Should I Become a Mathematician?

[SticksandStones]

Mathwonk: do you know what the current state of research into Topology is? I mean, is there still a lot of interest in the topic?

 

[mathwonk]

well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.

 

[Bourbaki1123]

Mathwonk,

You seem to give quite a bit of praise to Michael Artin's book on algebra. What do you think of his father Emil's book on the subject?

 

[mathwonk]

the only books i know of by the father are "galois theory" notes from notre dame lectures, and "geometric algebra". these books are great classics, but they are not as easy to read as mike's book. mike wrote his book for sophomore students whereas emil seemed to write his books for eternity. i.e. whoever can read them is welcome, and not one word is wasted.

i myself never could really learn from e. artin's galois theory book as it was too condensed for me. he also has some algebraic geometry notes from nyu but those also leave much to be desired from my viewpoint for learning ease. But it is almost sacrilegious to criticize anything written by e. artin, who is regarded with great awe by many people.

but i regard mike's books as much more user friendly.

but as i meant to imply, i am not aware of any books by e. artin strictly on abstract algebra. of course the great book by van der waerden is based on lectures of e. artin and e. noether. Is that what you mean by e. artin's book? I like it quite well and learned a lot from it as a student.

If that is representative of e. artin's lecture style then he was a very fine teacher. Indeed I have read in his own works that he always tried to write more than usual on the board when lecturing so that the student who was not following could recover the lecture from his notes. this struck me as admirable and i long followed this practice in my own lecturing.

 

[SticksandStones]

well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.

That's good to hear. I started reading Cairne's "Introductory Topology" and so far I've found it pretty fascinating. I can't wait to be able to take a class on it.

 

[mathwonk]

topology is the most fundamental branch of geometry. as such i believe it will always be one of the most fundamentally important topics.

the ideas developed in topology of ways to understand different types of connectivity, are absolutely crucial in all areas of mathematics.

the tool of cohomology, which is present in algebra, geometry, and analysis, received its greatest development within topology. Sometimes I think the greatest ideas in mathematics grew there.

that is probably unfair to analysis, but anyway.

 

[SticksandStones]

Your post went a bit over my head. :)

I really liked Abstract Algebra when I took it. It looks like group theory plays a roll in Topology, from skimming some things. Am I right in assuming this?

 

[mathwonk]

i am just saying that the ideas that were developed in the 30's, 40's and 50's within topology, like bundles, characteristic classes, and sheaves, and cohomology, grew outward and illuminated complex analysis and algebraic geometry in the 60's and 70's and are universally used now.

you are currently at the beginning, studying point set topology, but later when you study algebraic topology this will be meaningful.

 

[nomaris]

Mathwonk,

In the first page of the thread you said that a high school student should explore probability, linear algebra, calculus after having a thorough grasp of geometry and algebra. What constitutes knowing Euclidian geometry and algebra well?

 

[mathwonk]

i would say mastering harold jacobs' books on those topics are a minimum for a high schooler. if more ambitious you might search out smsg books from the 60's. say arent there numerous such recommendations in that thread? have you only read page 1?

 

[PhysicalAnomaly]

Do they really expect PhD students to learn 2 foreign languages in 3 years?

 

[Wretchosoft]

Do they really expect PhD students to learn 2 foreign languages in 3 years?

I don't see why this requirement would be intimidating. Two semesters in college is enough to teach the average student the basics of a language; with the generally higher capabilities of PhD students, I would imagine this time could be shortened. From there, it's just practice.

 

[alligatorman]

From what I've heard, the language exam is usually just to translate a mathematical paper from the language into English. I can't imagine that it's too difficult.

 

[The|M|onster]

Would Spivak's Calculus on Manifolds be a good reference text for a undergraduate course on multivariable analysis?

 

[maze]

Calculus on manifolds book is primarily useful for the exercises, which are quite good. The writing and explanation is too terse in my opinion, but some people swear by it.

 

[Vid]

I just took a course using the book and found it to be really good. Munkres Analysis on Manifolds is kind of like an expanded version of CoM and is really good as well.

 

[samspotting]

I am trying to prepare a good foundation for math. I am learning from a few sources but I will be proficient these areas from classes and books:

Real Analysis (Learned from pugh and baby rudin, and class)
Linear Algebra (Learned from Friedberg, Insel, Spence, and class)
Set Theory (Learned From Naive Set Theory)
Combinatorics (Learned from Class)

What is a good way to learn geometry? I never paid much attention to any of my high school math classes and never really got much out of it, besides the basic identities. It seems like it could be very interesting.

I was looking at Beyond Euclid's Elements, and was surprised to find Mathwonk as one of the featured reviews on amazon. Maybe he can offer some advice and input.

