Should I Become a Mathematician?

[Mépris]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

I'll keep this somewhere where I can find it, for when I study linear algebra next year. I think you laid out differential equations in the same manner in another post. I will try to come to a conclusion on my own first, though.

This might be my favourite thread in this forum. It's the kind of thing that would have benefited me greatly back in high school when I started writing. We don't have very good libraries here, which made finding books a little harder. I don't think there's a very rigid sequence of books that one should read or study but there are some essential things that one should do in philosophy, history and politics and literature, if they are interested in writing. At any rate, all this is to say that I've observed that guidance, if available in high schools (I've been to three!), is usually poor, from someone who just does not care. That's why I like this thread.

Plaritotle, instead of paying for such a course, you could learn programming using Python in your free time from OCW Scholar. It's a clearly laid out course, so you shouldn't be encountering too many bumps. My intention is to do a little of this every day as from June.
(link

 

[Plaristotle]

Ah, that's some good advice, chiro. I haven't taken any yet, but I will eventually be taking at least a few courses in which R or MATLAB will be used.

Mepris, I'll definitely consider learning Python using OCW. Thanks for the link!

 

[noobilly]

I'm wondering about abstract algebra. I was doing Calc 1-3 and although I had to put in plenty of effort, it still made sense to me. But now when I look at some notes on abstract algebra it doesn't seem to be comprehensible to me at all. Am I missing some kind of prerequisite here, or have to plug some kind of hole in my mental process?

 

[homeomorphic]

I'm wondering about abstract algebra. I was doing Calc 1-3 and although I had to put in plenty of effort, it still made sense to me. But now when I look at some notes on abstract algebra it doesn't seem to be comprehensible to me at all. Am I missing some kind of prerequisite here, or have to plug some kind of hole in my mental process?

It might help to get comfortable with proofs, elsewhere. You can try naive set theory, first, or something like that. You have to get used to the abstraction of modern math.

Also, probably most math books/notes are overly formal/unmotivated/boring, so they don't convey how to think about the subject very well, especially for a beginner.

I have heard Pinter's abstract algebra book is pretty well-motivated. Another interesting one is Nathan Carter's Visual Group Theory. Symmetry by Hermann Weyl is another. Once you have groups down, you can try Galois Theory, by Ian Stewart (he starts with subfields and subrings of the complex numbers which are a very good motivating example for the general case and arise naturally in the context of Galois theory). The more formal books have their merits, despite being very inadequate in some respects.

 

[noobilly]

Thanks, I think you hit the nail there - I'm not very great at grasping abstract concepts. For calculus i usually have to work through a few concrete examples to get the mechanics down, then think of an intuitive explanation, before I can look back at the proof and really "get it". So this learning style won't work as I move up in math?

 

[homeomorphic]

Thanks, I think you hit the nail there - I'm not very great at grasping abstract concepts. For calculus i usually have to work through a few concrete examples to get the mechanics down, then think of an intuitive explanation, before I can look back at the proof and really "get it". So this learning style won't work as I move up in math?

Maybe something like that could work sometimes, but it will need adjustment. Its different. The end goal in calculus is usually to calculate things. With abstract algebra, the end goal is to prove things. So, you can't start with calculations.

I never really learned that way, so it's hard for me to say how it could be adjusted. I always wanted to understand everything first, and then use it, rather than the other way around. In undergrad, I usually just liked to read and convert all the proofs into pictures that I could visualize (or sometimes just moving symbols around in my mind's eye for algebra--but there are many pictures that help, too, in algebra). If the proof was too tough, I would realize it halfway through this process, give up and just try to understand the statement and how to use it, postponing an understanding of why it was true. Then once I made everything obvious enough for myself, I would tackle the problems. I did pretty well, but I don't know if that's the "right" way to do it or not.

A principle that I came up with long ago was this: If you think about anything long enough, it will eventually make sense. So, that's what I did. Just think until it made sense. Every time. In more advanced math, sometimes, you have to think a long time before it clicks.

