Hardest upper undergraduate pure maths subjects?

[pivoxa15]

Which undergraduate pure maths subject do people find the most challenging?

I thought I might just list the three major areas, analysis, algebra, topology.

 

[NathanExplosion]

I don't have an answer, but I'm interested to see people's answers as well...

 

[math_owen]

In my classes, some how, the physicists always say the one they're in is hardest

Honestly though, a lot of factors come into play. I would say, who's taught all the courses up until those counts big time, and more importantly your interests.

For myself analysis has been hardest, and that's the exact opposite of what is common at my school. Most people as undergraduates have a harder time with their first algebra. Algebra comes quick to me - mostly because I love using that type of thinking.

 

[JasonRox]

The hardest one is probably the one in which you got the lowest grade.

I got A's in each of them, so I don't really know. I would say that Analysis felt like the hardest.

 

[daveb]

Topology felt hardest, but that's probably because I actually took classes in Analysis and Algebra (got A's) but tried to learn topology on my own (still not quite sure I've even got the fundamentals down!)

 

[cipher_42]

Quality of teaching is definitely a big factor in how hard a class seems. Teaching aside though, I think analysis was the one that clicked the least with me. Each subject has a different way that you have to approach the proofs, a different set of skills, a unique mindset. For me, getting the hang of epsilon-delta proofs and all that jazz took the longest. But then again, it was the first of those 3 that I took, so maybe I just gained more confidence after that.

 

[AsianSensationK]

Definitely analysis. It's hard to get used to that kind of thinking.

Slowly but surely, all the topics from that class are starting to make sense. Especially as I've seen them applied in other fields, like statistics.

 

[InbredDummy]

Real Analysis as in Measure Theory is probably the most difficult undergrad math course, on average.

 

[cristo]

I have to say the hardest part of my undergraduate course was a module on Differential Geometry. I'm not sure which particular category that fits into though.

 

[gravenewworld]

Logic. nothing will test your abstraction ability more than logic, hands down.

 

[pivoxa15]

I have to say the hardest part of my undergraduate course was a module on Differential Geometry. I'm not sure which particular category that fits into though.

I was thinking 3 year level maths subjects.

In my uni they teach differential geometry at honours or 4th year level. They also have algebraic topology at that level as well. Would you call the latter algebra or topology? I have heard that that course was the hardest in his studies by one academic.

As for me, analysis is hard also. I hate it when I think about it geometrically which means things get smaller and tighter at the exact places where the interesting maths is. The notion of infinity arises in analysis so you somehow have to fit that into your picture. So it's hard to picture it. I think my geometic approach may not be the best. That is why they use epsilon and delta.

I haven't done proper topology yet. From what I have seen, it is mainly set theory which can get you a bit if notions of infinity arises.

Algebra seems clearest. I feel it is the best maths subject to learn deductive reasoning. Once you have it then things become easier. Also learning the definitions seem hard at first as they may be less intuitive than definitions in analysis and topology where you can use pictures to think about things more.

 

[JasonRox]

In my uni they teach differential geometry at honours or 4th year level. They also have algebraic topology at that level as well. Would you call the latter algebra or topology? I have heard that that course was the hardest in his studies by one academic.

From what I have learned from it so far, I'd call it topology since it deals with topological issues and not algebra. Algebra is just used to solve problems.

 

[pivoxa15]

Looks like not much votes from our resident maths experts at PF. You could also intepret the question as which is the least easiest if you found them all easy.

 

[mathwonk]

i say analysis, but it may be rudins fault, since the analysis profs mostly use his book, and it is one of the least user friendly of all maths books. i.e. analysts are among the few remaining profs who take pride in presenting the material in a way that pleases themselves and not the student.

but i think pivoxa expressed my intrinsic reason perfectly about the subject getting smaller and tighter exactly where the action is. after years of practice, it seems the soloution is to stop short, accept that fact, and just be content that for every epsilon, all the rest of the action takes place inside an epsilon disc one has no need to imagine the center of.

for me topology was eaisest, algebra next, and analysis hardest. somehow algebraic topology is considered hard too, but the differential version of topology is much easier. to get into algebraic or differential topology without tears, anything by andrew wallace is recommended.

so again maybe it depends on the presentation. milnor is also great. for analysis i like dieudonne, even though he tries purposely to make it harder by drawing absolutely no pictures. but maybe his intent was to force the reader to draw them.

riesz nagy is also excellent but old fashioned. but then i have given all these recommendations in my mathematician thread, which of course is now too long for anyone to read.

