What does a mathematician do all day?

http://ask.metafilter.com/233883/What-does-a-mathematician-do-all-day#3387868: "A lot of them are college professors, and do the same kind of thing that other professors do." I was talking with a grad student the other day in mathematics, and I was surprised to learn that professors pay other people to read papers, and call them "research assistants." As a (former) CS grad students, this idea of reading papers all day sounds like fun, but I have no idea how this translates into collaboration on a new paper. I suppose it's easier if you focus on developing new fields of math rather than the Big Unsolved Problems. Like how Claude Shannon was a pretty smart guy and all, but another factor into his seemingly incredible string of algorithms and theorems is that he was the only person mining those veins at the start.

PS: My husband asked me to come back and clarify something about http://ask.metafilter.com/233883/What-does-a-mathematician-do-all-day#3387925http://ask.metafilter.com/233883/What-does-a-mathematician-do-all-day#3387925 above. He said that the bit of his day that, as a physicist, looks different from a mathematician's is the bit where he lifts weights at the gym. He claims physicists are all big and buff and mathematicians are pale and weedy.

Wow. Thank you all so much. This is fascinating stuff and very, very helpful. I'm writing about Abraham Wald for those who were curious, and I have a mountain of information about how mathematicians helped the U.S. government and the military during World War II, which is the other focus of what I am working on. Again, this is immensely helpful, thank you for letting me peek into your lives.

Yes, escabeche! I can only turn towards the board counter-clockwise and away from it the opposite way. If I do a full turn the room looks odd to me and I get flustered. I also take what my wife calls "old man walks" around town for 2-3 hours a week to get a new perspective on problems. You'll find much of mathematics is getting stuck on something, then seeking "fresh eyes" for it. That includes talking to others (collaboration is really important), turning your head slightly, or changing your environment.

I am a pure mathematician and a tenured professor. Today we have a bunch of visitors, so I'll attend three hour-long seminar talks by profs from other places (two from U. Sydney, one from U. Illinois. Chicago) and in the morning I'm meeting with another visitor from U. Illinois Urbana-Champaign to talk about some problems at the interface with topology and number theory we're both interested in. The actual way this meeting will look is that we'll sit in a coffeeshop and drink coffee and probably at some point take out pieces of paper and write on them, and it might be that one of us will open a laptop or tablet and we'll look at other people's papers, or it might be that my colleague will show me some data he's generated because he's 1000x as good a programmer as I am (mathematicians vary widely on how much code they write; some might spent a serious chunk of their time developing software for studying mathematical entries, others (like me) write occasional short sage or python or maple programs casually, some never program at all.) I'll also read twenty or thirty Ph.D. applications and make my recommendations about whether we should admit them; continue reading a paper one of my graduate students has written and scribbling comments all over it and then try to read my comments and write him e-mail asking him to clarify certain points so that I can convince myself he's proved what he says he has proved; compile the scores our grad students got on the various parts of their algebra qualifying exam and submit to the department the list of who passed and who failed. What almost all of this will look like, from the outside, is either: * I'm sitting in a room listening to someone talk at the blackboard and asking quetsions; * I'm drinking coffee with someone and talking with them and maybe writing in paper; * I'm sitting at my laptop writing e-mail. This is the great trichotomy of math. OK, there's staring-into-space time, too, but I don't think I have that much of that blocked out today. Actually, I'm not really much of a starer into space. Every mathematician has their own physical correlative of deep thought; some stare, some rock, some go for long walks. Me, I'm a pacer. Small circles, counterclockwise.

A lot of the above is what I do so I won't repeat it. I rarely use a computer as a theoretical mathematician, outside of teaching students how to use it for their own work. I stare off into space a *lot*, to the point that my wife tells me that she knows I'm working when I'm not talking. I'm usually juggling a few problems in my head at any given time and hoping to be inspired by something that leads me to a solution. That usually happens when I'm in the shower, and hopefully doesn't happen when I'm driving (from personal experience I can tell you that doing math while driving is dangerous business). I also draw lots of examples and do calculations on the board, then stand as far back from it as I can across the room or even in the hall and turn my head a little like when my dog hears a whistle on tv. It rarely helps as much as I hope it will.

My grandfather was a mathematician. While he shaved, he would mentally factor telephone numbers. I have no idea if that is typical.

This NOVA documentary might be helpful, http://www.pbs.org/wgbh/nova/proof/: For over 350 years, some of the greatest minds of science struggled to prove what was known as Fermat's Last Theorem—the idea that a certain simple equation had no solutions. Now hear from the man who spent seven years of his life cracking the problem.

pwnguin, it sounds like you're asking why reading papers is part of the process for writing new papers? The way to make progress is sometimes to find a method or result that someone else developed, and apply it to a problem you've been working on. You find those methods and results by reading papers (which you find by searching, or seeing their titles/abstracts in the arXiv, or through word of mouth within the network of people who are working on the same problems). If you're working on a bunch of problems and you're training grad students to do math research, you might assign them to read a bunch of papers and report back if they find anything useful that could turn into a publication/dissertation for them, or be applied to your problem(s). Another reason to read papers is to find out what results are already settled, and which ones are still unsolved, so you know where to direct your energy.