Originally Posted by Mahratta

3X03: "Basic Complex Analysis" (the pun gets old fast) by Marsden. Boring book, I found it takes all the fun -- i.e. the visualization -- out of complex analysis, and it's not even that rigorous...

But it was better then the one they used in the 2011 course.

Has anyone taken the modelling course? (Was Math 2E03 but now Math 3MB3)... If so how was it?

Originally Posted by Biochem47

Has anyone taken the modelling course? (Was Math 2E03 but now Math 3MB3)... If so how was it?

Yes I want to know this too, and why did it become a 3rd year course? lol

Also, has anyone taken math 3U03? combinatorics? what kind of combinatorics do they focus on?

Originally Posted by Eternal Fire

Yes I want to know this too, and why did it become a 3rd year course? lol

I checked out the first few lectures of 2E03 last year. It looked like a great course (obviously more on the applied side), but very few had taken a differential equations course or a probability course (which are 2nd year courses). That's probably why it's been moved to 3rd year; the prof doesn't need to teach all of the DEs and stats, which means the course is probably much more interesting as a 3rd year course too.

Also, has anyone taken math 3U03? combinatorics? what kind of combinatorics do they focus on?

From what I've heard, it's a weakly proof-based course (how weakly depends on your prof); you won't simply learn combinatorial methods, but will also prove basic results in combinatorics (ex. Hall's theorem). In general, the 3rd year discrete math courses (combinatorics and graph theory) are supposed to be bird courses, so if you want to take it, get in while you still can; it may actually fill up.

Dr Hart is teaching it next year, so you will probably see lots of cool stuff. He's a logician, so he may introduce some bits of order / lattice theory too. It's generally hard to tell with these elective upper-year courses (i.e. everything but 3A, 3X, 3E, 3EE, 3F, 3FF with more flexibility for the last few than the first two), since it's heavily dependent on what the prof wants to teach.

3U03 - (Generalized) Pigeonhole Principle, Set Theory and Generating Functions, all with the goal of "Combinatorially" proving things.

Sample homework problem:

Ex. Suppose there are n steps, and you can climb the steps one or two steps at a time. Show that the number of different ways you can climb the steps is F_n, where F_n is the n'th Fibonacci Number.

Proof: Let G_n be the number of ways to climb n steps. We show that G_n = F_n.

The last step taken is either a single step, or double step.

Case I - Single step:

Then there are exactly G_{n-1} ways to get to the (n-1)st step, and exactly one way to 'finish' and climb the last one.

Case II - Double step:

Then there are exactly G_{n-2} ways to get to the (n-2)nd step, and exactly one way to 'finish' and climb the last two.

Hence the number of ways of climbing n steps is the sum of the two cases, since they are mutually exclusive. Namely, G_n = G_{n-1} + G_{n-2}.

It remains to show that G_0 = G_1 = 1. But this is obvious (if there are no steps, there's only one (trivial) way to climb them. If there is one step, there's exactly one possibility).

Hence G_n satisfies the fibonacci recurrence relationship, and G_n = F_n. [ ]




This type of argument is called a "combinatorial" proof...and it's the 'math muscle' that you exercise in 3U. Believe it or not, this course is helpful for any and all math related disciplines...I've even seen its proof techniques used by a Physics major.

Does anyone have any input on Math 3TP3 (Godel's incompleteness theorems)?

Also, did anyone else notice that topology, combinatorics and geometry are all offered at the exact same time? The latter two even occur in the same room so I hope they move one of them to a different time slot. I was interested in taking all three of them.

Originally Posted by oranges

Does anyone have any input on Math 3TP3 (Godel's incompleteness theorems)?

I've not taken it, but I've heard 3TP3 is a cool course. They read through Hofstadter's "Goedel, Escher, Bach" and proved the results that Hofstadter vaguely alludes to. From what some profs have told me, it's supposed to be a sort of "hook" course for logic; 4L03 is a bit intimidating, so they use 3TP3 to get people interested.

As the course title says, you'll be proving Goedel's famous incompleteness theorems; in a nutshell, every sufficiently strong formal system is either inconsistent or incomplete. This requires some pretty serious proof theory.

You'll be looking at a lot of cool stuff on the way in logic, but it varies significantly from year to year.

Two years ago, they covered a good amount of basic computability theory; Turing expanded on Goedel's results by proving (with a negative result) the decision problem, a special case of which is the famous halting problem. Chances are that you'll see some introductory model theory too, even if it's not introduced as such.

Incognito has taken this course, he can tell you more.

Also, did anyone else notice that topology, combinatorics and geometry are all offered at the exact same time? The latter two even occur in the same room so I hope they move one of them to a different time slot. I was interested in taking all three of them

They did that last year with algebra I and quantum computing (same time slot). Pretty annoying, I really wanted to take both. They should resolve at least the geometry and topology conflict, though; most people taking topology will want to take geometry too.

Originally Posted by oranges

Also, did anyone else notice that topology, combinatorics and geometry are all offered at the exact same time? The latter two even occur in the same room so I hope they move one of them to a different time slot. I was interested in taking all three of them.

Yessss, I did too. I emailed them about it hopefully they change it. I think that 3B is supposed to be first semester b/c it was first sem for the last two years I believe--but idk.