 

[Group_Complex]

homeomorphic, did you take differential geometry as an Undergraduate? If so, can you recommend some good texts?

 

[homeomorphic]

I guess I was a 1st year grad student when I took differential geometry. I don't know that I would recommend the book we used. I'm not sure which book I would recommend for that. Visual Complex Analysis has a good section on it (actually, the author, Tristan Needham is rumored to be working on a differential geometry book that I'm sure will be mind-blowing), but it doesn't get into much detail. Also, there's a very similar discussion in Geometry and the Imagination. Probably do Carmo's book is okay for curves and surfaces, but I don't like his Riemannian Geometry book, which is sort of the sequel.

Some of these things, I just sort of learned from sources all over the place, and I never really bothered to track down the best book out there. Too little time.

 

[Group_Complex]

I guess I was a 1st year grad student when I took differential geometry. I don't know that I would recommend the book we used. I'm not sure which book I would recommend for that. Visual Complex Analysis has a good section on it (actually, the author, Tristan Needham is rumored to be working on a differential geometry book that I'm sure will be mind-blowing), but it doesn't get into much detail. Also, there's a very similar discussion in Geometry and the Imagination. Probably do Carmo's book is okay for curves and surfaces, but I don't like his Riemannian Geometry book, which is sort of the sequel.

Some of these things, I just sort of learned from sources all over the place, and I never really bothered to track down the best book out there. Too little time.

Why did you end up going into topology rather than differential geometry? It seems from my own humble studies of these subjects that whilst there is visualisation involved in Topology, it seems to be of a higher degree in Differential Geometry.

 

[dkotschessaa]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

This has stuck in my head since you posted it. (You always do that, mathwonk!)

One particular sticking point in the calculus sequence was a large chapter on series and sequences, divergence, convergence, etc. It was probably one of the most difficult subjects. But it also seems to be where some of the really interesting mathematics is, and where you can study cantor sets and such.

I was thinking of spending a good deal of time on my own reviewing and researching in this area, maybe even putting together a guide for undergraduates that I want to bring to the tutoring center next semester.

I was thinking along the same lines. If I understand that topic very well I will understand calculus in general much better - much of which can be understood in terms of riemann sums (even the definition of an in integral, which we cover but don't really explore).

Do you think this would be a good area to delve into WRT to what you just said above? I'll be done with the main calculus sequence this semester.

-Dave K

edited to add: Perhaps the ultimate aim would be to really understand Taylor/Maclaurin series.

 

[homeomorphic]

Why did you end up going into topology rather than differential geometry? It seems from my own humble studies of these subjects that whilst there is visualisation involved in Topology, it seems to be of a higher degree in Differential Geometry.

Actually, topology seems more visual to me. I do geometric topology, so topology of manifolds. Specifically, low-dimensional manifolds, in my case, and more on the visual side of it. It's not just a question of topology or geometry--some topologists are more visual than others, and the same goes for geometry. A lot of geometers are really into ugly calculations. You can't judge from what it's like just starting out in the subject.

I do enjoy hyperbolic 3-manifolds, though, which involves a bit of geometry. And by the way, Thurston's book on that subject is a good place to get started on geometry, once you have the prerequisites for it.

 

[mathwonk]

dk that sounds great. convergence is fascinating. here's a little problem that got my attention when i taught honors calc; suppose you have a riemann integrable function f on the interval [a,b], and you integrate it from a to x, to get an other function g. how do you characterize g? well if f is continuous, then g is the unique differentiable function with derivative equal to f and with g(a) = 0.

But what if f is only integrable? Then it turns out that actually f must be continuous except on a set of measure zero, and that g is differentiable with derivative equal to f wherever f was continuous. Moreover g is not just continuous but Lipschitz continuous on [a,b]. I fact g is the only Lipschitz continuous function on [a,b] with g(a) = 0, and with g'(x) = f(x) at every x where f is continuous.

