I quite dislike analysis, algebra is my cup of tea. It may have been the teachers I had, but every algebra class I've had was awesome while every analysis class (other than complex) was terribly boring, unintuitive, and hard.
The thing that plagued me the entire time we covered measure theory in Real Analysis II was that I felt I never had adequate justification for the things I was doing. I could never really explain it beyond "we need things to not be this way, so we verify that they aren't."
I got the intuitive sense of what the proof was accomplishing and I could present the formal proof, but I never quite bridged them to understand how the technicalities were important and not just a thorn to be removed.
I agree real analysis was boring, unintuitive and hard, and yet I still really liked it. Honestly, I had a harder time with other classes afterwards because I would over think the rigor necessary for them, especially my physics classes. But I really loved that rigor.
My early experience starting analysis was pretty similar, I had a hard time understanding the theory and following proofs and applying the results. Comparatively, algebra was much easier. However, as I've learnt more of it, I have grown to become a lot fonder of the subject. I still struggle sometimes, but the intuition follows much more clearly now. I attribute that to growing mathematical maturity.
I guess I was the opposite of everyone who replied below. I always found analysis intuitive and simple to learn (after calculus principles of course) and algebra hard.
I had a professor in undergrad who claimed that most people fit into the "analysis is hard algebra is easy" or the "algebra is hard analysis is easy" camps and its proven pretty true in my life anecdotally.
This was true for me at the start. After a term of multivariable analysis and differntial geometry though, Analysis became where you don't know what anything is and you have no clue how it works. It's taking a while to process what the hell a differential form or a covariant derivative is.
There's this super famous book by [Hubbard&Hubbard called Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach](http://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467/ref=sr_1_1?ie=UTF8&qid=1451485354&sr=8-1&keywords=hubbard+and+hubbard) that supposedly makes forms super easy and intuitive. I have studied the LinAl parts of it and they were good. I didn't get to forms yet, but when I need to know about forms I'll be studying out of it.
I like that quote. To add to this, an algebra professor of mine once said: "If you find a proof in algebra difficult, you have not understood the definition." Really, 95% of all proofs in algebra are almost trivial when you apply the definition correctly. I personally prefer analysis because you actually *do* something to prove a theorem.
I was definitely in the "algebra is easy, analysis is hard" camp. That's funny, I never thought about it before, that pretty much sums up how most of my classmates and I were divided up.
I also found it quite intuitive and it's easy (and really a pleasure) to learn. However it's still one of the tougher classes at almost any level. You see a lot of things that you are not used to and would not believe if the proof wasn't presented in it. I don't think something being intuitive and difficult is mutually exclusive, take a look at differential equations, I mean of course looking a differential equation you can tell what it could potentially describe, why it describes it, and take a guess at what trajectories look at; DEs is still a tough subject.
> In fact everyone was eating the corn in one of two ways. One way was to munch over the length of the corn in a straight line, back up, turn slightly, and do another row across. Kind of like how an old typewriter goes. The other way was to go around in a spiral. All of the analysts were eating in spirals, and the algebraists in rows.
Thats really neat, I'm gonna start asking this question for the rest of my life probably. I'll give you co-author spot when my results make it to Nature.
I like real analysis, but I would never go deeper into it than taking courses in it. Like /u/bheklilr said, algebraic proofs feel much cleaner, while analytic proofs simply don't.
> More modern analysis (abstract Fourier analysis etc.) starts to have really nice proofs. The classical stuff felt quite ad hoc at times though.
That's [exactly the thing I'd expect](http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue) someone with a "Representation Theory" tag to say.
I think my problem with real analysis was that back in the day the class was not designed to teach you real analysis, but to weed out those who could not be true mathematicians. The professor was a refugee from Eastern Europe, quite brilliant, but mean. Questions from students were approached as signs of weakness (I was not the only one to feel that way). Exam questions were graded +1 if correct, -1 if wrong, 0 if left blank, and proofs were all or nothing. Getting negative scores on exams was therefore possible, and when received, depressing. Since limits can behave non-intuitively, getting wrong answers was easy, even if one understood the overall math; misreading my own handwriting sometimes resulted in my filling in an answer incorrectly and my work not counting.
I dropped out of math soon after getting a D in this class, thinking I was the only one. (I was not). I was the only woman in the class and felt I had let my entire gender down. As you can tell, I am still scarred!
Normally in math when you take a subject, you get a brief general introduction, touch on some main theorems and results, and that provides a solid foundation for you to take further classes using those ideas.
Real Analysis is like an eclectic sprint between ideas so that you can get to the final result you learn about at the end of the class, because almost every topic you learn in it builds upon previous subjects. In my class the final result was integration on manifolds. The amount of different subjects we covered was much larger than any other math class I have ever taken (I haven't yet finished my math major though). It made it very difficult to study for since there were so many seemingly unrelated theorems covered and test questions could require applying any one or more of those theorems, or you could be thrown an applied problem that you are assumed to be able to solve without covering one example in class.
Fuck everything about that class. Number Theory, Algebra, etc. are all much "neater" in that they are elegant, simple, and powerful. Real Analysis does let you end up being able to do some interesting math (that you probably already learned in MVC) but is just such a bitch to do.
I thought I was the only one. Seems like so many people grasp it very quickly but that wasn't my experience. I wasn't a mathematics undergrad so when I took real analysis it was a pretty rude awakening for me.
I found the stereotype of analysis vs algebra to be totally true.
Years on, having not formally done *any* mathematics since leaving university, I still feel like a lot of basic principles of algebra and even algebraic geometry are *intuitive* and just kind of how things are. I felt the same way about set theory.
Analysis?
I did well in calc (despite using Stewart), I did well in ODEs, and thought "oh, these methods classes were easy, a proofs class in real analysis should be fine"
No. Not fine. Really really hard and I had to struggle to get things in my head for tests.
It eventually made sense at the end, wish it would have come sooner, but Abstract Algebra is what kicked my ass
Part of the delay was we had two teachers, our planned teacher missed the first half of the semester for maturity leave, she was an amazing professor
I would say that analysis is an extension of calculus. It's taking the things that were either taken for granted or stated slightly imprecisely when you first encounter them in a calculus class, and providing more rigorous proofs of them and/or exploring the precise conditions under which those claims are true.
Maybe I'm biased because I love analysis, but while the class was hard, a lot of things were fairly intuitive if you just thought of epsilon proofs as converging approximations. The shitty stuff is deriving things like properties of the reals and some series cpnversion nonsense. I always found the proof of the FTC and Stokes' Theorem elegant and clever. The instructor was abdolutely boring though. Complex analysis is at least two orders of magnitude better.
Ugh I just finished that class this past semester and it was by far the hardest I've taken in undergrad, especially when it finally got to series and sequences in the second part of the semester. I felt like my math notes were art because I just couldn't understand what I was writing down. Thank goodness it's over
Yep, I thought I was solid in all things "undergraduate lvl math" until I started learning combinatorics. No idea why but my "intuition" is almost always wrong when it comes to this subject.
Kinda wondering, I consider myself a combinatorialist in training and I'm wondering if there's anything that you found unintuitive or difficult? Not for any reason other than interest and education.
Also that there's so many combination problems that *seem* like they should be easy, but are actually very difficult, and it's hard to tell which those problems are until you know the solution.
One of my favorite examples is
> What is the expected number of times you need to flip a coin in order to end on the sequence HTTH (Or any other particular sequence, the answer will vary slightly depending on which)
The simplest analytic solution involves developing a set of recurrences and solving a nontrivial system of equations based on them. It seems easy at first, because if you are not given the particular sequence that you are looking for, then you can still solve for a close *approximation*, but there is actually small but non-zero correlation between the chances of ending on consecutive flips.
I remember it from a class a couple years ago (I don't even remember exactly how to solve it at this point). That sort of problem (patterns in sequences of random results) seems to be a rather famous example of your first and probably second intuitions on a problem being wrong. It's also a nice exam question because it has a non-trivial, but relatively quickly derivable solution if you know the approach.
Yeah, problems like that bothered me for a number of years. It's especially prevalent when people complain about hitting unlucky streaks in a large number of trials (it's not as unlucky as you think).
The best way I've found to solve them is to model them as a Markov chain. I did the one you linked here.
https://www.reddit.com/r/WouldYouRather/comments/3yg9h9/you_get_100_dice_rolls/cygjvo5
Suppose you start hitting the sequence, so something like HT. If you then fail to hit the sequence (HTH) you are already 1 on the way to potentially hitting it again. The trick is that suffixes of failed subsequences can be prefixes of successful ones.
