Advice on a variety of
topics -- note I'm not responsible for any negative outcomes, but I will take
responsibility for anything positive. These are rough notes and are being
supplemented / expanded, but I figured I'd post in case it's of use to anyone.
If you read this and
find it useful, or have comments, please email me at
sjm1@williams.edu;
also general advice on a
variety of subjects:
https://aimath.org/careers-in-academia/
Topics:
General Advice
Classes
Job Interviews
Research/Thesis
Giving a Colloquium
Writing a Paper
Talks
Grad Schools / GREs
Letters of Rec
- General advice
-
If all you have is a hammer,
pretty soon every problem looks like a nail.
This is one of my favorite quotes. There are a lot
of ways to look at it. One is that if you only know one way of attacking a
problem, you'll be forced to try and fit that method to whatever you have. A
better view, in my opinion, is that you should move to the land of the
screwdrivers. If you attack the same problem as everyone else using the same
methods and techniques arising from similar training at the same time,
you'll be in a race for results. If, however, you can go to a related field
where they don't know your methods, you could be the first to apply them to
these problems, and you have the potential to make real advances. I've done
this several times in my career, applying number theory methods to
marketing, for example.
- Thank people.
If someone has done a great job for and really helped
you, let them know. Talk to their boss, or department chair. Write a letter
of support for them. Don't let good deeds go unrecognized. This is
especially true for junior people. Think of it this way. How many times have
you called a company with an issue. Do you want a competent person who can
help you and find a way to resolve the issue, or the opposite? If you call a
helpline and the person does a good job, ask for their manager and praise
them. You want people like that to stay!
- Very hard to say no, very hard to ask for help,
very hard to delegate: Three things many of us have tremendous
difficulty doing; we are there for our friends, but we often don't ask for
help when we need it. We hate saying no to people but there is a danger in
trying to do to much, and not doing things well. As you get older you will
have more and more responsibilities; find ways to delegate, if possible pay
students to help. Not only will this assist you, it is valuable training for
them!
- Be generous with credit, praise and thanks:
Thank the people who help you; say so to them, but if possible do something
to thank them. If you like it when people praise you and thank you for
the work you do.....
- Be formal at first: Be careful about
how you refer to people. *I* don't care, but others do, and if you don't
know err on the side of formality. Thus I'm Professor Miller or Doctor
Miller, not Mr. Miller or Steve if you're initiating an email to me. There
are many people who are upset when incorrectly addressed; they worked hard
for the title, give it to them. If you're not sure, go online and look them
up. The situation can be complicated; people often sign a message
informally but prefer a formal address back. I find it's safest to stay
formal with someone at a higher rank until they order me not to. For
example, when I earned my PhD one of my advisors ordered me to call
him by his first name; now that I had PhD he wouldn't accept anything else!
It was uncomfortable for me, but something I needed to do. It was much
easier when I moved elsewhere and met new people at a similar rank above me;
I was coming in as a professor and thus it was natural to use their first
name; it was hard to call my professors from college and grad school by
first name. Thus sometimes the address changes over time. When in doubt be
cautious and wait to have someone tell you 'call me X' (or use roundabout
language, such as 'your husband told me...' when referring to your
significant other's father to their mother!).
- Best graduation speech I've ever heard / read:
[[Click
here for the link -- Michael Uslan, Indiana University ]]
This is full of great advice. It's not enough to be smart. You need to be
noticed. Another favorite quote of mine is that if you're one in a million,
then there are a thousand like you in China. You need to stand out from the
rest of the crowd. There are many ways to do this; most of them do not
involve you eating alone, or working alone. Get out. Get known. Talk to
people. [[Click
here for other great graduation talks.]] As links often fail or go
down, here are alternate ones for Uslan's talk:
(https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/)
(ted talk version here:
https://www.youtube.com/watch?v=7LUg-7zDWHo&t=534s).
- Work hard. Get things done. Don't
put off till tomorrow what can be done today. If something enters your
inbox, try to get it out, or at least some of it done. Build a reputation
for excellence and speed. All tasks are opportunities. The great coach Woody
Hayes elaborates on a good work ethic in one of his graduation speeches,
available
here.
- Have an advocate: Make sure there is
at least one person in your organization who knows you well and will support
you. We all make mistakes; it's nice to have someone who has say who will
speak in your defense. (Of course it's better to not make mistakes, but....)
- Good file names. Never send a file
named something like result.pdf, paper.pdf, conf.pdf, or cv.pdf. That's a
horrible name, and a recipe to be lost in someone's data system. Have a good
name. Like JacksonMillerPhan_Toeplitz01.pdf (start 01 and not 1 as this is
better for alphabetical programs, as otherwise 10 comes before 2). You want
people to be able to find your file quickly and easily on their system, and
a good name helps. It should include your name (and group mates if there are
some), something related to the work, and a number ranging from 01 to 99. I
usually increase the first digit with major changes. You should save older
drafts, as sometimes you might go back and add material you've cut.
- Always be 5 minutes early to meetings.
Things happen. Be there. Professors and professionals' times are
often hard to come by -- if you're late it can throw off an entire day.
Sadly, the flip side is NOT true -- professors are often not ready for you
when they said they would be. Don't just sit around twiddling your thumbs --
always have something on you to work on, either in your mind or written
down. Have scrap paper and a pen. Use the time productively. (I had some
free time once, and used it to work on a math competition exam that I had to
write months later; this'll save me time later on).