But it is NOT enough to just assume g is continuous with those other properties, you have to assume the stronger Lipschitz continuity.

I.e. there can exist a continuous g, with derivative equal to f at every x where f has a derivative, and g(a) = 0, and yet g is not the integral of f. Such examples are constructed using Cantor functions.

I.e. we can have a Cantor function g that is continuous everywhere and differentiable with g'(x) = 0 except on a closed set S of measure zero, and g(a) = 0, and yet g(b) = 1, Then g is not the integral of the function f which = 0 except on S where it equals 1, since that integral is identically zero.

The point is: how do you generalize the mean value theorem to cover a function g that only has derivative zero off a set of measure zero? How much more do you need to assume so that g is a constant? Ordinary continuity won't do.

 

[dkotschessaa]

I'll have to look into that mathwonk. It seems that I'm right about this being a central topic with a lot of fascinating ideas. Of course this is also where we get introduced to Euler's formula and some derivations of pi. Lots of great stuff there.

Final stretch. This coming week is my last week of the semester before finals. Though at this point I feel I can no longer measure in "how much time" (which is not much) but in terms of "how much stuff do I have to do/read/study." (Which is a lot).

-DaveK

 

[battousai]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

 

[Group_Complex]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

You must also be able to do most of the problems from a good text on abstract algebra. Simply watching videos will not be enough.

 

[battousai]

You must also be able to do most of the problems from a good text on abstract algebra. Simply watching videos will not be enough.

of course. that's what i meant by "successfully complete".

 

[dkotschessaa]

Since I'm still in undergraduate until at least Fall 2014 and not getting any younger, I'm starting to contemplate whether I should apply directly for a Phd. program.

Disadvantages: The time commitment. It's not over until it's over, or until (as I just read an old thread here, mathwonk says "until the fat guy says it is.") It'll also be an extraordinary jump in the kind of stuff I'm doing - but I think I'm mature enough to make that jump.

However if after 4 years, for some reason I can't finish, I won't have much to show for it.

Advantages: may save some time over FIRST doing a masters THEN a Phd. It might actually be funded rather than me having to pay (if I understand correctly), even if not very much, which is still more than what I get now, which is "zero minus tuition."

What am I missing, or where am I wrong?

 

[mathwonk]

the only thing wrong with going to school is burnout. (except for poverty). So if you are well motivated right now, it makes sense to me to go straight for PhD, assuming you are prepared for that. But noticing again that you are not yet a senior, that decision should probably be made a little later, when you know more how much love you still possesses for "the life".

 

[mathwonk]

by the way, in line of fascinating stuff about convergence, you might take a look at "Counterexamples in Analysis" by Gelbaum and Olmstead. Incredible kooky examples in there. Lots of fun.

 

[dkotschessaa]

Yes, I'm planning way ahead right now. Basically information gathering.

Semester is wrapping up. Going to look at some of this stuff we've been talking about over the summer, and hopefully do some tutoring as well.

-DaveK

 

[nickadams]

Do you guys think "working memory" is what determines ones' math ability? Working memory is defined as "a brain system that provides temporary storage and manipulation of the information necessary for such complex cognitive tasks as language comprehension, learning, and reasoning."

http://www.ncbi.nlm.nih.gov/pubmed/1736359

http://www.nytimes.com/2012/04/22/magazine/can-you-make-yourself-smarter.html?pagewanted=allMy working memory is very poor and I was wondering if it would be worthwhile to try to improve it? Or will continuing to do math improve it do you think?

Thanks

 

[chiro]

Do you guys think "working memory" is what determines ones' math ability? Working memory is defined as "a brain system that provides temporary storage and manipulation of the information necessary for such complex cognitive tasks as language comprehension, learning, and reasoning."

http://www.ncbi.nlm.nih.gov/pubmed/1736359

http://www.nytimes.com/2012/04/22/magazine/can-you-make-yourself-smarter.html?pagewanted=all

My working memory is very poor and I was wondering if it would be worthwhile to try to improve it? Or will continuing to do math improve it do you think?