 

[ircdan]

The hardest math class I took was combinatorics/graph theory.

 

[pivoxa15]

The hardest math class I took was combinatorics/graph theory.

So you took classes in all three areas in the option but still found combinatorics/graph the hardest?

 

[pivoxa15]

i say analysis, but it may be rudins fault, since the analysis profs mostly use his book, and it is one of the least user friendly of all maths books. i.e. analysts are among the few remaining profs who take pride in presenting the material in a way that pleases themselves and not the student.

but i think pivoxa expressed my intrinsic reason perfectly about the subject getting smaller and tighter exactly where the action is. after years of practice, it seems the soloution is to stop short, accept that fact, and just be content that for every epsilon, all the rest of the action takes place inside an epsilon disc one has no need to imagine the center of.

for me topology was eaisest, algebra next, and analysis hardest. somehow algebraic topology is considered hard too, but the differential version of topology is much easier. to get into algebraic or differential topology without tears, anything by andrew wallace is recommended.

so again maybe it depends on the presentation. milnor is also great. for analysis i like dieudonne, even though he tries purposely to make it harder by drawing absolutely no pictures. but maybe his intent was to force the reader to draw them.

riesz nagy is also excellent but old fashioned. but then i have given all these recommendations in my mathematician thread, which of course is now too long for anyone to read.

What do you think of Spivak's Calculus as a book to learn analysis from? It looks basic enough but it is a bit wordy I found. However the other extreme might be Rudin's book which is not wordy enough? Is that its major problem? Or does it skip too much stuff? I probably need one that has minimal words but lays out the maths completely and fully without any abbreviations. I wonder if such a book exists?

Even though a lot of people think analysis is difficult. I tend to get the picture that it is a 'lower order' subject compared to algebra and topology. The latter two seems more prestigous than analysis? Is analysis less abstract than the latter two?

 

[ircdan]

So you took classes in all three areas in the option but still found combinatorics/graph the hardest?

Yes and yes. I didn't like the subject and I think that's why I had a hard time.

 

[math_owen]

I think Spivak's Calculus is an amazing book - but I don't like the Bourbaki method that Rudin sticks to. I think, though, that Rudin would be great if you've already read Spivak's for the problems he has. I think, the "beauty" of the terse bourbaki method is mostly due to the fact you're suppose to fill in all the gaps and do all the problems.

A more modern book is Bartle's. Though, I would save some money, and go for an older 2nd edition, b/c IMO it's written better.

Do yourself a favor, throwout that word 'prestige'. Just chuck it, forget it even exists. It's a silly game, played by insecure people. Study what you love and never listen to the "prestige".

And though, I may never be an analyst, Terrance Tao is one. And analysis is one of the oldest most developed forms of modern mathematics. It's impossible to prove which is used the most, but it appears that analysis may be the most useful. So I say it deserves the same respect as any math, whether that be Stats or Category Theory or Mathematical Physics.

Down with this hampering by prestige. What do you like?

edit: all math is abstract if you take it far enough

 

[morphism]

Although Spivak is a great book, it is definitely NOT suitable as an analysis text. For example, it contains no metric space theory. You're better off getting a real analysis text (pun not intended ). I second mathwonk's remarks about Rudin; it's only good if you've seen all the material before, and want a slick reference. Some of my favorite analysis texts are: Intro to Topology and Modern Analysis (Simmons), Elements of Real Analysis (Bartle) [note: there is also Intro to Real Analysis by Barte; this is not the same book], Elements of Integration (Bartle), and Functional Analysis: An Intro to Banach Space Theory (Morrison).

And no: analysis is definitely not less abstract than algebra or topology. Actually, the subjects get intertwined down the road, e.g. Banach algebras are algebras that are complete normed vector spaces (i.e. Banach spaces), we can define a measure (the Haar measure) on the Borel algebra generated by the compact subsets of a locally compact topological group, etc.