- Be prepared. Don't walk into a
meeting unprepared. If possible, research whomever you think you'll see.
Know the basic facts about the company or grad school, so when you ask a
question you can ask a good question. (Yes, asking good
questions is a way to impress people). Don't ask how many graduate students
are in the program. That's a simple fact you can easily find. Ask what the
department does to help graduate students find their advisors. When I have
job interviews (either for myself or as part of a hiring committee), I have
a file on all the people I'm going to meet. I save it as a pdf on my
smartphone, and quickly glance at it before they arrive. I have a few bullet
points so I can quickly glance and recall. When I was interviewing for
Williams, I only knew one person who was definitely on the hiring committee.
The department has under 20 people; I looked everyone up, took notes about
their classes, research and interests, and was prepared to talk to whomever
I met.
- Have different versions of who you are prepared
depending on whom you meet and how much time you have!
You should have a 20-30 second, a minute to two
minutes, and a five minute spiel on who you are that you can rattle off
SMOOTHLY on a moment's notice. You should have multiple versions of each of
these, so if you end up talking to an expert in your field you can give some
meaty details, while if you meet someone with no background at all you can
find a way to connect. It's fine to ask a quick question to get a sense of
someone's background so you can pitch properly (are you familiar with
classical mechanics? ...).
- Be sociable. Not surprisingly,
people like to be around people they like. You should be able to talk to
almost anyone for 15 minutes on most subjects. I'm a cultural philistine,
but I remember a job interview where I had lunch with someone from HR who
was an art history major. I'm not a big fan of art, but I know a little
(this is a value of a liberal arts education). I talked about Eakins, who
secretly projected pictures onto his canvasses to save time. We ended up
having a very nice conversation.
- Go to conferences. Meet people.
Don't sit in your hotel room or scrimp on lunch. Build connections.
Introduce yourself. If you're with me I'll help and initiate a lot of this.
When you apply for jobs / grad school you want to be more than a name on a
list; you want a personal connection, you want to be able to send an email
to someone saying how thrilled you are to be under consideration at their
place.
- Technology fails. Have multiple backups.
NEVER have an important file in just one place. At the very least, each week
you should back up everything. I often back up major files a few times an
hour, as I *HATE* having to do the same thing twice. I know someone who lost
an entire book they were writing. All that survived was a printout. It was
not fun.
- Related to the above, have multiple copies of
your talks for a conference. Have it on your computer, email the
organizer, post on the web (so you can click on it from any browser -- if
you don't have this ability send me the file and I'll post it on my
webpage), and have a copy in your inbox. Flashrives are another good option.
You'll be amazed how many people don't do this and problems ensue. Some of
my students at [X] thought I was paranoid and ignored me. At the joint
meetings their computer didn't work, it wasn't on the web, they didn't have
it on a flash drive, .... I eventually rescued a copy off their hardrive,
but they lost a lot of time and looked bad. Another time I was at a
conference and again, no backups and the person's computer froze. Had to
quickly fix it enough to transfer the file. Things happen, be prepared.
Along these lines, you should have your slides set up and ready to go well
before your talk. Try loading them on the computer early to make sure things
work.
- Spacing: when writing tex with me, I
like to space things out a bit to make it more readable. Thus, in an
equation please use \ = \ and not =.
- Ask questions. Ask each other, ask
me, ask yourself. think about what you're assuming -- it might be facts, or
that you have to approach a problem in a given way. One of the most
important parts of research is asking good questions. This should be a
constant process. If you do this all the time, you'll frequently find new
problems to work on. Whenever you read a paper ask if all the assumptions
are needed. Ask if it can be extended. See if it reminds you of anything....
- CVs / web: You should have a good
CV. It should be clear, easy to follow. You may want to tailor it based on
where it's going (i.e., if you're applying to a liberal arts college you
might emphasize your teaching, whereas if you're applying to a big state
school you might put your research first). Your papers / talks /
slides should be on the web. You should have a webpage, or have things
posted on the arXiv or some site. Make it easy for people to find out about
you.
- Confirmations: Always leave windows
open until you get the confirmation from orders / purchases / registrations.
Even better is to take a screen shot or paste and send to yourself as an
email. do for travel, for registration, as a reminder to do something, ....
- Shaving: if you are going to shave
your face, shave many
hours before an interview or talk -- if you shave too close to the time and
cut yourself, you'll have to give a talk with bandaids. (I was recently at a
nice hotel, and didn't have my razor as I'd flown recently and took it out. I
got one from the front desk, and cut myself multiple times. When I went
downstairs to ask for bandaids they gave me a huge number, saying the razors
are cheap and many people are cutting themselves. While this is a solution
for them to the problem, it's a sub-optimal one!)
-
Fidgeting: Try not to fidget. There are many things to watch for,
from playing with your fingers to your hair to bouncing around in your seat.
- Stay calm: Don't lose your cool. A
friend of mine had an interview in a public place, and nearby someone had a
heart attack. He was calm, had someone call 9-11, walked over to help. The
hiring was impressed!
- Email (lot of points here!)
- Email with care:
Once you send an email, it is out of your control. You
must assume that the worst possible person for you to see that email will
see it, and at a time when they are maddest at you! Be careful. If you're
responding to something and you don't want it to go further, explicitly say
so. Try to have one main point per email; don't accidentally reply to one
email to start a new chain and unwittingy share other information. Don't put
in confidential information if you can avoid it. Look carefully at who the
email is being sent to; use bcc rather than cc when appropriate so people
cannot reply to all. I and friends have many times sent emails with
questions, points or concerns to one person and been a little surprised to
see that email forwarded in its entirety to many others. Think about this
ahead of time when you email!