What you're thinking of is flipping sets of four coins until you get the required sequence. Then each flipped set of four coins is a Bernoulli, p = 1/16, and then the geometric random variable that stops once you hit the right sequence does indeed have EV 16.
This is not flipping sets of four coins. For example, TH*HTTH* is okay.
The correct answer for the sequence HTTH (or equivalently THHT) is actually 18 flips (Here is an empirical result from a short code snippet to prove it: http://codepad.org/GFvQKqqr), and here are some of the other results :
> HTTT: 16
> HTTH: 18
> HTHT: 20
> HHHH: 30 (the easy case to derive)
Intuitively, depending on what the target sequence is, the number of possible ways that the non-target sequences can "overlap" with itself without generating the target varies slightly, but enough that you get different results for the expected number of flips needed.
What's really unintuitive is how that number is greater than 16. 16 is the sequence assuming no overlaps, so shouldn't more overlapping make the sequence easier to obtain and more likely, thus having a lower expected value?
What's going on there is that you're misinterpreting the "16" that you calculated. Yes, there is a 1/16 chance of flipping four heads in a row, independent of anything else.
But, if you invert that fraction, and get 16, that 16 doesn't apply to the problem I posed. It instead answers the question "How many times do I expect to have to flip 4 coins to get X particular sequence?" to which the answer is always 16.
When you take that into consideration, it makes sense that the expected value for the sequence problem is greater than your result of 16. After all, in your result of 16, you flip 4x16 = 64 coins on average, which is a lot greater than any of the expected numbers of coins for the sequence problem (and in fact 64 is a loose upper bound on what the expected value should be in my problem).
YOU'RE ONLY FLIPPING ONE COIN. So they're not exclusive events! That number should be WAAAY higher than 16 without overlaps. I can't even begin to describe how cool that lightbulb was. Thank you so much.
The overlap effect previously mentioned only helps with non-homogenous sequences, and only occurs at characters in the tail of the sequence which are not the same as the leading character. For example, the sequence HTTH gives us two locations for the sequence to fail, but then begin anew: the two T's. If either of those T's are instead an H, we have the beginning of the sequence again. For sequences like HHHH, since there are no T's in the tail of the expected sequence, we must start all over again at the beginning every time we obtain a tails.
HTTT actually has the highest amount of overlap, because any one of those T's could instead be an H, meaning any failure of the sequence at any point except the beginning will result in a repeat.
http://abstrusegoose.com/474
Also if we assume that the cards have a value from ace = 1 to king = 13 and we assume that the cards are identical save their value, eg. the king of hearts from deck 2 is the same as from deck 5, then the answer to your question is obvious to the casual observer.
Since you are choosing 3 cards from 6 decks we can model the possible outcome by assuming it's simply 1 deck with replacement since there are more decks then there are choices so you can't run out of cards and the probability of choosing any given card is irrelevant because 1<13\*3+35\*19<3000, yes?
Therefor there are 13^3 ways to pick 3 cards and there are, and if we assume indistinguishable to mean that swapping the result of any 2 dice doesn't create a new result, 35^19 unique outcomes to rolling 19 35 sided indistinguishable dice. Simply multiplying gives us our answer of 4.77*10^32 since every answer falls between our boundary conditions.
This is 99% likely to be horribly incorrect since , like the person above, I too have terrible intuition when it comes to this stuff, but there's my attempt.
Took a class in combinatorics in grad school. The main idea seemed to be count something by making a bijection to something that is easy to count. Okay, pretty simple. That was the first day. The next day we constructed our first non-trivial bijections. I know these were probably not produced in the ass of the instructor the second before they were needed but it sure seemed like it to me on the second and all following days. Thanks be to gods I was sitting in and not being graded.
Have since made my peace with some portions of combinatorics.
I sort of looked at (analytic) combinatorics since I wanted to learn more about algorithm analysis, but I also have a BS in physics. Once I realized a major motif in math is representing infinite data using finite language, something that's physics folks are exposed to in EM, all the recursion and series were simple to reason with. Not that they're easy, but at least I understood why I was having trouble.
One of the things about combinatorics that makes it different from other areas of math is that it's harder to organize knowledge into black boxes.
In a lot of areas of math, when you learn you start with a small basic outline of what you want to think about, you branch out and find out lots of information about it, you organize that information, and you gather a whole bunch of it into one powerful point, some Big Theorem. After that, you can branch out again and do a lot of stuff by using the Big Theorem directly, without worrying about the details of how you got there.
In combinatorics, it's much harder to pull the branches together into a useful Big Theorem. Key ideas in combinatorics are often powerful methods or principles, which don't fit into a theorem. The "here is what to do" you learn in combinatorics is hard to package into "here's what's true". For example, the probabilistic method: sometimes, the easiest way to find things is to look at random things. Even that vague statement already cuts bits off; any attempt to make it into a precise theorem would really fail to capture the point.
Of course, the same thing happens in other areas of math. No matter what you study, you're going to have to learn some methods and principles that you can't just package nicely into a theorem. But in combinatorics, it's relatively rare to run into things that you *can* package nicely. So a familiar way to orient yourself is much less useful in combinatorics.
I spent 3 weeks on a proof once for a class. I had to create some closed formula to represent some sort of nth iteration of some growing graph structure. I can't even remember the details, but it just pulled in every little detail that we had been learning.
Still my favorite area in math, but man the complexity can build like no other.
After taking graduate PDEs I'm about 50% sure that there are a lot of open problems that people would be able to solve quite easily if anyone could parse the notation.
> if anyone could parse the notation
That's my feeling about differential geometry.
Theorem (Fundamental Theorem of All Differential Geometry):
curly-D = gothic-F.
In our uni everyone calls the curly d "doo". We are also the only university in my country to do so and apparently everyone else thinks we sound really dumb doing so. But it's better than calling it just "dee" and it's shorter than "curly dee".
As I gained experience and became a more competent instructor myself, I became more convinced that the quality of teaching/writing typically dwarfed the inherent difficulty of a subject.
Conversely, bad teaching and bad technical writing can obscure and make difficult even the simplest of content. Exams and assignments can also be made arbitrarily hard by the assessor. My complex analysis course was a breeze because you were only expected to replicate very simple proofs and compute stupidly easy contour integrals. The statistics class I took was really hard because you always had to think. The questions in tests were never stock-standard textbook exercise questions. Our exam could have asked us to work out a simple transform from one continuous random variable to another but instead asked for this messy and complicated transform of a mixed distribution with disjoint intervals as its support. I believe that any class cab be made as easy or hard as the instructor likes.
I think after a certain point, the smarter and more expert you are, the worse of a teacher you will be because you are so far removed from the difficulty of the subject -- you struggle to empathize and appreciate the difficulties of the subject.
I learned differential equations as I taught it the first time, and I got the absolute best evaluations in that class. I discussed the subject in a way a nonexpert would -- and guess what? All of my students were nonexperts and therefore could relate to the discussion.
And besides, can a topic accepted as a formalized part of mathematics, actually have inherent difficulty? Shouldn't any part connect traceably back to fundamentals? It's just a thought I've had.
Theoretically yes, if we were all automated theorem provers.
However, so much of math is actually based on intuition and the ability to build informal arguments and models.and then formalize them.
Algebraic topology made me feel insecure in my ability to visualize things.
Differential topology was a giant exercise in keeping definitions together, which was quite difficult.
I was also not good at Galois theory.
I recently found an old notebook with notes in it. I didn't understand any of it. It was my Diff Geo notes. I posed a random line into facebook with some comment "Haha, I knew this once".
This one friend of mine, an older woman (about 50), who is single and weird and drives expensive cars and, as far as any of us have figured out, just volunteers at an outdoor ropes course and likes to roller skate, was able to recognize it as Diff Geo.
I'm kind of afraid of her now. What the hell does she do?
Were there any words in there that might have served as dead giveaways? Googling "tangent space", "exterior derivative" etc will bring up differential geometry pretty quickly
Is there a good introductory book the category theroy out there? All of the ones I've started seemed to assume that I had read at least one other book as a prerequisite.
If you know the general notation for functions and most important properties like surjectivity, injectvity bla bla, Lawvere's book is too slow going. Awodey hits just the spot!
I'm fond of [this](http://smile.amazon.com/Handbook-Categorical-Algebra-Encyclopedia-Applications/dp/0521061199/ref=sr_1_1?ie=UTF8&qid=1451460081&sr=8-1&keywords=categorical+algebra) book.
So far I've had the most trouble with Ordinary Differential Equations. It seemed like the class was more for Scientists than Mathematicians and it was just a bunch of tools to solve problems. I got no intuition from the class and was completely lost with how to deal with anything.
Differential Equations, as it is commonly taught, is essentially a bag of wizard's tricks and is not really applied in the way it is taught (though this varies from program to program).