Thanks

I don't know about working memory, but I would say that if you push yourself to as far as you can go personally, then you will probably be very surprised how far you actually get.

The thing about learning and memory per se is that there is no real consensus on both in terms of how they work, why they work and so on.

Sure there are little bits of insight here and there, but the thing is that it's not something that is easy to generalize in a simple way as of yet and if there was (especially for learning), and it was known then teachers and pretty much everyone in general wouldn't be arguing and debating and the process of learning would be very much streamlined.

I know that there are things like the IQ workouts and so on, but really if you want to develop a skill you got to work at it period and for mathematics this meanings thinking about mathematics, reading mathematics, doing mathematics, talking to other people about mathematics and basically expending time and energy in some way on things related to a particular focus of mathematics.

But even then, the thing is also that if you isolate yourself too much on what you 'think' mathematics is vs what mathematics actually is in all its unbounded context, then I personally think you will be missing a large part of the picture.

When you see the entire world through your mathematical lense I gaurantee you will see things that you won't see in greek letter equations in a textbook or formal proofs. It's important to realize this because it's amazing how much is out there and if you spend all your time trying to look for the answers only in one place, then you will probably be missing out on a lot.

Also with regard to comprehension, if you want to improve that then comprehend. One recommendation I have is to answer questions that people ask in the forums: this is a great way to improve comprehension of a subject.

With language, my best suggestion is to read (and read widely) as well as to write. Anything that forces you to organize, plan, and execute your thoughts for different audiences will help you immensely in this regard. Don't just read stuff by the same author or in the same style: read things with many styles and many themes. Listen to a wide range of people who organize and portray their thoughts differently. Force yourself to take the time to purposely have to comprehend something specifically for that person.

As for reasoning, again pay attention to how people reason and not just one group of people. Look at how layman reason, how mathematicians/statisticians reason, how lawyers reason, and how people who have been doing something for many many years reason about things that they have been involved with for a long time.

You can get some good guidelines from mathematics, statistics, logic and philosophy, but remember that if you want some good advice and good reasoning about something, ask someone who has been doing it for a while and is actively engaged in something. The thing is that an expert will be able to see what's really relevant and even if you had good reasoning skills, reasoning on assumptions that are either invalid or completely unknown to yourself is not much use. Also be aware of uncertainty and it's role in reasoning and how you treat reasoning.

 

[battousai]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

bump

 

[dkotschessaa]

Do you guys think "working memory" is what determines ones' math ability? Working memory is defined as "a brain system that provides temporary storage and manipulation of the information necessary for such complex cognitive tasks as language comprehension, learning, and reasoning."

http://www.ncbi.nlm.nih.gov/pubmed/1736359

http://www.nytimes.com/2012/04/22/magazine/can-you-make-yourself-smarter.html?pagewanted=all


My working memory is very poor and I was wondering if it would be worthwhile to try to improve it? Or will continuing to do math improve it do you think?

Thanks

I keep running into mathematicians that say they have a terrible memory, and these are also the "where did I put my keys" people. That also happens to be working memory.

I think you can find ways to compensate. In my case, I WRITE ABSOLUTELY EVERYTHING. If an equation goes from (-4^2 + 9) my next step is not (25) but (16+9), and THEN (25) I can't do stuff in my head and hold numbers there at the same time. As a result, my work is very easy to follow and my professors appreciate this.

-DaveK

 

[dkotschessaa]

by the way, in line of fascinating stuff about convergence, you might take a look at "Counterexamples in Analysis" by Gelbaum and Olmstead. Incredible kooky examples in there. Lots of fun.