-
Reply promptly / time delayed emails:
You should respond to most emails within a day, even if you are
just acknowledging receipt and too busy to respond in full, let people know
you got it. If you are unable to do so (say traveling, ill, ...) you should
set up an automatic response so people emailing you know it was received but
it could be a few days before you get to it. Numerous times things fall
through the cracks, person A emails person B to do X and then forgets about
it; if person B is unavailable A can move on to someone else, but only if
they know to do so. The older you get the more demands you have on your time
- it's easy to forget things (I try to keep my inbox to just a page so I can
quickly look and see what I need to do, but I often figure I'll deal with X
when the person I emailed responded, so if I don't get a reply I may forget
to do X).
-
Time Delayed Emails: I strongly
encourage you, especially in the zoom era, to set up time delayed emails
about events. This is a great way to make sure your inbox isn't too
cluttered, and to bump things to the top of the queue, and to make sure you
don't forget a task. If you have a meeting it's great to get a reminder. If
you have to write a referee report, you get a reminder when it is getting
close to being due. If you are meeting someone.... You get the point. You
can set multiple reminders at the same time, maybe you get one emailed to
you at the start of the day as a reminder, and another an hour before.
- Check your spam: Sadly sometimes
important emails end up in your spam folder. At the very least you should
check it once a week to make sure you haven't missed anything.
- Reply to all appropriate parties:
Sadly a lot of times people reply only to the sender when it concerns
everyone on the email (for example, comments on a paper or a talk). If it is
a topic that concerns everyone on the list, reply to all so everyone stays
informed.
- Time Delayed Emails: I strongly
encourage you, especially in the zoom era, to set up time delayed emails
about events. It is amazing how many things fall off your radar, having an
email that bumps things to the top is great. You can also use this to keep
your inbox down to a manageable size, moving things to the ether to return
later.
- One or two points per email: I often
have just one maybe two main points in an email. It makes it easier for
filing, and if there are too many topics you are discussing it's possible
that in a reply one is missed.
- Classes
- Know why you are taking the class!
What's your goal for the course? Why are you taking it? Is it to learn
material needed for something else? Is it because you're interested in
making this your life's study? Do you just like the professor and would take
a class on lint if they offered it? If you know why you're here, you can
have a better experience, you can seek out options or additional problems.
- General takeaways from the course:
What did you learn? Where do you expect to use it again? How does it relate
to things you've seen? I've started a compilation of key points / ideas /
techniques from classes I've taught; [[click
here for the file of takeways from classes.]]
- Be prepared! Things happen. Read the
book and build up a reserve. Start reading before the semester begins if
possible. Read ahead in the book so you'll know what the professor is going
to lecture. For example, in math I find it hard to see a new concept, digest
it, and understand theorems and examples all at once. If, however, I've
worked hard preparing for class and have mastered the definition, then the
lecture or talk is much easier to follow.
- Form study groups. You're not taking
this class online; you're at a college, with many bright people around you.
Take advantage. Talk to them. I usually had study groups of just 2 people,
rarely 3 or 4; I found if the group is too large its more socializing than
work, but different styles work for different people.
- Find a balance: Often my study
partner and I would work for a bit, grab dinner or see a movie, and then
work some more. I'm all in favor of working hard, but don't go so long or
strong that you burn out. Recharge.
- Gain useful skills. You should know
math, and statistics and a programming language. You should also be able to
write well.
- Do problems. Don't just do the
assigned homework problems. Try doing a few more to make sure you really
understand the material, or to explore it in more depth.
- Make sure the professor knows you.
You never know when you'll need a letter of recommendation. Or the professor
might have an opportunity for you that's a perfect fit. If you try to get to
know well at least one professor a semester, if you are successful just half
the time you'll have 3 or 4 people who know you well and can write good
letters of recommendation. The worst letters are those from professors who
just summarize the transcript.
- Speak up in class. Don't be silent.
If you ask a question then yes, there's a chance you'll look foolish, but
odds are half the class wants it asked. Years ago I was teaching a class and
I occasionally used clickers. I asked how many people were uncomfortable
with some material. No one responded. I asked if everyone was fine and if we
shouldn't have a review. No response. I then used clickers and said I
wouldn't report the result to the class, but how many want a review: 43%
wanted one. (If old students are looking at this, let me know if you think
this was your class!) The lesson is you're often not alone, but even if you
are: SO WHAT! The goal is to learn the material. If you tell the
professor when you don't know something, the class can adjust and help you.
Also, your question might spark an interesting conversation, or a research
project, .... I gave a lecture to the MA state teachers association on the
importance of asking good questions; [[click
here for the video (actually it's a variant that was given in one of my Calc
III classes).]]
- Constantly study. You should spend
at least an hour every two days on a class. Don't wait till the final comes.
Constantly work. Ask for practice problems to prepare for exams. Or, even
better, create your own practice problems! When I was a student I
occasionally predicted exam questions (this is easier the better you know
the professor). Things in life always happen.
- Job Interviews
- Prepare: Read as much as you can
about the company and the field.
- Honesty: Don't be afraid to say you
don't know something or ask for time to think. You can say that's a good or
interesting question, and ask for a moment to collect your thoughts.