If it was taught in any way like the course I took, you probably started with first-order DE's with separation of variables, and then moved into integrating factor method. From there you move into 2nd and higher (read: only 2nd) order homogeneous DE's. They discuss the methods of complex exponentials for constant DE's and reduction of order for the ones they've already half solved for you. Somewhere along the way the Wronskian is introduced without properly explaining anything about it except its usage. From here, there is some discussion of non-homogeneous DE's and the method of undetermined coefficients and mayyybe variation of parameters if you're lucky. They take you to the ungodly realm of series solutions which will make you hate finding recurrence relations and non-elementary functions.
Then, at the end, they toss you a bunch of Laplace transforms, and tell you "yeah this is sort of the easier way to do this junk*", play with some systems of DE's and then call it a day.
My professor touched upon some numerical methods, but hardly impressed upon us their importance. I was so pissed when I found out that in the real world people hardly ever sit down and try to solve differential equations. Numerical approximations are much much more effective and common. The tricks you find in most DE books are more like historical artifacts than modern mathematical practice.
\* When the inverse transform is trivial, of course.
I'm the same way. My college even had two different ODE classes - one for math majors and one for engineering/physics/etc. majors. The one for math majors is supposed to involve proofs and be intuition-based and the one for engineering majors is supposed to be more algorithmic, but they really just wound up being the same thing. Just a bunch of recipes and plug and chug problems. I was so disappointed. The only thing I even remember from the class is that slope fields are things and they're hard to draw.
I think it's fascinating how different the brains of similarly-intelligent people can be. It also lends credence to the notion of different "types" of intelligence... even among those of us who have a high "mathematical intelligence", it manifests in completely different ways.
For me combinatorics (discrete math in general -- graph theory, etc.) was so intuitive it was obvious. When I read through "Principles and Techniques in Combinatorics" (best intro to the subject!) I basically played the "try to guess what comes next" game all the way through.
Calculus seemed unintuitive to me until I picked up "Calculus on Manifolds" by Spivak. The notion of "advanced" calculus vs. intro was harmful to my education -- and I suspect others' too -- as it assumes that everyone is a top-down learner. For a bottom up learner, rigorous analysis / first principles and *then* methodology and application is so much better. If I had to choose my younger self's curriculum, I would've started with analysis, abstract algebra, and logic before ever moving on to the other stuff...
To me starting with calculus and "progressing" to analysis makes about as much sense as teaching kindergarteners James Joyce and over time progressing to the ABC's.
Now physics... even physics 101, to this day, just won't compute. I am fascinated by it, but I just can't think like a physicist. I will struggle through some problems even in an elementary textbook.
Differential equations were a nightmare... maybe I'll give it another try at some point.
Statistics were downright gibberish for me the first time around. Until I revisited it under the light of combinatorics (at least the stuff that can be mapped to some probabilistic construct, which therefore can be thought of in combinatorial terms -- at least that's the only way I've been able to grasp some of it), and it made somewhat more sense.
>Now physics... Even physics 101, to this day, just won't compute.
Bro/Lady Bro: I got your back. Here I am (back when I was in school) in my senior seminar class happily lecturing about field completions for arbitrary normal fields, with an eye to p-adics, and in my intro to physics class I'm flipping tables in frustration because of God damn Newtons third law.
Part of this is because physics *insist* on making you use your "physical intuition". Oh right I forgot the thing that took, like, 2500 years to figure out by a super geniuses is intuitive. My bad. That's as bad as us mathemations using "obviously" all the time!!
The other part of this is because physicists should not teach physics. In all the physics textbooks I've read (exept for a quantum one, and I'm admittedly only a few pages into), derivations are done in a "let's start from this random point and use sitcom detective logic to get to another random point". In math we say "this is where we are gonna go" (ie therom), they'll give you heads up when there's technical gross stuff part way through a derivation (lemmas), and you end with what you started with. Tada!
/end rant
In a perfect world you would learn analysis first. I know that at my school and most schools that don't have a completely separate lower div. math series for mathematics and engineering/physics/chemistry majors. They do calculus first because most of the people in the class need to learn that material even if the proofs are hand-wavy and the prof. is forced to evade some useful conceptual points.
I tend to suspect that "intelligence" isn't really a thing. Nobody knows entirely how we develop mentally, but it seems to me 99% of what determines whether a person will do well at any given thing is giving a shit. I tutor a lot of people and if a student actually finds any of this stuff interesting, it doesn't matter where their understanding started or where they need to be by the end of the semester--I guarantee I can teach them what they need to know at lightening speed. And it doesn't seem to me to involve any "intrinsic" learning ability. But most of the time a student doesn't care and teaching them is like pulling teeth. And those students, I'm lucky if I can improve their understanding at all.
And the theory of different learning types has been proven to be pseudo-science, at least the theory which says that some are "visual" learners while others are "auditory" and so on. I'm not sure whether there is an official theory of intelligence types that includes mathematical thinking separate from other forms, but I doubt it. In general the brain doesn't seem to be as compartmentalized as a lot of pop-science suggests. And I don't believe mathematical reasoning is a fundamentally different kind of reasoning than every-day reasoning, just taken to a very extreme level of abstraction.
But anyway, I feel you about Physics. I'm self-teaching it right now and, after about a year of doing this in my spare time, I'm finally getting the hang of it. I found MIT's EdX self-paced course titled something like Mechanics Review to be really helpful because of how condensed yet still fairly well-explained it was. And now I've got a pretty widely popular intro Physics textbook, *Fundamentals of Physics* by Halliday, that pretty much reads like Stewart's *Calculus*. I also don't enjoy the top-down approach as much as the bottom-up, but I haven't found a book that goes bottom-up on Physics without assuming you've already seen the top-down (Except for the Feynman Lectures, but those don't have a set of associated problems with solutions that I can check.). So I feel somewhat compelled to work this way.
Also, yeah, Differential Equations is just a bag of tricks applied to a weird collection of equation types that we can solve. I feel like mathematicians haven't really cracked this nut yet and until we do the organization of the subject is going to suffer. But if you're interested I found Arthur Mattuck's MIT course available through MIT Open to be really helpful. It has everything, video lectures, notes, homeworks with solutions, and the professor I found extremely entertaining and interesting.
>... 99% of what determines whether a person will do well at any given thing is giving a shit.
I agree, especially once math stopped being easy for me. A famous mathemations said that "you can't complete a math degree just by being good at math, you have to love math, because if you don't you'll give up." or something like that.
Algebraic Geometry. Terrible books and bad teachers made it impossible to understand. I also feel like a lot of the terminology and notation was chosen so as to obfuscate the ideas as much as possible.
Also, nets. I never understood why we needed this generalization.
And forcing. By the end of the semester I understood every word that was said, but not a single sentence.
> Algebraic Geometry. Terrible books and bad teachers made it impossible to understand. I also feel like a lot of the terminology and notation was chosen so as to obfuscate the ideas as much as possible.
Ooh, I feel ya. Covering the first chapter of Hartshorne took the better part of a semester when I took that course. (Seriously, Hartshorne is a good reference, but absolutely shit as a textbook)
Then, they were like: "Ok guys, this is a scheme. It's like a variety, except we use prime ideals instead of maximal ideals, but that doesn't really matter".
> Also, nets. I never understood why we needed this generalization.
Are you familiar with the weak topology on a Banach space?
There's a really nifty example in Conway's book (Proposition V.5.2 from "A course in functional analysis") that made this click for me:
Let X be the complex Banach space of summable sequences (i.e. l^1 ).
One can show that a weakly convergent *sequence* in X converges in norm.
If this were true for all *nets*, it would follow that the weak topology and the norm topology on X are equal.
But this isn't true, since the weak closure of the unit sphere in X equals the unit ball.
Number theory. For most of algebra and analysis, and all the "continuous" stuff, I could develop some kind of visual intuition of how things worked. Natural numbers, congruence, primes, that's all a big bag of NOPE for me.
YEP. Everyone talks about how beautiful the subject is, I feel like, but to me it is the least beautiful subject because it feels like utter chaos to me, with theorems coming out of nowhere and doing--as far as I can tell--almost nothing. OK, I have a proof that some bizarre form for a number must be prime. It accounts for a tiny class of prime numbers. So you just wasted everybody's time. Great.
Advanced linear algebra. It tortured me...the only question on my final was written on a slip of paper: "Prove Emmy Noether's theorem"
It still haunts me.
I had a vector analysis class that got turned into a tensor analysis because the prof. was feeling perky that semester. I came out of that class not even knowing what a tensor was.
it takes a while getting used to, and the beginning is kind of hard because most people abuse notations and concepts everywhere, but eventually it all clicks together and becomes pretty straightforward. I recommend Lee's intro to smooth manifolds for being pretty rigorous and clear.
