for those interested.)  Complex analysis is an absolutely beautiful subject when couched in geometric terms and is accessible to anyone with a bit of calculus under their belt.

This was also a change of approach to learning for me.  Usually, I get generally interested in a subject, pick a textbook, and read it cover to cover, doing the problems as I go.  This time, however, my explorations in complex analysis were motivated by a very specific application, so I dove right into the middle of the book to extract the particular pieces I needed.  Two great results:

1. I’m honing in on a proof of our version of Levinson’s theorem using tools from complex analysis.
2. I became so enamored with complex analysis and Needham’s book in particular that I’ve spent the whole weekend hopping from chapter to chapter learning all sorts of interesting things about the topological properties of complex functions.

I’ve noticed that I seem to learn much more quickly with an application in mind like this.  Further investigations are needed, but this might hint at a more efficient and productive approach to learning for me – first, find a particular problem that requires the tools you’re interested in and then, go off and learn them.  This is fun because every new topic I stumble upon in Needham’s book gives me new insights into the problem I’m working on.

I still want to take a crack at a proof using my original approach as well, as I might be able to get it to work with a little more insight from my adventures in complex analysis.  It would be really cool to prove our theorem using two entirely different approaches, each of which gives a unique insight into the problem.

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By far the best of the solid state textbooks I’ve found. Whereas Kittel hops from model to model with little explanation, Ashcroft devotes entire chapters to the merits and failings of the free electron gas, periodic potentials, mean field theory, and so on. In a field that can seem as arbitrary as the dating preferences of teenage girls, such clarifications are crucial.

Still, Ashcroft does assume a grasp of graduate-level quantum mechanics at times. There’s fame, fortune, and wild and sexy physicist parties waiting for he or she who writes a great solid state text accessible to undergrads.

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Those who follow the pack waste days wrinkling their foreheads at the long, winding, historical path through quantum mechanics that David Griffiths leads his unsuspecting followers on. Those who know better skip the foreplay and face the glorious intellectual burden that are the axioms of quantum mechanics in just the second chapter of Shankar.

Shankar’s introductory chapter on the mathematics of quantum theory is the best out there. It was my saving grace after getting bogged down in a quantum information book for which I went in unprepared. If you too have lept into papers or books only to be baffled by the mysterious dances of Dirac brackets and quantum operators, let Shankar be your guide.

Shankar is also great for the independent learner. He embeds problems within the text so if you’re reading on your own, you can feel as though you’re engaged in the development of the theory (but don’t kid yourself, you’re not that smart).

I don’t see why this book isn’t far more popular. It’s not only clear enough for undergrads (and, more specifically, clearer than Griffiths historical tour), but it covers many of the topics you’d want to see in any introductory grad level course too.

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