- Be prepared for opportunities: if
they ask about your favorite class, material, activities, have something
ready. Have one or two stories where you seized initiative
and did something, where you worked well in groups. Have a story where you
failed / did poorly, how you dealt with it, and how you've learned and
grown.
- Practice! Do a practice interview.
Have someone play the job of an interviewer. Career services will often do
this. Don't walk in cold to your first interview.
- Say thank you. People are busy;
recruiting is important, but it takes a lot of time. Say thank you. If
possible / appropriate, email them a thank you afterwards, discussing
something interesting from your conversation / time with them.
- Video Interviews. If you are doing a
video / phone interview, you can have information / reminders of topics to
discuss out of view of the camera. Use this! Great to be able to quickly
glance and remember items you wished to bring in.
- Watch the time. Be mindful of how
busy everyone is when answering. Be careful about going on too long and in
too much detail in response to a question. Try to read the audience - are
they following and enjoying? Are they beginning to woolgather (some people
are better at hiding this than others!). If you're asked to discuss your
research get a sense of what level they are looking for - are you in a room
of experts or do you need to take a few minutes to set the background? You
might have a mixed group and they might say for this question talk to the
expert in the room and ignore the rest of us (one of the best questions I
ever got was to pretend my interviewer was a world expert and just talk
shop). Make sure you look at / talk to all members of the audience.
- Advice article from friends.
https://afolsom.people.amherst.edu/rnoti-p1651.pdf
- Research / Thesis
- Many of the comments here are similar to comments for
taking a class. When it comes time to write it up, see my comments on
writing a paper.
- Why are you doing this? Know your
reason for your research. Are you trying to decide if you want a career in
research in general? In this field in particular? Are you trying to start a
research program that will be productive for years to come? The answer to
these questions can affect the problem your choose to study.
- Pick a good advisor. What makes an
advisor good can be different for different people. Find someone you get
along with, who has a style conducive to your needs. Are you someone who
needs a lot of supervisions and frequent, long meetings? Are you someone who
a few minutes of conversation suffices? It's often nice to work with someone
who has a lot of students, as then you have a lot of people to talk to;
however, you also have the danger of having to fight for the professors
time, or competing with them for niche jobs.
- Ask questions. Ask students doing
similar work, ask me, ask yourself. think about what you're assuming -- it
might be facts, or that you have to approach a problem in a given way. One
of the most important parts of research is asking good questions. This
should be a constant process. If you do this all the time, you'll frequently
find new problems to work on. Whenever you read a paper ask if all the
assumptions are needed. Ask if it can be extended. See if it reminds you of
anything.... Many of my best papers came from taking a result I'd heard /
seen and asking about something related which no one had thought to study.
This then made me the first one there, and while others may be stronger, if
they're not looking at the problem....
- Present your work. Write it up. Give
talks. Too many people do good work which never sees publication or never is
presented. While a lot of the fun is solving the problem, if you don't bring
your results to the community not only do you not receive credit for what
you've done, but others can't build on it.
- Have many people know of your work.
You'll almost surely need letters of recommendation. If possible, have
several people familiar with your work. This way you can get more letters;
also, you might find common ground and collaborate on something.
- Giving a colloquium
- All Williams math/stats majors need to give a
30 minute colloquium. See the advice
below for talks, but remember that there is a large spread of
background, abilities and motivation among the students (i.e., many will
never take another math class). Make sure you know your audience.
- Before you write your talk, decide what you
want the audience to take away from it. When they leave and are
asked what the talk was about, what will they say?
- Templates (for papers and talks and
posters and Mathematica) are available
here.
- Writing a paper
- Templates (for papers and talks and
posters and Mathematica) are available
here.
- Here is a good version of a trimmed paper of mine
that will give you a nice template for writing a paper:
papertemplate.tex
(you should still see the other files if you
want a longer explanation on how to use / write in TeX).
- TeX Code: If you want the TeX file
of any paper I've written, all you have to do is click on a link on my
papers page and change the extension from .pdf to .tex.
- Biggest TeX Mistake: You should
NEVER hard-code a label. Thus never write "Theorem 1.1" or "Equation
(2.4)", as those numbers could change if you add another theorem or equation
earlier in the paper. All labels should be dynamic and done using \label.
This is one of the worse possible
TeX mistakes you can make. If you
hard-code things and items are added it becomes an enormous time sink trying
to fix things.
- Words to try to avoid when writing:
Watch out about using "will" (I have to fight against overusing it!), "in
fact", "I" (use we).
- Have all equations labeled. I know a
lot of people don't label most equations in their paper. The problem with that is that, even if an equation is not important in the paper, it may be
important for someone else, and it's MUCH easier to refer to it if it's
labeled. Thus, label all equations.
- Biggest TeX Mistake: You should
NEVER hard-code a label. Thus never write "Theorem 1.1" or "Equation
(2.4)", as those numbers could change if you add another theorem or equation
earlier in the paper. All labels should be dynamic and done using \label.
This is one of the worse possible TeX mistakes you can make. If you
hard-code things and items are added it becomes an enormous time sink trying
to fix things.
- Grammar: All sentences must end with
a punctuation mark, even if it ends with an equation / math. Thus, if it
ends with a math equation, put a period or some such. I believe I have a
sentence in a paper ending "!!.", which was fun. Also do not start a
sentence with math (also don't start with "And", "But"). Sentences start
with capitals; if you have a sentence starting with $\lambda$ do you
capitalize? Better to just avoid. Write i.e., not i.e. (i.e., follow with
comma). Avoid the word 'will' when you can.