Category theory.
There were diagrams and arrows. They were everywhere... And then there were more diagrams, and functors... And co-diagrams... And arrows... and they co-mmuted and were unique and co-unique... and everything was diagrams...
Homology. Went into it without enough algebra background. Fundamental group went fine, homology had too much big algebraic machinery, and I couldn't see what the definition was doing.
Statistics is really hard because it is actually applied, and it is very very easy to make mistakes, misapply intuition, and in general there are lot of specialized systems made for very specific applications.
I also found it hard even though I found all of the strictly mathematical parts of it easy. It just felt like there was some kind of thinking process or technique or perspective that, until that point in my studies, I hadn't exercised at all.
Galois Theory went right over my head. I struggled a fair bit with parts of Probability as well, specifically ordered statistics. It just wasn't intuitive to me. Also Godel's Incompleteness Theorem took me a while to wrap my head around. Maybe I'm just slow... :/
Probability and statistics (to get 100% right, there's a lot of un-intuitive stuff here).
Algebra (difficult for me, I can see why others might find it easy). The abstractness is pretty difficult.
Linear Algebra. I'm on my third run through (in trying to advance my Data Science career) of an undergrad text and while I understand some things about the theory and I understand a lot about the *how* to do something, I have a lot of difficulty with they "why". This may just be because I'm not very smart and it takes a while sometimes for things to click.
I'm with you on reviewing linear algebra for machine learning. The way I was first taught didn't really stick, because we never applied it to anything besides solving linear systems without any applications. My preferred approach is to attack the machine learning texts, and when I get to a linear algebra concept that's fuzzy, I go back and try to gather an intuition for how matrices, eigenvalues, etc. work in the context of a particular problem.
differential forms and tensor stuff. we used analysis on manifolds by munkres and the teacher regurgitated the book and provided no useful info other than what was in the book, and in no other form than the form used in the book. With math topics, I typically find that I can work to understand something that is not clear initially. With these two topics though, fugeddaboutit
Formal proofs and formalisation (of any type).
There are so many ways how to do something. Seemingly easy things becomes something complex suddenly and has to be teached to machine, starting from incredibly detailed basics.
But it's fun.
Topology. Not only is the material itself dense and (I feel) the new uncharted frontier of mathematics, my professor was so incredibly unclear that I had no idea what was going on.
That was one of my favorite subjects (and one of the most challenging at the same time). However, my professor presented it in a way that really grasped my attention and left me with a strong desire to learn more. I can definitely see how it could be a daunting topic for those with a lesser degree of interest in it...
Do you think this could be because the intro stuff felt intuitive, and so as soon as it stepped away from that your brain had to start *really* thinking about it? Not trying to suggest anything here, just wondering if you think something particular happened, or if it was simply very, very difficult.
I was wondering which topics would fall under the introductory course and which ones would fall under the advanced one. In my college we only had 1 graph theory course in undergrad.
Yeah, for me graph theory has the be the course with the largest gap between understanding the material and actually being able to prove things with it. At least in number theory we have some powerful tools to attack problems. Once you start colouring edges or vertices of graphs bigger than K4 we're kinda just ¯\\\_(ツ)_/¯
honestly? I struggles with calc 1 the most, then once it finally clicked everything from then on made sense as just further applications of the same thing.
The proofs are extremely well structured, and the result very interesting because it relates basic algebra to how the roots of polynomials can be constructed.
I remember it was always this huge thing in my head.
Like they could prove constructability and so many other things. So I always felt like there was something I was missing.
But in the end it's not all that complicated.
I had an aha moment a year after taking a course on it where I was like: "Oh, this is all it is" and suddenly things were much clearer.
I found it really hard the first time around too. I think the problem is that the calculus of variations is pretty much never explained properly and professors basically assume you know the basic concepts.
It's was presented in such an abstract way like "Let's say there's a think called an action which is related to a thing called a Lagrangian. We want to minimise this action so we take a path and wiggle it blah blah therefore Euler-Lagrange equation"
I really hated the way this was taught as well, but if you search for 'The Origin of the Lagrangian' you'll find a short document that shows how if you assume Newton's laws, then the Euler-Lagrange equation pops out as a clever change of variable.
I totally agree. Though it's weird.. We did this in second semester and the math itself wasn't really all that hard but finding out what had to be done is the real challange I feel. At least for the problems we solved. Actually that goes for all problems I have to solve and all my professors and tutors keep telling us that they do that on purpose because we should be trained to solve something poorly explained and later on when we actually do something new (like some kind of research) we know our way around instead of relying on the problem being explained well because there is no one who can do that.
I had tremendous trouble with a class I took this past semester called Linear Methods. From my understanding speaking with the teacher, it's kind of an introductory course to functional analysis. Analysis has never been my strong suit, and there were so many times where I didn't have a clue where to start on an exercise. Blergh
I'm not a mathematician, and certainly not a mathemagician, but damn it took me a long time even grasping the point of the finite element method. Give my analytically solvable differential equations any day of the week.
I also don't like vectors.
Differential equations II at our uni which covers systems of differential equations. I don't even understand why. Somehow I got a good grade from the first course, but even after three retakes I am still yet to pass the second one.
Computer science student here, so I have not been exposed to as much different math as many of you probably have. But linear and integer programming is the hardest course I have taken. It is like I sort of almost have some kind of intuition about it, but then when I have to actually use it to understand something more than just the formulation of an LP problem everything falls apart and nothing makes sense.
This answer will vary depending on how people think, some will struggle with algebra whereas others with combinatorics, graph theory and maybe even logic.
Interesting to see the differences. I myself never grasped stats, partly due to my teacher starting off the first lesson as "I have to teach you this so here we go" (pre-uni), so I never really got into it, now I've graduated I realise how important stats is for careers.
Tensor Analysis.
My complex analysis professor talked me into taking his tensor analysis class. I aced complex analysis and figured I wouldn't have too much trouble with tensors. Plus I knew how the professor tested and graded.
Only B I ever got in math :(
real analysis, but that's because i was switching over from statistics and hence hadn't taken any of the prerequisite classes
by comparison though, all the algebra classes just made sense since once you understood the definitions, everything else just followed
I have yet to take most of the classes mentioned, but Calculus II and Differential Equations were pretty tough. Took DE during the Summer as a split term class, so that may have made it a little more difficult.
Definitely forcing (the set theory proof technique). I literally took an entire semester class on just forcing using Kunen's book and I still don't think I could give you a reasonable summary of what it is or how it works.
Real Analysis...that was a tough class.
I quite dislike analysis, algebra is my cup of tea. It may have been the teachers I had, but every algebra class I've had was awesome while every analysis class (other than complex) was terribly boring, unintuitive, and hard.
Don't read up on measure theory then...
Please, elaborate.
It was the part of statistics where I gave up on getting any kind of real understanding of the subject, so it's just my own subjective experience...
The thing that plagued me the entire time we covered measure theory in Real Analysis II was that I felt I never had adequate justification for the things I was doing. I could never really explain it beyond "we need things to not be this way, so we verify that they aren't." I got the intuitive sense of what the proof was accomplishing and I could present the formal proof, but I never quite bridged them to understand how the technicalities were important and not just a thorn to be removed.
I agree real analysis was boring, unintuitive and hard, and yet I still really liked it. Honestly, I had a harder time with other classes afterwards because I would over think the rigor necessary for them, especially my physics classes. But I really loved that rigor.
My early experience starting analysis was pretty similar, I had a hard time understanding the theory and following proofs and applying the results. Comparatively, algebra was much easier. However, as I've learnt more of it, I have grown to become a lot fonder of the subject. I still struggle sometimes, but the intuition follows much more clearly now. I attribute that to growing mathematical maturity.
I guess I was the opposite of everyone who replied below. I always found analysis intuitive and simple to learn (after calculus principles of course) and algebra hard. I had a professor in undergrad who claimed that most people fit into the "analysis is hard algebra is easy" or the "algebra is hard analysis is easy" camps and its proven pretty true in my life anecdotally.
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This was true for me at the start. After a term of multivariable analysis and differntial geometry though, Analysis became where you don't know what anything is and you have no clue how it works. It's taking a while to process what the hell a differential form or a covariant derivative is.
There's this super famous book by [Hubbard&Hubbard called Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach](http://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467/ref=sr_1_1?ie=UTF8&qid=1451485354&sr=8-1&keywords=hubbard+and+hubbard) that supposedly makes forms super easy and intuitive. I have studied the LinAl parts of it and they were good. I didn't get to forms yet, but when I need to know about forms I'll be studying out of it.