As for me, the two hardest math courses I've taken were probability and graph theory.

 

[quasar987]

Probability was tough. The theory is very easy but some problems are just barbaric. But I guess in any subject you can make up problems that are very difficult, that's partly why I did not vote on your poll. I don't see a good criterion on which I could say such and such is hardest. Besides, I haven't even taken algebra yet! (and I have 3 years under my belt, it's preposterous, I know)

 

[jdstokes]

quasar987,

Meh, I'm third year and I haven't taken nor intend to take any time soon, a single analysis course. I did do Metric spaces, however.

By the way, sorry to be a thread imposter but no one gave me any advice on my last post. Who can rate these in descending order of usefulness/importance in theoretical physics?

Functional analysis
Partial differential equations
Algebraic topology
Algebraic geometry
Commutative algebra
Representations of the symmetric group

(n.b. they are all at fourth year pure maths level)

 

[pivoxa15]

Probability was tough. The theory is very easy but some problems are just barbaric. But I guess in any subject you can make up problems that are very difficult, that's partly why I did not vote on your poll. I don't see a good criterion on which I could say such and such is hardest. Besides, I haven't even taken algebra yet! (and I have 3 years under my belt, it's preposterous, I know)

Well the title and theme of the thread is a comparison of the difficulties of pure maths subjects only. It may be possible that some are intrinsically more difficult than others because they may be naturally less intuitive like analysis. For me, analysis so far has been a bit hand wavy which is not a good sign. However some parts of algebra I have found very solid although my understanding of modules is also hand wavy.

The thing is analysis seem to be less fundalmental than algebra and topology? In that things in analysis seem to start with a lot of things intuitively assumed. Or is it that I haven't learned analysis properly.

 

[pivoxa15]

When we talk about Rudin's books, are we talking about his "Real and complex analysis" book?

Or "Principles of mathematical analysis."

 

[morphism]

Probability is a pure math subject. (In fact, probability could fall under the subheading of measure theory, i.e. analysis.)

And if analysis is handwavy, then can I ask you what analysis you've seen, and what exactly you found to be handwavy?

[I was talking about Principles, btw. I haven't read Real & Complex enough to formulate an opinion about it yet.]

 

[CompuChip]

In my experience, Topology is a very hard subject. It starts out easy, but when it comes to quotient topologies and fundamental groups, the subject gets harder and (for me) less intuitive.

Differential geometry (manifolds, 2-forms, algebras) is also one of the more abstract subjects which I think are harder to understand.

I've also heard that a lot of people have trouble with measure theory, though I can't agree with that.

 

[d_leet]

When we talk about Rudin's books, are we talking about his "Real and complex analysis" book?

Or "Principles of mathematical analysis."

Real and Complex Analysis is supposed to be a graduate level text, so if we are talking in the context of an undergraduate curriculum we're most often more likely speaking of Principles.

 

[quasar987]

The thing is analysis seem to be less fundalmental than algebra and topology?

I suppose it depends on your point of view. Historically, analysis came first. The ancients wanted to make calculus rigorous, what I'm guessing happened is that while trying to do that, they realized that it might be a good idea to make a definition out of the types of sets that we know as open and closed, because they popped up time and time again in the thms. At one time, they must have realized that some properties of R^n (or of metric spaces in general) could be formulated without any mention of sequences! For instance, x is an accumulation point of A <==> there's a sequence in A\{x} converging to x <==> any open nbhd of x has a non-empty intersection with A\{x}. Less trivially, a set A is compact <==> every sequence in A has a subsequence converging to a point of A <==> any open cover of A has a finite subcover.

This must have suggested to them that there could be a class of results about metric space that are true in a more general setting than metric spaces. This more general setting is of course what we know today as a topological space, of which metric spaces are now only a subcategorie for which the set of opens (the topology) is metrizable.


So as it turns out, analysis ends up being spawned from a very special case of topological space and a very special case of ordered field (i.e. the unique complete ordered field). So undoubtedly, in the great hierarchy of math starting with formal logic, set theory, and up, analysis is less fundamental as it appears higher on the tree and do topology & algebra.