- Spacing: I find it helps to space
things out in equations. Don't write \begin{equation} x < y \end{equation}
but \begin{equation} x \ < \ y \end{equation}. By adding the extra spaces it
makes it much easier for people to read. Very important: when writing in TeX,
do not have a blank line before and after an equation unless you want extra
spacing. If the equation is part of the sentence, you don't want extra
indentation after it, which will happen if an equation is followed by a
blank lines.
- Make sure everything is aligned properly.
I have a lot of aligning commands, such as doing two or three cases. You can
get the code for that in the templates / papers.
- Shortcuts: Commands you use a lot
should be shortcutted to save time. Thus I always write \be for \begin{equation}.
One nice consequence is that if you decide to change your notation you just
have to fix it in one place.
- Finding bugs in TeX: If you are
having issues compiling, put in \end{document} and move around until you get
something that compiles, then move till it doesn't and then you've
determined the error. Better is to keep compiling every 1 to 3 slides to
find mistakes. Also change the file extension now and again so you have an
earlier version that does compile.
- The introduction (or the abstract) is the last
part of the paper written. It should be fairly complete and
include the main result. So you can refer to it later, you can have a very
short intro at first and have the theorem stated there so you can have it
labeled. Most people don't read past the title; I view it as exponential
decay (or worse!) from title to abstract to introduction to paper. The
intro should describe the history of the problem, define terms, state
the main result, and then if the paper is say 10 pages or more give a rough
outline of what is going to happen and when.
- Theorems should be complete and essentially
self-contained. If there are any assumptions state them. If you
use GRH, say so in the intro. Some people write things earlier in the paper
saying 'In the rest of the paper we always assume GRH.' While it's fine to
do that, it's still a good idea to state the assumption again in the
theorem. A lot of people dn't carefully read all of a paper, and are looking
for a result. Help them; it doesn't cost much, and it makes the theorem
self-contained. Of course, you shouldn't be defining TOO many terms in a
theorem. You can trust the reader to have read earlier in the paper to see
the notation. Sometimes people write "notation as above". When you
give a proof, if you can try to give the big idea as to WHY it's true.
- For the most part mathematicians are casual
about a bibliography, so long as it's consistent. Only twice in
my career have I had real issues with a journal about a bibliography; in
both I held my ground and they backed down. For me and most of my friends /
colleagues, the point of a bibliography is to make it easy to find the
article. You can see how I do bibliographies in any of my papers; here's the
code: \begin{thebibliography}{9999} \bibitem[ST]{ST} M. D.
Springer and W. E. Thompson, \emph{The distribution of products of
independent random variables}, SIAM J. Appl. Math. \textbf{14} (1966)
511--526. \end{thebibliography} (the 9999 in the beginning is telling the
program how much space to set aside). I like to use author last name
initials to refer to papers. What's nice about that is if you're doing say a
paper on MSTD and see a reference MO, you know it's Martin and O'Bryant.
- I like to refer to bibliography entries by
authors last names (or last initial) rather than number. I find
if I'm reading a paper and I see the reference [ILS] I know what it is, but
[12] means nothing. I don't use BibTeX; I've found there are occasional
issues when having two files so I just have the bibliography directly
included. I typically grab what I need from an earlier paper in the
sequence.
- I think it's great to have everything
hyperlinked. This paper shows you have to have your email
addresses linked, as well as internal links within the paper. Make it easy
for the reader. Make it easy for them to contact you, or if they need to
refer to earlier in the paper to find that.
- Always acknowledge funding support,
whomever you talked to, and have a full
bibliography (the last two help journals find referees).
- How much of a calculation to show?
'Elementary' calculations are often suppressed from papers, which I find
sad. You can and should include them in a longer, friendlier version that's
on the arxiv, and you can refer to that; however, if you really need a
formula you should give a reference to something in the PUBLISHED
literature, or say it's a trivial proof and sketch the key ideas.
- Biggest TeX Mistake: You should
NEVER hard-code a label. Thus never write "Theorem 1.1" or "Equation
(2.4)", as those numbers could change if you add another theorem or equation
earlier in the paper. All labels should be dynamic and done using \label.
This is one of the worse possible TeX mistakes you can make. If you
hard-code things and items are added it becomes an enormous time sink trying
to fix things.
- You want to use good names when labeling.
I like labels such as eq:equationname and lem:lemmaname, as then it's quite
clear if I mean an equation, a lemma, ....
- NEVER have a figure in a paper without giving
it a label and referring to it by the label. Don't say "As the
figure below indicates", as figures are often moved in journal typesetting.
- Including code: If you want to
include code, \begin{verbatim} ... \end{verbatim} is a good way to add
programming code.
- Write up as you go. If you wait till
the end it's overwhelming. As you get results write them up. This is what TV
shows are made for. Also, when you try to write things up carefully you
often find small mistakes / issues.
- Citation advice. From E. Kowalksi:
https://web.williams.edu/Mathematics/sjmiller/public_html/mathematical_citation_guidelines.pdf
(see also
https://www.citationgenerator.com and
https://www.citationmachine.net/resources/cite-sources/ )
- References. There are many different
styles you can use for references; I don't really care too much what is
used, but you need to be consistent. Refer to each journal the same way,
authors the same way.
- Email advice. Make sure you include
your email address on all your publication, so it's easy for people to find
you. If you have a permanent email address use that as well as your current
one.
- Date advice. No, not the useful
kind; make sure you include the date on your papers. This helps you keep
track of what version you're viewing.