Originally Posted by Biochem47

Yessss, I did too. I emailed them about it hopefully they change it. I think that 3B is supposed to be first semester b/c it was first sem for the last two years I believe--but idk.

Yeah, that's exactly what I was thinking... That would make the most sense. But oh well, hopefully they fix it soon.

Wow, they actually fixed all of the conflicts. 3B and 3U are at different times now. And 3FF got switched to a different time slot as well.

My second term schedule looks really heavy now, though...

Originally Posted by oranges

Wow, they actually fixed all of the conflicts. 3B and 3U are at different times now. And 3FF got switched to a different time slot as well.

My second term schedule looks really heavy now, though...

Noooo now 3B is at the same time as Stats 4F .... ughhhh

Yeah, random rearrangements are so terrible. Math 4B03 was always 2nd term now 1st , Math 3B03 was 1st now 2nd. Now I'm in dilemma whether to take Math 2S03 or Math 3B03 as they both "prerequisites" for Math 4B03 . I feel like Math 2S03 is the most important but Math 3B03 seems really fun. Math 4B03 probably will use Spivak's little book but I can't imaging reading it without "hardcore" linear algebra. Any thoughts about Math 4B03?

^^Take 3B as preparation for 4B. Definitely.

2S is technically preparation for 3E (any 'hardcore' linear algebra you need, should be covered in 2R), and although you require some group theory when studying manifolds, it is a direct expansion/generalization of the concepts presented in 3B03.

In any case, I'd recommend taking 3E before 4B anyway (which trumps 2S03 in this regard). When studying such 'monsters' as the de Rham cohomology in 4B (which is an exact sequence of abelian groups), it's helpful to know what the heck an exact sequence/an abelian group is (which you won't do in 2S).

EDIT: I also want to point out that 4B is a highly theoretical, "Pure" math course. It's not "applied" or "calculus like" in any way, despite the name. The theory presented is an in-depth analysis of stokes theorem (whose 'special cases' are the stokes/greens/divergence theorems you learn in second year calculus). But there isn't a whole ton of computation involved in the course.

EDIT 2: We used Manfredo Do Carmo's book for the course, with a supplementary source (Flanders). Spivak's book has nice explanations but horrendous practice problems. You may be wise to read spivak and do carmo to help solidify the concepts. Something to consider.

Originally Posted by Differential

Yeah, random rearrangements are so terrible. Math 4B03 was always 2nd term now 1st , Math 3B03 was 1st now 2nd. Now I'm in dilemma whether to take Math 2S03 or Math 3B03 as they both "prerequisites" for Math 4B03 . I feel like Math 2S03 is the most important but Math 3B03 seems really fun. Math 4B03 probably will use Spivak's little book but I can't imaging reading it without "hardcore" linear algebra. Any thoughts about Math 4B03?

If your goal is to take 4B, then take 3B instead of 2S (I'd listen to Incognitus on this one rather than me, as his specialty is topology).

That said, 2S is an immensely useful course; the work I've been doing recently heavily involves group theory and functional analysis, and I've found that 2S has been more useful to me than either 3E or 3EE (with 3EE better than 3E).

I found 3E03 to be very basic; there's nothing in that course that one would have any trouble grasping by themselves. On the other hand, there are some very sophisticated ideas in 2S03 -- anyway, when I took it, we covered the basics of group theory, and many of the theorems we proved were special cases of group theoretic theorems, which could be easily recovered with a bit of work.

Think about it this way: 3E03 gives you a basic intro to groups, and 3EE to rings and fields. Linear algebra is actually a special case of the theory of modules, which are abelian groups defined over rings (vector spaces are "finite dimensional" -- I put this in quotes because there is no easy module-theoretic generalization of dimension -- modules over fields). This is why linear algebra still remains somewhat mysterious (from a "first-principles" perspective) even after learning about groups, rings, and fields.

Personally, I got the most out of 2S, even though I didn't realize it immediately after taking the course. After taking 2S and 3EE, I started to realize what was actually going on with linear algebra. In mathematics, the things that seem easiest -- linear algebra, calculus, etc. -- are actually rather complicated in generality.

However, this may be because the prof teaching it (Dr Boden) crammed a lot of material in; other profs may not do the same. If he teaches it again, I highly recommend taking it at some point. If you can't, then at least read through Valenza's linear algebra textbook, it will really solidify the subject for you.

Also, you won't need "hardcore" linear algebra to read baby Spivak; 2R should suffice.

Yeah, I wasn't sure about the content of 2R and 2S, so thank you for clarification. I have no idea about the content of 3B but I read 1st chapter of Do Carmo's diff geo book and it was pretty good.

So I think I will go with 3E+3B and pick up w/e is needed along the road. When I say applied, I don't mean braindead courses like 2C or Calc 3 but rather something like analysis of PDE's or mathematical physics.While they are "applied" streams, I hardly call them "calculus like" .

My background: should have working knowledge of real analysis(3A),some 4A,2C,2R, 2X/XX(folland) by the end of summer,know basic topology of R^n(3 C's, HB, BW) and if I've some time try to learn some 3E ( I like herstein).

Thanks for suggestions and I'm definitely looking forward for next year.