But on the other hand, analysis is where it all started, because our world is described by the unique complete ordered field, not by some magma! It is in the study of the reals that we found the motivation to extend some of their features to more general settings. It would be foolish to start studying topology without having met with the concrete topology of R^n first... topology would seem much too abstract and unmotivated!

 

[InbredDummy]

Well the title and theme of the thread is a comparison of the difficulties of pure maths subjects only. It may be possible that some are intrinsically more difficult than others because they may be naturally less intuitive like analysis. For me, analysis so far has been a bit hand wavy which is not a good sign. However some parts of algebra I have found very solid although my understanding of modules is also hand wavy.

The thing is analysis seem to be less fundalmental than algebra and topology? In that things in analysis seem to start with a lot of things intuitively assumed. Or is it that I haven't learned analysis properly.

yeah you couldn't be more wrong, probability is a pure math subject and is usually lumped in together with measure theory and integration theory. what you do with the theory is what makes it applied math, but the theory of probability is a pure mathematics subject and is very rigorous.

 

[pivoxa15]

yeah you couldn't be more wrong, probability is a pure math subject and is usually lumped in together with measure theory and integration theory. what you do with the theory is what makes it applied math, but the theory of probability is a pure mathematics subject and is very rigorous.

But probability is an intrinsically applied maths subject. Just like Quantum Mechanics is intrinscially applied. It happens that there is an abstract formalism of QM but does that make QM a pure maths subject?

 

[mathwonk]

spivak is not an analysis book, it is only called one by people who have elarned calculus without any theory, it is a beginning calculus book with theory.

but if you do not know the stuff in spivak, it is a good place to start.

for intro to analysis, i recommend simmons. it seemed clear to me back in 1970 when i first encountered it.

rudins real and complex book is an advanced analysis book, excellent, but based on what to me is a flawed premise, that real and complex analysis are closely entertwined and should be learned together.

i myself find them to have very different flavors.

but do not be afraid to look at any book, no matter who says it is ahrd. you have tod ecide for yourself what book is right for you.

i once avoided reading the book homological algebra for years, by cartan and eilenberg because someone told me it is was hard, only to find it extremely clear and easy to read.

when i told the person, they checked their sourcde and had actually misunderstood the assertion, it was that the book seemed "tedious" to someone else, which is often another way of saying carefully written and detailed.

 

[InbredDummy]

But probability is an intrinsically applied maths subject. Just like Quantum Mechanics is intrinscially applied. It happens that there is an abstract formalism of QM but does that make QM a pure maths subject?

until you have taken a pure math probability theory course and a measure theory course, we can't really have this discussion.

 

[hockey777]

Combinatorics was my hardest.

 

[mathwonk]

our students do worst in analysis, hands down. possibly because we used rudin's very user unfriendly book.

 

[Coto]

quasar987,

Meh, I'm third year and I haven't taken nor intend to take any time soon, a single analysis course. I did do Metric spaces, however.

By the way, sorry to be a thread imposter but no one gave me any advice on my last post. Who can rate these in descending order of usefulness/importance in theoretical physics?

Functional analysis
Partial differential equations
Algebraic topology
Algebraic geometry
Commutative algebra
Representations of the symmetric group

(n.b. they are all at fourth year pure maths level)

I'm doing my masters in applied mathematics (undergrad in physics). This is just my experience when dealing with analytical mechanics.

Functional Analysis - Gives fundamental results that will really help benefit your understanding of PDEs.

Partial differential equations - Almost every fundamental physical equation is a PDE of some kind and at some level. The draw back of PDEs is their dependence on a coordinate system, which as you can guess becomes a problem when dealing with subjects such as GR.

As a side note, welcome to Tensor Analysis, Differential Geometry...

Algebraic Topology - Haven't encountered anything further than an introduction to topology, and haven't needed anything further or seen any higher physics using it.

Algebraic Geometry - Likewise.

Group Theory - Probably the most useful of algebraic abstractions for physics. Lie groups, symmetric group, fundamental results relevant to particle physics, analytical mechanics, quantum mechanics...


Hope this helps.