- Write clearly. I heard a commercial
which is a great example of the need for good, clear writing. It was an
interesting ad about litter boxes for cats. Here's my favorite part: Your
cat deserves the perfect litter box, and so do you.
- Consistency. Especially if you have
many co-authors, make sure you are consistent. Use similar language
throughout. Refer to things the same way. Try to speak with one voice. If
you refer to mathematicians only with last names at one point don't use
first names as well later....
- Labeling. Please do not label
Definition 1, Definition 2, Theorem 1, Lemma 1, Definition 3, Lemma 2, Lemma
3, Definition 4, Theorem 2.... That makes it very hard for a reader to
quickly navigate through the paper to find what you are referencing. Label
by section and then number, and go up sequentially. So it would be possibly
Definition 1.1, Definition 1.2, Theorem 1.3, Lemma 2.1, Definition 2.2,
Lemma 2.3, Lemma 3.1, Definition 3.2, Theorem 3.3....
- Biggest TeX Mistake: You should
NEVER hard-code a label. Thus never write "Theorem 1.1" or "Equation
(2.4)", as those numbers could change if you add another theorem or equation
earlier in the paper. All labels should be dynamic and done using \label.
This is one of the worse possible
TeX mistakes you can make. If you
hard-code things and items are added it becomes an enormous time sink trying
to fix things.
- General Writing Comments:
- Avoid using 'will', state what you are doing!
- Space out LHS and RHS of equations: \ = \ (do not do this in
paragraphs).
- Avoid using \frac if you can in a paragraph.
- Use \cdots for multiplying, \dots for a list.
- Have HIGH LEVEL quick summaries of what you are doing at the start of
sections -- there is a lot of algebra / manipulations, help the reader.
- Biggest TeX Mistake: You should
NEVER hard-code a label. Thus never write "Theorem 1.1" or "Equation
(2.4)", as those numbers could change if you add another theorem or equation
earlier in the paper. All labels should be dynamic and done using \label.
This is one of the worse possible
TeX mistakes you can make. If you
hard-code things and items are added it becomes an enormous time sink trying
to fix things.
- Giving a talk / Going to
a talk / Chairing a Session
- Know your audience.
Who is going to be there? What is their background?
What do they want to get out of it? What do you want them to get out of it?
Is this a job talk, and do you want to impress them with your technical
mastery, or perhaps with your ability to convey technical information. You
should know what your objective is.
- When you are young, every talk you give is a
job talk. You don't know who will sit on
a graduate admissions committee, or an NSF funding panel, or am postdoc
hiring committee. I received a personalized request to apply for a job in
large part because a professor in that department heard me give many talks
over a year at conference after conference, and was impressed with the work
I did with my students, and how we presented it.
- Finish before time is up!
If you have 20 minutes, aim for 18. No one complains about ending early.
Don't go lower than 16 minutes.
- Equipment:
- Laser pointer: Have a good laser
pointer. It should have a
green not red light (green films better), and good ones can count down for
time (this is important as you may not be able to see the clock in
the room, if it even has one, and you don't want to be constantly looking at
your watch like you're bored). Here is one:
Logitech Professional Presenter R800, Presentation Wireless Presenter with
Laser Pointer Green (no promises the link will work when you're
reading!).
- Interfacing: If you are projecting,
make sure you can project from your computer. Do you have all needed cables?
I don't have a mac but I travel with one of their adapters as lots of times
people with such computers are not prepared. Have a flashdrive with your
slides there so you can quickly move to another computer if needed (often a
computer is already set up in the room). Have your slides on the web and
know the link (email the link or slides to you) so you can quickly grab them
that way.
- Templates (for papers and talks and
posters and Mathematica) are available
here.
- Prepare your slides well in advance.
Do not wait to the last minute as many of you are doing. Things happen. As
soon as you know you are giving a talk, START WORKING ON YOUR SLIDES. Make
finishing them a top priority, post them on your webpage. Again, things
happen. As you progress you will not typically have profs checking each
slide and making sure things are good (though I'm happy to provide this
service for a few more years). Don't wait to the last minute.
- Finding bugs in TeX: If you are
having issues compiling, put in \end{document} and move around until you get
something that compiles, then move till it doesn't and then you've
determined the error. Better is to keep compiling every 1 to 3 slides to
find mistakes. Also change the file extension now and again so you have an
earlier version that does compile.
- Make sure you have good, descriptive names.
This means have a good file name, and have good notation. People will
be trying to learn what you're doing on the fly; simple terms help.
- Start off with a template that DOES compile
-- you'll have slides from talks you've given with me. Before you change
anything really important, save what you have. Look VERY carefully to see
how things format. If equations are being cut off, make sure you fix. Make
sure all the information you want to appear on a slide does (sometimes it
gets cut off). Make sure figures are as large as possible and readable. Trim
English text. Be concise. The more verbage, the more the audience is reading
the slides and not listening to you. Be concise. Space out lines....
- Do not number equations unless you are going to
refer to that equation. Don't introduce symbols / terminology
unless you really need them. The more you do, the harder for the audience.
Remember they're getting the definition then, trying to process it and
you've already moved on. This is different than a paper -- they can't
easily go back and refer to a previous slide (note this is an advantage of a
blackboard talk, where the text is up for longer).