Is there anythink else that math majors should know before moving on? One very interesting book that caught my eye is Inequalities by hardy, littlewood, and polya. It looked intense though, is that book my level?

 

[mathwonk]

i liked calculus of several variables by wendell fleming.

as i said in my review, hartshorne's book is an excellent guide to euclid.

 

[quasar987]

IMO, "Inequalities" is a reference book, as opposed to a book you read from back to back... Say you're stuck on a problem and realize that if you had some kind of inequality then it would work... you go look in "Inequalities".

 

[mathwonk]

do you think these proof questions are too hard?

I.A i) Recently, my only guests for Thanksgivings have been turkeys.
ii) No mathematicians fail to solve crossword puzzles faithfully.
iii) The only faithful crossword puzzle solvers I know are my recent Thanksgiving guests.
Conclusion (using all the hypotheses):

IB. i) The Americans who exploited the Hawaiian natives ended up doing quite well.
ii) Some American missionaries who came to Hawaii originally to do good, started pineapple plantations.
iii) The pineapple planters in Hawaii exploited the natives’ land and labor extensively.
Conclusion(using all hypotheses):

IC. i) I consider money not spent enjoyably, to be wasted.
ii) I have had little joy out of anything lately other than comic books.
iii) An intelligent person does not waste money.
Conclusion(using all hypotheses):

ID. i) Dr. Smith has discovered the most wonderful beach.
ii) Some things are really fine, but nothing is as fine as the sand at the beach.
iii) If a person discovers something really fine, he should bury his head in it.
Conclusion(using all hypotheses):

 

[Bourbaki1123]

do you think these proof questions are too hard?

I.A i) Recently, my only guests for Thanksgivings have been turkeys.
ii) No mathematicians fail to solve crossword puzzles faithfully.
iii) The only faithful crossword puzzle solvers I know are my recent Thanksgiving guests.
Conclusion (using all the hypotheses)::

By i)&iii) the stuffing is drugged, don't eat it.


IB. i) The Americans who exploited the Hawaiian natives ended up doing quite well.
ii) Some American missionaries who came to Hawaii originally to do good, started pineapple plantations.
iii) The pineapple planters in Hawaii exploited the natives’ land and labor extensively.
Conclusion(using all hypotheses)::[/QUOTE]
Some American missionaries ended up doing quite well

IC. i) I consider money not spent enjoyably, to be wasted.
ii) I have had little joy out of anything lately other than comic books.
iii) An intelligent person does not waste money.
Conclusion(using all hypotheses)::[/QUOTE]

If I were intelligent, then I would buy comic books.

ID. i) Dr. Smith has discovered the most wonderful beach.
ii) Some things are really fine, but nothing is as fine as the sand at the beach.
iii) If a person discovers something really fine, he should bury his head in it.
Conclusion(using all hypotheses):[/QUOTE]

What if its a tar beach? Or a rock beach? If fine means the same thing in all of its uses, and is defined as to mean granulated, or ground to a very small scale , then Dr. Smith should bury his head in the sand iff the beach mentioned in i) is the beach mentioned in ii), else, we cannot say that the beach in i) even has sand, so ii) and iii) have no bearing. If, however, we take fine to mean; good, wonderful, grand, then Dr.Smith should bury his head in the beach. Now, if the beach in ii) is the same beach, or has sand as fine as the beach in ii), we conclude that Dr.Smith should indeed bury his head in the sand at the beach. Otherwise, perhaps he might be equally well off burying his head in some rocks or seaweed.

Now what if fine has two distinct meanings? Am I meant to exhaust all possibilities?

 

[mathwonk]

no one seems to notice the qualifier in B that renders it similar to a famous quote: "The American missionaries, who originally came to Hawaii to do good, ended up doing well".

and in C), wouldn't it be "...only comic books"?

I give up on D. I think the conclusion is that humor and tests do not mix, or humor and mathematicians do not. or more accurately, to cite another famous quote:

"I knew Lewis Carroll, Lewis Carroll was one of my favorite authors. ... Dr. Smith, you are no Lewis Carroll."

 

[uman]

Sorry Dr. Smith, what quote are you referencing?

Also, A) should be "all mathematicians are turkeys".

 

[Bourbaki1123]

A) is not all mathematicians are turkeys. ii) says that all mathematicians solve puzzles, its not necessarily true from that, that all puzzle solvers are mathematicians. My conclusion would be that thanksgiving is for the birds.