Thanks. Summer fun. :)

 

[dkotschessaa]

I don't know about working memory, but I would say that if you push yourself to as far as you can go personally, then you will probably be very surprised how far you actually get.

The thing about learning and memory per se is that there is no real consensus on both in terms of how they work, why they work and so on.

Sure there are little bits of insight here and there, but the thing is that it's not something that is easy to generalize in a simple way as of yet and if there was (especially for learning), and it was known then teachers and pretty much everyone in general wouldn't be arguing and debating and the process of learning would be very much streamlined.

I know that there are things like the IQ workouts and so on, but really if you want to develop a skill you got to work at it period and for mathematics this meanings thinking about mathematics, reading mathematics, doing mathematics, talking to other people about mathematics and basically expending time and energy in some way on things related to a particular focus of mathematics.

I wish someone would do a study on this, but I swear that just doing mathematics trumps all these other "brain booster" on the market, which may just be mathematics in disguise. I could even see "math therapy," though people would no doubt be terrified of it.

They've done studies that show that learning a language does this.

-DaveK

 

[Jimmy84]

Greetings I am attempting to self teach myself a major in math, my major is in physics. I'm doing analysis right now and after having a bit of a hard time with Rudin's definitions n theorems I'm starting to study the book of Apostol. I think Rudin is not the best when it comes to self teaching analysis or an introduction to it. I was wondering if anyone could recommend me a book on complex analysis and about functional analysis as well. I heard Kreyszig is good but long so perhaps it is a lot for me.

Im also interested in reading about operator algebras I was wondering what is the mathematical background needed for that?

I do enjoy hyperbolic 3-manifolds, though, which involves a bit of geometry. And by the way, Thurston's book on that subject is a good place to get started on geometry, once you have the prerequisites for it.

Furthermore I'm interested in hyperbolic 3 manifolds, I'm curious what are the perquisites to start reading Thurston's book? and does hyperbolic 3 manifolds have applications in physics?

Thanks in advance.

 

[MathematicalPhysicist]

I keep running into mathematicians that say they have a terrible memory, and these are also the "where did I put my keys" people. That also happens to be working memory.

I think you can find ways to compensate. In my case, I WRITE ABSOLUTELY EVERYTHING. If an equation goes from (-4^2 + 9) my next step is not (25) but (16+9), and THEN (25) I can't do stuff in my head and hold numbers there at the same time. As a result, my work is very easy to follow and my professors appreciate this.

-DaveK

The more you practice maths, the more you remember the stuff you're doing and can apply it to other questions. For example you did remember that 4*4=16 and 3*3=9, I know it looks trivial, but I heard people saying that they don't remember how much is 8*6 (or was it 8*7).

So it all depends on how much you're acquainted to something and practice it, in the end you remember it.

 

[homeomorphic]

Greetings I am attempting to self teach myself a major in math, my major is in physics. I'm doing analysis right now and after having a bit of a hard time with Rudin's definitions n theorems I'm starting to study the book of Apostol. I think Rudin is not the best when it comes to self teaching analysis or an introduction to it. I was wondering if anyone could recommend me a book on complex analysis and about functional analysis as well. I heard Kreyszig is good but long so perhaps it is a lot for me.

For complex, there's Visual Complex Analysis, and for functional and a lot of other topics, maybe Robert Geroch's Mathematical Physics. You can skip towards the end where he covers measure theory and then functional analysis. He gets right to the point and doesn't delve that deep into the subject, but it's also pretty intuitive.

Im also interested in reading about operator algebras I was wondering what is the mathematical background needed for that?

Probably just a course in functional analysis, but I don't know that much about it.

Originally Posted by homeomorphic View Post I do enjoy hyperbolic 3-manifolds, though, which involves a bit of geometry. And by the way, Thurston's book on that subject is a good place to get started on geometry, once you have the prerequisites for it.

Furthermore I'm interested in hyperbolic 3 manifolds, I'm curious what are the perquisites to start reading Thurston's book? and does hyperbolic 3 manifolds have applications in physics?