- Practice the talk many times. Know
how much time you have, and have entrance / exit points so you can add /
delete material as the time pressures dictate. You do not want to go over
your allotted time. If you have 20 minutes, aim for 16 to 18. Check the room
before speaking. if there isn't a visible clock, do you have a watch you can
see? My laser pointer / clicker has a timer. When I gave a 30minute talk
starting at noon, I set the timer for 34 minutes and started it at 11:55, so
I would be able to see it's working BEFORE the talk started...
- Practice the talk in the room you'll give it
projecting from the computer you'll use. This isn't always
possible, but good if you can. You can position chalk and erasers where you
want. Also sometimes a pdf that works on your machine doesn't work on
another; I was recently at a talk where the pdf stopped displaying properly
partly through the talk. Fortunately the speaker had it on a flash drive; I
had her toss me the drive, quickly loaded onto my computer, and we projected
from mine.
- Have multiple backups. Email
yourself, the organizer, have on your computer / flash drive (both!) and
post of the web. Technology fails. If you have a Mac make sure you own an
adapter so you can connect. If possible have your own clicker (preferably a
green light, as that records better, and a timer that counts down in case
you cannot see a clock).
-
Don't go into too much detail. For
talks, it's very hard the less time you have. There's a strong temptation to
try to do too much. You're excited, and rightly so, about the work you've
done. That said, if you don't have too much time there's a real danger in
trying to say too much. Start by motivating the problem and explaining the
main result. If you're lucky, you have some time to briefly describe the
method of proof. Go for key ideas -- no one wants to see a lot of algebra on
the board.
-
Don't overload the slides / talk.
Building on the previous point, do not have too much information on each
slide. Use the pause command or some such to slowly reveal information and
keep people's attention focused. If you have a lot of cases to prove in the
paper, do not do all in the talk - concentrate on a representative example.
Thing BIG PICTURE.
- Have your slides in multiple places
-- on my webpage, in your inbox, on a flash drive. Things happen! Have a
clicker.
- Practice talks by standing up and, if possible,
having the slides projected. It's amazing how different it is
standing up then sitting down. Go to the room you're speaking in. See how
it's set up. Position erasers and chalk / markers in key places.
- Have a glass of water nearby --
sometimes it's nice to be able to take a quick sip to calm yourself down.
- Make sure your slides include your contact
info, and how they can email you / get the slides.
- I don't like slides that have the text in light
writing and then darken as you progress through the talk. I find
I often am straining to read the lightened text.
- Rota advises have one main point. Make sure
they know what you did. Clearly state the result, explain the
notation.
- Go for the big picture. Don't worry
about giving the proof in detail, or boring them with pages and pages of
algebra. Give them a sense of what you did and why it worked, but don't
flood them with too much algebra. SOME is good, and can impress people as to
the nature of the calculation / proof; no one, though, wants to see you
doing algebra in public. If you can do a special case do it, or maybe
concentrate on a typical term.... If you can, have good images / pictures /
movies.
- Uses pauses appropriately. Don't
have the slides TOO busy. You will practice many times and you will know it
-- don't make the mistakes I did when I started by putting every word up
there. It's good to know it's there, but it makes for a bad talk. Slides
should be TERSE. Just the key items. You'll know what to say.
- Have a good summary page. Reiterate
what you did and how you did it, why people care, .... This is your chance
to have them leave with an appreciation.
- Have a final 'thank you' page,
thanking the audience for their time and thanking the sponsors.
-
Load files / video / audio,
Oftentimes there are time delays as your talk is processed in real-time.
Download audio/video so you can play from your local computer and not have
to rely on a good net connection. Go through each slide so it is loaded in
memory; this can be important with images / video / audio.
- Advice from Others, Here are some
words of advice from some great speakrs.
- Going to a talk:
When you go to a talk, you should pay attention. Try
not to be too obvious about doing your work. Sometimes it's fine, sometimes
not; look around and see what others are doing. If you are going to do your
own work during the talk, you might want to sit in the back. In addition to
learning new material, a talk is an opportunity for you to impress the
speaker and the audience. You should try and ask good questions that will
help you and the others understand what was done. I try to begin at least
one new collaboration at every conference or campus visit I do, and
frequently it begins by asking good questions at talks of my colleagues.
Here are some questions that can be asked in almost all situations.
- Look at the main theorem -- if they haven't discussed, ask what happens
if they remove a condition. Is it still true? Is the proof harder?
- Ask about generalizations. What would be needed to extend to higher
dimensions? To a general number field? To higher degree curves? To composite
moduli?
- If the proof isn't given or discussed, ask for the main idea or an
illustrative example.
- If it isn't discussed, maybe ask about applications. Are there other
problems where these results can be used?
- Ask if quantities can be made explicit. What are the values for the
constants? What is large enough in the statements? Can the computations be
done in a reasonable time on a computer?
- Is the main result optimal? Ask if they believe a better result might
hold, or if the bound is sharp.
- Make sure you are wearing your name tag so people
know who you are!!!
- Chairing a session: Below is some
advice on chairing a session.
- Introduce the speaker. welcome audience, say names (not institution), if
unsure how to say a name ask the speaker beforehand. Don't take too much
time.
- Make sure the speaker doesn't go over (give warnings at say 5, 2 and 1
minute). If they are looking like there is no chance of ending on time you
can stand
up and move to the front to send a STRONG signal. You need to keep people to
the time allowed, as people want to move from room to room.
- You are responsible for making sure every talk has a question. If no one
from the audience has one, YOU must have one. Thus you must pay enough
attention to
ask a question. it could be "is this condition necessary", or "can you
extend this to...", or "can you give an example". You want people to think
someone
cares...