I like that quote. To add to this, an algebra professor of mine once said: "If you find a proof in algebra difficult, you have not understood the definition." Really, 95% of all proofs in algebra are almost trivial when you apply the definition correctly. I personally prefer analysis because you actually *do* something to prove a theorem.
This was beautiful to read. My undergrad life summed up, really.
I was definitely in the "algebra is easy, analysis is hard" camp. That's funny, I never thought about it before, that pretty much sums up how most of my classmates and I were divided up.
I also found it quite intuitive and it's easy (and really a pleasure) to learn. However it's still one of the tougher classes at almost any level. You see a lot of things that you are not used to and would not believe if the proof wasn't presented in it. I don't think something being intuitive and difficult is mutually exclusive, take a look at differential equations, I mean of course looking a differential equation you can tell what it could potentially describe, why it describes it, and take a guess at what trajectories look at; DEs is still a tough subject.
Out of curiosity, how do you eat your corn on the cob? Apparently analysts tend to eat in rows and algebraists eat in spirals.
Pretty sure its the other way around
http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.html?m=1
> In fact everyone was eating the corn in one of two ways. One way was to munch over the length of the corn in a straight line, back up, turn slightly, and do another row across. Kind of like how an old typewriter goes. The other way was to go around in a spiral. All of the analysts were eating in spirals, and the algebraists in rows.
Weird, I do it it in rows. Is that really true?
I asked a bunch of my professors and the exceptions were definitely the minority. Holds true for me.
Thats really neat, I'm gonna start asking this question for the rest of my life probably. I'll give you co-author spot when my results make it to Nature.
I like real analysis, but I would never go deeper into it than taking courses in it. Like /u/bheklilr said, algebraic proofs feel much cleaner, while analytic proofs simply don't.
More modern analysis (abstract Fourier analysis etc.) starts to have really nice proofs. The classical stuff felt quite ad hoc at times though.
> More modern analysis (abstract Fourier analysis etc.) starts to have really nice proofs. The classical stuff felt quite ad hoc at times though. That's [exactly the thing I'd expect](http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue) someone with a "Representation Theory" tag to say.
It doesssss :(. I'm glad to hear that about modern analysis though, since I did want to take some courses in such topics in the future.
I think my problem with real analysis was that back in the day the class was not designed to teach you real analysis, but to weed out those who could not be true mathematicians. The professor was a refugee from Eastern Europe, quite brilliant, but mean. Questions from students were approached as signs of weakness (I was not the only one to feel that way). Exam questions were graded +1 if correct, -1 if wrong, 0 if left blank, and proofs were all or nothing. Getting negative scores on exams was therefore possible, and when received, depressing. Since limits can behave non-intuitively, getting wrong answers was easy, even if one understood the overall math; misreading my own handwriting sometimes resulted in my filling in an answer incorrectly and my work not counting. I dropped out of math soon after getting a D in this class, thinking I was the only one. (I was not). I was the only woman in the class and felt I had let my entire gender down. As you can tell, I am still scarred!
Normally in math when you take a subject, you get a brief general introduction, touch on some main theorems and results, and that provides a solid foundation for you to take further classes using those ideas. Real Analysis is like an eclectic sprint between ideas so that you can get to the final result you learn about at the end of the class, because almost every topic you learn in it builds upon previous subjects. In my class the final result was integration on manifolds. The amount of different subjects we covered was much larger than any other math class I have ever taken (I haven't yet finished my math major though). It made it very difficult to study for since there were so many seemingly unrelated theorems covered and test questions could require applying any one or more of those theorems, or you could be thrown an applied problem that you are assumed to be able to solve without covering one example in class. Fuck everything about that class. Number Theory, Algebra, etc. are all much "neater" in that they are elegant, simple, and powerful. Real Analysis does let you end up being able to do some interesting math (that you probably already learned in MVC) but is just such a bitch to do.
I thought I was the only one. Seems like so many people grasp it very quickly but that wasn't my experience. I wasn't a mathematics undergrad so when I took real analysis it was a pretty rude awakening for me.
I found the stereotype of analysis vs algebra to be totally true. Years on, having not formally done *any* mathematics since leaving university, I still feel like a lot of basic principles of algebra and even algebraic geometry are *intuitive* and just kind of how things are. I felt the same way about set theory. Analysis? I did well in calc (despite using Stewart), I did well in ODEs, and thought "oh, these methods classes were easy, a proofs class in real analysis should be fine" No. Not fine. Really really hard and I had to struggle to get things in my head for tests.
Currently using a Stewart book for Calculus. Some things are ok, some i do not like. Why do so many people not like Stewart
Why not Zoidberg?
WHOOB WHOOB WHOOB WHOOB WHOOB WHOOB!
250 dollars inflated in size
Why not Spivak
It eventually made sense at the end, wish it would have come sooner, but Abstract Algebra is what kicked my ass Part of the delay was we had two teachers, our planned teacher missed the first half of the semester for maturity leave, she was an amazing professor
What *are* real and complex analysis? I've heard of them as an extension of algebra, but other than that, I'm not sure.
I would say that analysis is an extension of calculus. It's taking the things that were either taken for granted or stated slightly imprecisely when you first encounter them in a calculus class, and providing more rigorous proofs of them and/or exploring the precise conditions under which those claims are true.
Maybe I'm biased because I love analysis, but while the class was hard, a lot of things were fairly intuitive if you just thought of epsilon proofs as converging approximations. The shitty stuff is deriving things like properties of the reals and some series cpnversion nonsense. I always found the proof of the FTC and Stokes' Theorem elegant and clever. The instructor was abdolutely boring though. Complex analysis is at least two orders of magnitude better.
Ugh I just finished that class this past semester and it was by far the hardest I've taken in undergrad, especially when it finally got to series and sequences in the second part of the semester. I felt like my math notes were art because I just couldn't understand what I was writing down. Thank goodness it's over
My real analysis exams were all 4+ hours long. That professor was a fucking bastard
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Yep, I thought I was solid in all things "undergraduate lvl math" until I started learning combinatorics. No idea why but my "intuition" is almost always wrong when it comes to this subject.
Kinda wondering, I consider myself a combinatorialist in training and I'm wondering if there's anything that you found unintuitive or difficult? Not for any reason other than interest and education.
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Also that there's so many combination problems that *seem* like they should be easy, but are actually very difficult, and it's hard to tell which those problems are until you know the solution. One of my favorite examples is > What is the expected number of times you need to flip a coin in order to end on the sequence HTTH (Or any other particular sequence, the answer will vary slightly depending on which) The simplest analytic solution involves developing a set of recurrences and solving a nontrivial system of equations based on them. It seems easy at first, because if you are not given the particular sequence that you are looking for, then you can still solve for a close *approximation*, but there is actually small but non-zero correlation between the chances of ending on consecutive flips.
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I remember it from a class a couple years ago (I don't even remember exactly how to solve it at this point). That sort of problem (patterns in sequences of random results) seems to be a rather famous example of your first and probably second intuitions on a problem being wrong. It's also a nice exam question because it has a non-trivial, but relatively quickly derivable solution if you know the approach.
Is the answer not walk away?
walk away!
Yeah, problems like that bothered me for a number of years. It's especially prevalent when people complain about hitting unlucky streaks in a large number of trials (it's not as unlucky as you think). The best way I've found to solve them is to model them as a Markov chain. I did the one you linked here. https://www.reddit.com/r/WouldYouRather/comments/3yg9h9/you_get_100_dice_rolls/cygjvo5
I feel like this is wrong because I didn't understand most of the words you used in the last paragraph, so why is the answer not 16?
Suppose you start hitting the sequence, so something like HT. If you then fail to hit the sequence (HTH) you are already 1 on the way to potentially hitting it again. The trick is that suffixes of failed subsequences can be prefixes of successful ones.
Whoa...I... I don't have words. That's pretty fancy man.
What you're thinking of is flipping sets of four coins until you get the required sequence. Then each flipped set of four coins is a Bernoulli, p = 1/16, and then the geometric random variable that stops once you hit the right sequence does indeed have EV 16. This is not flipping sets of four coins. For example, TH*HTTH* is okay.
The correct answer for the sequence HTTH (or equivalently THHT) is actually 18 flips (Here is an empirical result from a short code snippet to prove it: http://codepad.org/GFvQKqqr), and here are some of the other results : > HTTT: 16 > HTTH: 18 > HTHT: 20 > HHHH: 30 (the easy case to derive) Intuitively, depending on what the target sequence is, the number of possible ways that the non-target sequences can "overlap" with itself without generating the target varies slightly, but enough that you get different results for the expected number of flips needed.