Thanks in advance.

Maybe just point set topology, but I'm not sure. He (and co-author, Levy) doesn't assume that much. Probably covering spaces, too.

 

[Monocles]

I was wondering if anyone could recommend me a book on complex analysis and about functional analysis as well. I heard Kreyszig is good but long so perhaps it is a lot for me.

Im also interested in reading about operator algebras I was wondering what is the mathematical background needed for that?

For functional analysis, there are two main topics: geometry/topology of infinite dimensional vector spaces and the properties of operators on those spaces. For the former I really like A Course in Functional Analysis by Conway, and for the latter I really like Theory of Linear Operators in Hilbert Space by Akhiezer and Glazman. Another great reference is the series of books by Reed and Simon.

For operator algebras, there is a very exhaustive series of three books by Takesaki called Theory of Operator Algebras. Since you mentioned doing physics as well, you would probably like Operator Algebras and Quantum Statistical Mechanics I and II by Bratteli and Robinson. The first volume is almost entirely about C*- and von Neumann algebras, with the second one focusing on their applications to quantum statistical mechanics. Some books on noncommutative geometry have a good section on operator algebras as well (as well as A Course on Functional Analysis mentioned above).

Some functional analysis is needed to start learning about operator algebras, but nowhere near an entire course. If you are comfortable with undergraduate level real analysis, algebra, and topology then you could start learning about operator algebras and fill in the functional analysis knowledge as you go. If you know some quantum mechanics then that is enough functional analysis to get started.

 

[dkotschessaa]

My favorite math professor offered to do some work with me and one other student in Summer B. He'll be teaching a pre-calc class, but he will be spending a lot of time doing nothing for the other hours he's there. So we're going to pick a book (one of those undergraduate texts in mathematics) and go over it about an hour a week. I'm super excited about this, and super privileged. Wowee wow. Can't let him down.

 

[Mépris]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

---

Does anyone have experience with the math departments at these colleges:
- Berea College
- Carleton College
- Reed College
- UChicago
- Colorado College
-Grinnell College
- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

 

[dkotschessaa]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

Sorry, I guess that's not universal. It's just the second summer session. There's summer A, (six weeks) summer b(six weeks) and summer C (10 weeks, overlapping both).

- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

I'm at USF. I've heard it's selective, but then they let me in!

Certainly not cold and bleak here. Quite the opposite. Shorts and sandals weather most of the year.

As you can see from my previous post, I love our math dept. I am here really by circumstance (moved to Florida to get married) but extremely happy with USF. There are lots of opportunities here to get involved in research as well. It's one of Florida's top 3 research universities.

Where are you coming from?

-Dave K

 

[Mépris]

I can see that, yes! I wasn't certain if it was USF or another Florida college that you were at. (you might have mentioned it in another post - I thought FIT or Florida Atlantic)

What texts do they use for the calculus sequence?

---

The Analysis notes of Terence Tao look amazing. I'm on a rather long study break for coffee and tried to read the first few pages.
http://terrytao.wordpress.com/books/analysis-i/

On the course page are additional notes on logic and naive set theory.

 

[dkotschessaa]

James Stewart's Essential Calculus, Early Transcendentals.

Web component here: http://www.stewartcalculus.com/media/6_home.php

Generally a very disliked book, I have to say, at least by we mere undergrads. The book seems to be pared down from earlier editions to be more "concise," which actually makes it very hard to read if you're coming at it for the first time (which I was.) The earlier editions are much more readable. For one that is more mathematically literate than I was I think it's probably a fantastic book. I've just finished the three course calc sequence but I'll probably be digging into the book for years.Also, the online component is good but under-utilized. (In another attempt to pare down I guess he put stuff online.) People don't know it's there, so it doesn't get used.

You seem like you have more experience so you probably won't have a problem. What year are you, or are you a grad student?

-Dave K