- Grad
schools / GREs / Math Research
- GREs:
- When to take: Take in October of
senior year if not earlier. This way if you have a bad score you have a
chance to take it again. I know at least 3 people who went to lower ranked
programs precisely b/c of a bad score.
- Get your timing down -- do lots of
practice tests and see how many problems you're answering. It's a timed
exam, many people are rushed, and all problems are worth the same.
- Don't do a problem just b/c you can do it
-- it's a time pressure exam; do the problems you can do correctly AND
quickly first, then go back for the rest.
- Illustrative example: Remember the goal is
not to spend time solving problems, but to eliminate all answers but one
quickly. For example, imagine you can paint a house in 2 days, and I can
paint one in 3. How long does it take us together? You could solve it (x/2 +
x/3 = 1 so x = 6/5), or you could notice that the correct answer has to be
somewhere between 2/2 and 3/2, and usually the GRE will only give one answer
in this range.
- Remember the point of the problems:
Frequently you can eliminate four wrong answers faster than you can find the
right exam. See the example above.
- Review all math: the GRE has a lot
of problems on calculus (Stokes and Greens theorems), basic group theory and
basic real analysis. While there are problems on complex analysis,
differential equations and so on, a lot of people have trouble with the
material from earlier in their college career because it's been a long time
since they've done them.
- Sending scores: Some programs don't
require a score. If your score isn't great and you have the option, don't
send it. The problem is it's often unclear for the large number of people
who score in the great middle what's a `good' score. Frequently the GREs are
used as a quick check to make sure your classes were legitimate ones; a bad
GRE score can raise warning flags, and cause programs to look at your file
with a far more critical eye.
- Explaining scores: If you get a bad
score, let at least one of your letter writers know so they can explain it
away. For example, years ago a school called me up as they were interested
in one of my students, but were worried by a somewhat low score. I told that
school the student was in the top 500 in the Putnam the previous year, and
all issues (regarding the score!) evaporated! So, get good Putnam scores and
let your writers know about it!
- Additional lesson: I had written in my
letter that this student was active in math competitions and problem solving
and had scored in the top 500 on the Putnam, but it was missed. There are so
many applications to read that people will occasionally miss things. That's
why it's good to address issues, if there are any, in multiple places to
maximize the chance someone sees it.
- Bring needed documentation:
Years ago one of my students forgot to bring his driver's
license, and was not allowed to take the exam. Check multiple times to make
sure you have everything you need (this isn't voting in elections -- here
they care about making sure you are you and will deny you entry if you don't
have a valid ID). Do you want water to drink? Do you have enough pens /
pencils?
- Resources: There are plenty of
resources online to review material. I've started a compilation of key
points / ideas / techniques from classes I've taught; [[click
here for the file of takeways from classes.]]
- Grad School:
- Advisors: How did they find out
about possible advisors and how did they choose their advisor? Is it easy to
find out about different professor's' research? What is a person like as an
advisor? Do they have long meetings with students, or quick high level
talks? What approach do you think will work well for you?
- Jobs: How are the grad students
doing on the job market? (Find out who they are through
Math Genealogy, and then
google them to see where they end up).
- Social life: Do the grad students
seem happy? Do they hang out together? What does the common room look like
in the department? Is it active? Are people playing games? Talking math? Do
people go to tea?
- Seminars: What are the seminar
series like? Are they appropriate to grad students? Are grad students
invited to dinners with speakers (and subsidized)? How many are there and in
what fields? Who are the outside speakers?
- Funding: How is the dept with
conference funding? Are students going to listen, or to present work? Have
they published any papers yet?
- Teaching: What is the teaching load?
Some schools give high loads early on; you want some but not overwhelmingly
large amounts.
- Math Research:
-
Asking me for letters of
recommendation
- Please make sure your files / part of the
application has been received *BEFORE* I send my letter. If my
letter arrives first, it's more likely to be lost than if it arrives second.
If your stuff comes in first they start a folder for you....
- Make sure you waive your right to view my
letter. If you don't, I *CANNOT* compare you to other students
who have been at the institution / program, and it makes the letter weaker.
- Please send me your CV and a statement of interest if you have it when
you ask for a letter. It's not essential as I know you well from the summer,
but sometimes I can work something in and emphasize it. If I can't write you
a strong letter, I'll say so. If you're asking someone else, it's okay to
ask if they can write you a strong letter for a given fellowship / grad
school / job.
- Please give me at least a week's notice from a
due date. That's enough so you don't have to worry about me
traveling and being without internet access, though usually a day is fine.
My record is under an hour for a rush, last minute application.
- I have drafts of letters written for many
people. I can adjust them based on where you are applying. I will
personalize the letters of rec to the grad school you are applying. I think
that sends a signal about how much I value and respect you (ie, not sending
a generic form letter to all schools, but taking the time to personalize b/c
I believe you are worth it).
- My policy is I *always* tell students *when* I
send letters and *when* I receive confirmation; it's up to you (if you wish)
to keep track. I usually get these done within a day; if a week
goes by and you haven't heard from me you should email me back.
If you read this and
find it useful, or have comments, please email me at
sjm1@williams.edu; you can email me
anonymously by using
ephsmath@gmail.com
(password is 1793williams).
Topics:
General Advice
Classes
Job Interviews
Research/Thesis
Giving a Colloquium
Writing a Paper
Talks
Grad Schools / GREs
Letters of Rec