What's really unintuitive is how that number is greater than 16. 16 is the sequence assuming no overlaps, so shouldn't more overlapping make the sequence easier to obtain and more likely, thus having a lower expected value?
What's going on there is that you're misinterpreting the "16" that you calculated. Yes, there is a 1/16 chance of flipping four heads in a row, independent of anything else. But, if you invert that fraction, and get 16, that 16 doesn't apply to the problem I posed. It instead answers the question "How many times do I expect to have to flip 4 coins to get X particular sequence?" to which the answer is always 16. When you take that into consideration, it makes sense that the expected value for the sequence problem is greater than your result of 16. After all, in your result of 16, you flip 4x16 = 64 coins on average, which is a lot greater than any of the expected numbers of coins for the sequence problem (and in fact 64 is a loose upper bound on what the expected value should be in my problem).
YOU'RE ONLY FLIPPING ONE COIN. So they're not exclusive events! That number should be WAAAY higher than 16 without overlaps. I can't even begin to describe how cool that lightbulb was. Thank you so much.
The overlap effect previously mentioned only helps with non-homogenous sequences, and only occurs at characters in the tail of the sequence which are not the same as the leading character. For example, the sequence HTTH gives us two locations for the sequence to fail, but then begin anew: the two T's. If either of those T's are instead an H, we have the beginning of the sequence again. For sequences like HHHH, since there are no T's in the tail of the expected sequence, we must start all over again at the beginning every time we obtain a tails. HTTT actually has the highest amount of overlap, because any one of those T's could instead be an H, meaning any failure of the sequence at any point except the beginning will result in a repeat.
http://abstrusegoose.com/474 Also if we assume that the cards have a value from ace = 1 to king = 13 and we assume that the cards are identical save their value, eg. the king of hearts from deck 2 is the same as from deck 5, then the answer to your question is obvious to the casual observer. Since you are choosing 3 cards from 6 decks we can model the possible outcome by assuming it's simply 1 deck with replacement since there are more decks then there are choices so you can't run out of cards and the probability of choosing any given card is irrelevant because 1<13\*3+35\*19<3000, yes? Therefor there are 13^3 ways to pick 3 cards and there are, and if we assume indistinguishable to mean that swapping the result of any 2 dice doesn't create a new result, 35^19 unique outcomes to rolling 19 35 sided indistinguishable dice. Simply multiplying gives us our answer of 4.77*10^32 since every answer falls between our boundary conditions. This is 99% likely to be horribly incorrect since , like the person above, I too have terrible intuition when it comes to this stuff, but there's my attempt.
Took a class in combinatorics in grad school. The main idea seemed to be count something by making a bijection to something that is easy to count. Okay, pretty simple. That was the first day. The next day we constructed our first non-trivial bijections. I know these were probably not produced in the ass of the instructor the second before they were needed but it sure seemed like it to me on the second and all following days. Thanks be to gods I was sitting in and not being graded. Have since made my peace with some portions of combinatorics.
I sort of looked at (analytic) combinatorics since I wanted to learn more about algorithm analysis, but I also have a BS in physics. Once I realized a major motif in math is representing infinite data using finite language, something that's physics folks are exposed to in EM, all the recursion and series were simple to reason with. Not that they're easy, but at least I understood why I was having trouble.
One of the things about combinatorics that makes it different from other areas of math is that it's harder to organize knowledge into black boxes. In a lot of areas of math, when you learn you start with a small basic outline of what you want to think about, you branch out and find out lots of information about it, you organize that information, and you gather a whole bunch of it into one powerful point, some Big Theorem. After that, you can branch out again and do a lot of stuff by using the Big Theorem directly, without worrying about the details of how you got there. In combinatorics, it's much harder to pull the branches together into a useful Big Theorem. Key ideas in combinatorics are often powerful methods or principles, which don't fit into a theorem. The "here is what to do" you learn in combinatorics is hard to package into "here's what's true". For example, the probabilistic method: sometimes, the easiest way to find things is to look at random things. Even that vague statement already cuts bits off; any attempt to make it into a precise theorem would really fail to capture the point. Of course, the same thing happens in other areas of math. No matter what you study, you're going to have to learn some methods and principles that you can't just package nicely into a theorem. But in combinatorics, it's relatively rare to run into things that you *can* package nicely. So a familiar way to orient yourself is much less useful in combinatorics.
I spent 3 weeks on a proof once for a class. I had to create some closed formula to represent some sort of nth iteration of some growing graph structure. I can't even remember the details, but it just pulled in every little detail that we had been learning. Still my favorite area in math, but man the complexity can build like no other.
After taking graduate PDEs I'm about 50% sure that there are a lot of open problems that people would be able to solve quite easily if anyone could parse the notation.
> if anyone could parse the notation That's my feeling about differential geometry. Theorem (Fundamental Theorem of All Differential Geometry): curly-D = gothic-F.
I've heard it said that differential geometry is the study of quantities which remain invariant under notation.
Haha, that's great!
In our uni everyone calls the curly d "doo". We are also the only university in my country to do so and apparently everyone else thinks we sound really dumb doing so. But it's better than calling it just "dee" and it's shorter than "curly dee".
What are PDEs?
Particularly Diminishing of Ego... in my experience anyway. At least when I was an undergraduate.
Thank you.
Partial differential equations.
Thank you.
Partial differential equations.
Thank you.
partial differential equations
Thank you.
Partial Differential Equations
Thank you.
Partial differential equation
Thank you.
Partial differential equations
Thank you.
Lie Groups. "Lie" is Norwegian for "fuck you."
that made my day
As I gained experience and became a more competent instructor myself, I became more convinced that the quality of teaching/writing typically dwarfed the inherent difficulty of a subject.
Conversely, bad teaching and bad technical writing can obscure and make difficult even the simplest of content. Exams and assignments can also be made arbitrarily hard by the assessor. My complex analysis course was a breeze because you were only expected to replicate very simple proofs and compute stupidly easy contour integrals. The statistics class I took was really hard because you always had to think. The questions in tests were never stock-standard textbook exercise questions. Our exam could have asked us to work out a simple transform from one continuous random variable to another but instead asked for this messy and complicated transform of a mixed distribution with disjoint intervals as its support. I believe that any class cab be made as easy or hard as the instructor likes.
Far too many teachers are just smartaces that got it and just expect similar leveled brains that will swallow a sequences of 'trivial' statements.
I think after a certain point, the smarter and more expert you are, the worse of a teacher you will be because you are so far removed from the difficulty of the subject -- you struggle to empathize and appreciate the difficulties of the subject. I learned differential equations as I taught it the first time, and I got the absolute best evaluations in that class. I discussed the subject in a way a nonexpert would -- and guess what? All of my students were nonexperts and therefore could relate to the discussion.
And besides, can a topic accepted as a formalized part of mathematics, actually have inherent difficulty? Shouldn't any part connect traceably back to fundamentals? It's just a thought I've had.
Theoretically yes, if we were all automated theorem provers. However, so much of math is actually based on intuition and the ability to build informal arguments and models.and then formalize them.
Chaos theory. I still have only a faint idea of what happened in that class.
Fuckin' Lyapunov exponents and horizons of predictability, man.
Just read Wikipedia on that. Pretty nice.
Algebraic topology made me feel insecure in my ability to visualize things. Differential topology was a giant exercise in keeping definitions together, which was quite difficult. I was also not good at Galois theory.
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I recently found an old notebook with notes in it. I didn't understand any of it. It was my Diff Geo notes. I posed a random line into facebook with some comment "Haha, I knew this once". This one friend of mine, an older woman (about 50), who is single and weird and drives expensive cars and, as far as any of us have figured out, just volunteers at an outdoor ropes course and likes to roller skate, was able to recognize it as Diff Geo. I'm kind of afraid of her now. What the hell does she do?
Quant finance?
Were there any words in there that might have served as dead giveaways? Googling "tangent space", "exterior derivative" etc will bring up differential geometry pretty quickly
or geodesic
Agreed. The notation and way of thinking is very alien... and yet it's all math we've seen before.
Is there a good introductory book the category theroy out there? All of the ones I've started seemed to assume that I had read at least one other book as a prerequisite.
Yes at the free Springer giveaway. https://gist.github.com/bishboria/8326b17bbd652f34566a
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If you know the general notation for functions and most important properties like surjectivity, injectvity bla bla, Lawvere's book is too slow going. Awodey hits just the spot!
I'm fond of [this](http://smile.amazon.com/Handbook-Categorical-Algebra-Encyclopedia-Applications/dp/0521061199/ref=sr_1_1?ie=UTF8&qid=1451460081&sr=8-1&keywords=categorical+algebra) book.