In earnest, I would say that the conclusion would be that all of your recent guests are faithful crossword puzzle solvers and turkeys. Since it isn't really implicated that all crossword puzzle solvers are mathematicians, what happens is that your guests and the set of all mathematicians are subsets of the set of all crossword puzzle solvers, and these subsets can still have a null intersection.

 

[uman]

By ii), all mathematicians are faithful crossword puzzle solvers. By iii), all faithful crossword puzzle solvers Dr. Smith knows are his recent Thanksgiving guests. By i), all these are turkeys.

I guess the conclusion should read: All mathematicians that Dr. Smith knows are turkeys.

 

[Tickitata]

I believe these are the answers (fun puzzles by the way! I wish my intro to proofs class used this idea)

1. Some mathematicians may be turkeys
2. Some American missionaries ended up doing quite well
3. Dr. Smith spends all his money on comic books
4. Dr. Smith should bury his head in the sand

 

[mathwonk]

ok please forgive me if these answers do not make sense. After all, I made up the answers before I made up the questions.

\my personal answers are:
1) all the mathematicians I personally know are turkeys.

2) Some Americam missionaries who went to hawaii to do good, ended up doing quite well.

3) If I am intelligent I will spend money only on comic books.

4) Dr. Smith should bury his head in the sand at his wonderful beach.

 

[mathwonk]

the other famous quote i refer to involved dan quayle, and ran roughly as follows:

"I knew Jack kennedy,.. Jack kennedy was a friend of mine,.. and you senator are no Jack kennedy!"

 

[mathwonk]

I am encouraged to post some more of my challenging exam questions: (provided you can handle them.)

In “A few good men”, after a marine named Santiago was killed by two soldiers of the colonel’s command, Tom Cruise cross - examined the colonel (Jack Nicholson) as follows: “Colonel, you told us you ordered Santiago transferred off the base because he was in grave danger, and that your men always do exactly as you tell them.” “That’s right”. “I just have one question: If you told them Santiago wasn’t to be touched, and if your men always do exactly what you say, then why would Santiago be in danger?”

A) Clarify Cruise’s implication, by giving the contrapositive of the statement “If you told your men he was not to be touched, then Santiago was not in danger.”
...

Later, Cruise elicited from the Colonel a list of items he had packed for a weekend trip, plus several phone calls he had made in preparation. Then he observed, “Colonel, you were leaving for two days, and you packed two bags and made three phone calls. I’m just puzzled, since according to you, at 5am the next morning, Santiago was leaving for the rest of his life, but he hadn’t called anybody, and he hadn’t packed a thing.”

B) Clarify this implication by giving the contrapositive of the statement:
“If Santiago knew he was being transferred off the base first thing in the morning, he would have made some phone calls or at least packed some clothes.”
...

C) Based on the contrapositives of these statements, what would you say Tom Cruise is implying the colonel did (or did not do)?
...

D) Do you think Cruise’s arguments raise sufficient reasonable doubt, to counter the prosecution’s charge that the two soldiers acted without the colonel’s approval, or do you think he needed to go after a full confession by the colonel? Why or why not?

 

[uman]

A) If Santiago was in danger, then you did not tell your men he was not to be touched.
B) Since Santiago neither made phone calls nor packed some clothes, he could not have known he was being transferred off the base first thing in the morning.
C) Cruise is implying that the colonel did nothing to prevent his men from killing Santiago.
D) Cruise does raise sufficient reasonable doubt: Although he does not obtain a direct confession from the colonel, assuming his premises (those listed in A and B above, as well as the implicit assumption that the fact that the Colonel did nothing to stop Santiago's murder is equivalent to his tacit approval) are true, he uses valid logic to lead from what the colonel did say to the conclusion that the Colonel's men acted with his approval. Assuming his premises are true, this is just as good as a full confession. His premises may or may not be true, but they are certainly plausible enough to meet the standard of "reasonable doubt" required for an acquittal.

 

[mathwonk]

so why in the movie did tom cruise go for the confession, at the risk of blowing the whole case?

 

[uman]

I don't know. I haven't seen it. What did you think of my answers? I'm fairly sure they're correct, but obviously your eyes are better than mine.

 

[mathwonk]

i like your answers. that's why i ask another question. my own opinion is that the forced confession is a dramatic device, which as you argue was not necessary for the judicial purpose of the trial.

 

[uman]

Or maybe the lay jurors (or whoever is judging the case -- I'm not familiar with how military trials work), having not been trained in mathematics and logic, are at risk of not grasping an indirect and therefore more subtle argument.

The idea that it's just a device to move the plot along is plausible too. Again, I haven't seen the movie.

Did you really give these questions on a test?

 

[mathwonk]

what, aren't they standard?just kidding... I get the "are you serious?" question a lot.