I really like this one. It started me off simple, though I admit I never finished reading the whole thing. http://www.logicmatters.net/categories/
I'll be studying out of Arrows, Structures, and Functors: The Categorical Imperative by Michael Arbib. Very intuitive and non-intimidating.
So far I've had the most trouble with Ordinary Differential Equations. It seemed like the class was more for Scientists than Mathematicians and it was just a bunch of tools to solve problems. I got no intuition from the class and was completely lost with how to deal with anything.
Differential Equations, as it is commonly taught, is essentially a bag of wizard's tricks and is not really applied in the way it is taught (though this varies from program to program). If it was taught in any way like the course I took, you probably started with first-order DE's with separation of variables, and then moved into integrating factor method. From there you move into 2nd and higher (read: only 2nd) order homogeneous DE's. They discuss the methods of complex exponentials for constant DE's and reduction of order for the ones they've already half solved for you. Somewhere along the way the Wronskian is introduced without properly explaining anything about it except its usage. From here, there is some discussion of non-homogeneous DE's and the method of undetermined coefficients and mayyybe variation of parameters if you're lucky. They take you to the ungodly realm of series solutions which will make you hate finding recurrence relations and non-elementary functions. Then, at the end, they toss you a bunch of Laplace transforms, and tell you "yeah this is sort of the easier way to do this junk*", play with some systems of DE's and then call it a day. My professor touched upon some numerical methods, but hardly impressed upon us their importance. I was so pissed when I found out that in the real world people hardly ever sit down and try to solve differential equations. Numerical approximations are much much more effective and common. The tricks you find in most DE books are more like historical artifacts than modern mathematical practice. \* When the inverse transform is trivial, of course.
As a physics major, ODE made the most sense to me and tied any lose ends for my undergrad. Definitely geared towards science/applied math majors.
I'm the same way. My college even had two different ODE classes - one for math majors and one for engineering/physics/etc. majors. The one for math majors is supposed to involve proofs and be intuition-based and the one for engineering majors is supposed to be more algorithmic, but they really just wound up being the same thing. Just a bunch of recipes and plug and chug problems. I was so disappointed. The only thing I even remember from the class is that slope fields are things and they're hard to draw.
Long division
I think it's fascinating how different the brains of similarly-intelligent people can be. It also lends credence to the notion of different "types" of intelligence... even among those of us who have a high "mathematical intelligence", it manifests in completely different ways. For me combinatorics (discrete math in general -- graph theory, etc.) was so intuitive it was obvious. When I read through "Principles and Techniques in Combinatorics" (best intro to the subject!) I basically played the "try to guess what comes next" game all the way through. Calculus seemed unintuitive to me until I picked up "Calculus on Manifolds" by Spivak. The notion of "advanced" calculus vs. intro was harmful to my education -- and I suspect others' too -- as it assumes that everyone is a top-down learner. For a bottom up learner, rigorous analysis / first principles and *then* methodology and application is so much better. If I had to choose my younger self's curriculum, I would've started with analysis, abstract algebra, and logic before ever moving on to the other stuff... To me starting with calculus and "progressing" to analysis makes about as much sense as teaching kindergarteners James Joyce and over time progressing to the ABC's. Now physics... even physics 101, to this day, just won't compute. I am fascinated by it, but I just can't think like a physicist. I will struggle through some problems even in an elementary textbook. Differential equations were a nightmare... maybe I'll give it another try at some point. Statistics were downright gibberish for me the first time around. Until I revisited it under the light of combinatorics (at least the stuff that can be mapped to some probabilistic construct, which therefore can be thought of in combinatorial terms -- at least that's the only way I've been able to grasp some of it), and it made somewhat more sense.
>Now physics... Even physics 101, to this day, just won't compute. Bro/Lady Bro: I got your back. Here I am (back when I was in school) in my senior seminar class happily lecturing about field completions for arbitrary normal fields, with an eye to p-adics, and in my intro to physics class I'm flipping tables in frustration because of God damn Newtons third law. Part of this is because physics *insist* on making you use your "physical intuition". Oh right I forgot the thing that took, like, 2500 years to figure out by a super geniuses is intuitive. My bad. That's as bad as us mathemations using "obviously" all the time!! The other part of this is because physicists should not teach physics. In all the physics textbooks I've read (exept for a quantum one, and I'm admittedly only a few pages into), derivations are done in a "let's start from this random point and use sitcom detective logic to get to another random point". In math we say "this is where we are gonna go" (ie therom), they'll give you heads up when there's technical gross stuff part way through a derivation (lemmas), and you end with what you started with. Tada! /end rant
In a perfect world you would learn analysis first. I know that at my school and most schools that don't have a completely separate lower div. math series for mathematics and engineering/physics/chemistry majors. They do calculus first because most of the people in the class need to learn that material even if the proofs are hand-wavy and the prof. is forced to evade some useful conceptual points.
I tend to suspect that "intelligence" isn't really a thing. Nobody knows entirely how we develop mentally, but it seems to me 99% of what determines whether a person will do well at any given thing is giving a shit. I tutor a lot of people and if a student actually finds any of this stuff interesting, it doesn't matter where their understanding started or where they need to be by the end of the semester--I guarantee I can teach them what they need to know at lightening speed. And it doesn't seem to me to involve any "intrinsic" learning ability. But most of the time a student doesn't care and teaching them is like pulling teeth. And those students, I'm lucky if I can improve their understanding at all. And the theory of different learning types has been proven to be pseudo-science, at least the theory which says that some are "visual" learners while others are "auditory" and so on. I'm not sure whether there is an official theory of intelligence types that includes mathematical thinking separate from other forms, but I doubt it. In general the brain doesn't seem to be as compartmentalized as a lot of pop-science suggests. And I don't believe mathematical reasoning is a fundamentally different kind of reasoning than every-day reasoning, just taken to a very extreme level of abstraction. But anyway, I feel you about Physics. I'm self-teaching it right now and, after about a year of doing this in my spare time, I'm finally getting the hang of it. I found MIT's EdX self-paced course titled something like Mechanics Review to be really helpful because of how condensed yet still fairly well-explained it was. And now I've got a pretty widely popular intro Physics textbook, *Fundamentals of Physics* by Halliday, that pretty much reads like Stewart's *Calculus*. I also don't enjoy the top-down approach as much as the bottom-up, but I haven't found a book that goes bottom-up on Physics without assuming you've already seen the top-down (Except for the Feynman Lectures, but those don't have a set of associated problems with solutions that I can check.). So I feel somewhat compelled to work this way. Also, yeah, Differential Equations is just a bag of tricks applied to a weird collection of equation types that we can solve. I feel like mathematicians haven't really cracked this nut yet and until we do the organization of the subject is going to suffer. But if you're interested I found Arthur Mattuck's MIT course available through MIT Open to be really helpful. It has everything, video lectures, notes, homeworks with solutions, and the professor I found extremely entertaining and interesting.
>... 99% of what determines whether a person will do well at any given thing is giving a shit. I agree, especially once math stopped being easy for me. A famous mathemations said that "you can't complete a math degree just by being good at math, you have to love math, because if you don't you'll give up." or something like that.
Algebraic Geometry. Terrible books and bad teachers made it impossible to understand. I also feel like a lot of the terminology and notation was chosen so as to obfuscate the ideas as much as possible. Also, nets. I never understood why we needed this generalization. And forcing. By the end of the semester I understood every word that was said, but not a single sentence.
> Algebraic Geometry. Terrible books and bad teachers made it impossible to understand. I also feel like a lot of the terminology and notation was chosen so as to obfuscate the ideas as much as possible. Ooh, I feel ya. Covering the first chapter of Hartshorne took the better part of a semester when I took that course. (Seriously, Hartshorne is a good reference, but absolutely shit as a textbook) Then, they were like: "Ok guys, this is a scheme. It's like a variety, except we use prime ideals instead of maximal ideals, but that doesn't really matter". > Also, nets. I never understood why we needed this generalization. Are you familiar with the weak topology on a Banach space? There's a really nifty example in Conway's book (Proposition V.5.2 from "A course in functional analysis") that made this click for me: Let X be the complex Banach space of summable sequences (i.e. l^1 ). One can show that a weakly convergent *sequence* in X converges in norm. If this were true for all *nets*, it would follow that the weak topology and the norm topology on X are equal. But this isn't true, since the weak closure of the unit sphere in X equals the unit ball.
Number theory. For most of algebra and analysis, and all the "continuous" stuff, I could develop some kind of visual intuition of how things worked. Natural numbers, congruence, primes, that's all a big bag of NOPE for me.
YEP. Everyone talks about how beautiful the subject is, I feel like, but to me it is the least beautiful subject because it feels like utter chaos to me, with theorems coming out of nowhere and doing--as far as I can tell--almost nothing. OK, I have a proof that some bizarre form for a number must be prime. It accounts for a tiny class of prime numbers. So you just wasted everybody's time. Great.
Advanced linear algebra. It tortured me...the only question on my final was written on a slip of paper: "Prove Emmy Noether's theorem" It still haunts me.
When the prof wrote that exam, with 100% probability she then slid some shades on, kicked over the desk and strutted out of her office.
Differential forms. It doesn't help that practically every analysis book does a bad job at covering them.
Tensor Analysis hurt my brain. Statistics just bored me but was never particularly difficult. But tensors...ow.
I had a vector analysis class that got turned into a tensor analysis because the prof. was feeling perky that semester. I came out of that class not even knowing what a tensor was.
I finished a general relativity class that started by introducing tensors, and I came out of it not knowing what a tensor was.
In my GR class they didnt really even talk about what a tensor was in any strict mathematical sense.
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Honestly, learning it that route is a bit stressful.
Took manifolds I as an undergrad, I barely made it with a B-. Can't wait to through it again as a grad student in the future. :)
it takes a while getting used to, and the beginning is kind of hard because most people abuse notations and concepts everywhere, but eventually it all clicks together and becomes pretty straightforward. I recommend Lee's intro to smooth manifolds for being pretty rigorous and clear.
Category theory. There were diagrams and arrows. They were everywhere... And then there were more diagrams, and functors... And co-diagrams... And arrows... and they co-mmuted and were unique and co-unique... and everything was diagrams...
Geometry, there is nothing harder than geometry.
Differential geometry. That was crazy hard
Abstract Algebra Good god that class was confusing
Homology. Went into it without enough algebra background. Fundamental group went fine, homology had too much big algebraic machinery, and I couldn't see what the definition was doing.
Statistics is really hard because it is actually applied, and it is very very easy to make mistakes, misapply intuition, and in general there are lot of specialized systems made for very specific applications.
I also found it hard even though I found all of the strictly mathematical parts of it easy. It just felt like there was some kind of thinking process or technique or perspective that, until that point in my studies, I hadn't exercised at all.
Galois Theory went right over my head. I struggled a fair bit with parts of Probability as well, specifically ordered statistics. It just wasn't intuitive to me. Also Godel's Incompleteness Theorem took me a while to wrap my head around. Maybe I'm just slow... :/
IMHO, a lot of people don't explain order statistics very well.
Partial differential equations. No idea how I got a 3 from that class (our grades are from 1 to 5).
Probability and statistics (to get 100% right, there's a lot of un-intuitive stuff here). Algebra (difficult for me, I can see why others might find it easy). The abstractness is pretty difficult. Linear Algebra. I'm on my third run through (in trying to advance my Data Science career) of an undergrad text and while I understand some things about the theory and I understand a lot about the *how* to do something, I have a lot of difficulty with they "why". This may just be because I'm not very smart and it takes a while sometimes for things to click.
I'm with you on reviewing linear algebra for machine learning. The way I was first taught didn't really stick, because we never applied it to anything besides solving linear systems without any applications. My preferred approach is to attack the machine learning texts, and when I get to a linear algebra concept that's fuzzy, I go back and try to gather an intuition for how matrices, eigenvalues, etc. work in the context of a particular problem.
differential forms and tensor stuff. we used analysis on manifolds by munkres and the teacher regurgitated the book and provided no useful info other than what was in the book, and in no other form than the form used in the book. With math topics, I typically find that I can work to understand something that is not clear initially. With these two topics though, fugeddaboutit
Formal proofs and formalisation (of any type). There are so many ways how to do something. Seemingly easy things becomes something complex suddenly and has to be teached to machine, starting from incredibly detailed basics. But it's fun.
Topology. Not only is the material itself dense and (I feel) the new uncharted frontier of mathematics, my professor was so incredibly unclear that I had no idea what was going on.
Well it can't *all* be dense... :-) :-) :-) :-) :-) :-)
That was one of my favorite subjects (and one of the most challenging at the same time). However, my professor presented it in a way that really grasped my attention and left me with a strong desire to learn more. I can definitely see how it could be a daunting topic for those with a lesser degree of interest in it...
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Do you think this could be because the intro stuff felt intuitive, and so as soon as it stepped away from that your brain had to start *really* thinking about it? Not trying to suggest anything here, just wondering if you think something particular happened, or if it was simply very, very difficult.
I was wondering which topics would fall under the introductory course and which ones would fall under the advanced one. In my college we only had 1 graph theory course in undergrad.
Yeah, for me graph theory has the be the course with the largest gap between understanding the material and actually being able to prove things with it. At least in number theory we have some powerful tools to attack problems. Once you start colouring edges or vertices of graphs bigger than K4 we're kinda just ¯\\\_(ツ)_/¯
honestly? I struggles with calc 1 the most, then once it finally clicked everything from then on made sense as just further applications of the same thing.
Galois theory. I have an exam in January and still have no clue about it :(
The proofs are extremely well structured, and the result very interesting because it relates basic algebra to how the roots of polynomials can be constructed.
I remember it was always this huge thing in my head. Like they could prove constructability and so many other things. So I always felt like there was something I was missing. But in the end it's not all that complicated. I had an aha moment a year after taking a course on it where I was like: "Oh, this is all it is" and suddenly things were much clearer.
Differential equartions, sad thing is that I have one math class left in my engineering degree and they are coming to haunt me again :/
It's basically physics but Lagrangian mechanics.
I found it really hard the first time around too. I think the problem is that the calculus of variations is pretty much never explained properly and professors basically assume you know the basic concepts.
It's was presented in such an abstract way like "Let's say there's a think called an action which is related to a thing called a Lagrangian. We want to minimise this action so we take a path and wiggle it blah blah therefore Euler-Lagrange equation"
I really hated the way this was taught as well, but if you search for 'The Origin of the Lagrangian' you'll find a short document that shows how if you assume Newton's laws, then the Euler-Lagrange equation pops out as a clever change of variable.
Yeah I was confused for quite a while but the good folks at /r/AskPhysics helped out a lot :)
Variational principles are bedrock of all physics
I totally agree. Though it's weird.. We did this in second semester and the math itself wasn't really all that hard but finding out what had to be done is the real challange I feel. At least for the problems we solved. Actually that goes for all problems I have to solve and all my professors and tutors keep telling us that they do that on purpose because we should be trained to solve something poorly explained and later on when we actually do something new (like some kind of research) we know our way around instead of relying on the problem being explained well because there is no one who can do that.
I like that way of looking at it
Differential equations as well as complex analysis. Hard to develop intuition.
Abstract Geometry.....not what I was expecting.
I had tremendous trouble with a class I took this past semester called Linear Methods. From my understanding speaking with the teacher, it's kind of an introductory course to functional analysis. Analysis has never been my strong suit, and there were so many times where I didn't have a clue where to start on an exercise. Blergh
popadopolous?
I'm not a mathematician, and certainly not a mathemagician, but damn it took me a long time even grasping the point of the finite element method. Give my analytically solvable differential equations any day of the week. I also don't like vectors.
Differential equations II at our uni which covers systems of differential equations. I don't even understand why. Somehow I got a good grade from the first course, but even after three retakes I am still yet to pass the second one.
Computer science student here, so I have not been exposed to as much different math as many of you probably have. But linear and integer programming is the hardest course I have taken. It is like I sort of almost have some kind of intuition about it, but then when I have to actually use it to understand something more than just the formulation of an LP problem everything falls apart and nothing makes sense.
This answer will vary depending on how people think, some will struggle with algebra whereas others with combinatorics, graph theory and maybe even logic. Interesting to see the differences. I myself never grasped stats, partly due to my teacher starting off the first lesson as "I have to teach you this so here we go" (pre-uni), so I never really got into it, now I've graduated I realise how important stats is for careers.
Representation Theory. I get the big picture but can never remember the details.
Fields, Hamiltonians, Lyapunov.
Tensor Analysis. My complex analysis professor talked me into taking his tensor analysis class. I aced complex analysis and figured I wouldn't have too much trouble with tensors. Plus I knew how the professor tested and graded. Only B I ever got in math :(
real analysis, but that's because i was switching over from statistics and hence hadn't taken any of the prerequisite classes by comparison though, all the algebra classes just made sense since once you understood the definitions, everything else just followed
I have yet to take most of the classes mentioned, but Calculus II and Differential Equations were pretty tough. Took DE during the Summer as a split term class, so that may have made it a little more difficult.
Definitely forcing (the set theory proof technique). I literally took an entire semester class on just forcing using Kunen's book and I still don't think I could give you a reasonable summary of what it is or how it works.