Should I Become a Mathematician?

[MotionStudy]

Any tips for a studying mechanical engineer who cringes at the thought of not using math on the job? I was very disappointed to hear working engineer's describe their average work day as "using calipers to measure parts" and "taking soil readings with a tensiometer." I've made a decision to never take a job that doesn't involve mathematical engineering analysis of creative design. I love the math and I love the development process. I guess it's relevant to add I enjoy physics equally, if not more so, than pure math. Would a minor in math be beneficial (I would enjoy it) to getting the job's I'm interested in?

 

[jedishrfu]

I've got a couple of math book references to contribute here for future reference.

Casual reading for the budding mathematician:

Math 1001 by Elwes -- covers many topics of interest in very short paragraph descriptions to catch your imagination.

The Math Book by Pickover -- covers many historical math discoveries in one page synopses. Pickover has other math titles of interest from Fractals to you name it...
- The Möbius Strip

Math in a 100 Key Breakthroughs by Elwes -- covers major math discoveries over the years

Sacred Geometry by Skinner -- covers how geometry is used in mystical ways by our ancient forefathers and mothers...

Magical Mathematics by Diaconis and Graham -- Magic and math a cool combination to astound and confound your teachers

The Story of Mathematics by Stewart -- resource for time travelers of the imagination

The Mathematical Mechanic by Levi -- uses physics to demonstrate the truth of mathematics

Magnificent Mistakes in Mathematics by Posamentier and Lehmann -- quite a few conjectures and theorems that were later proven wrong and resulted in new discoveries

Knots by Sossinsky -- a small treatise on the theory of mathematical knots

The Knot Book by Adams -- more theory and history of knots

Mathematical Mysteries by Clawson -- covers Godels proof in greater detail than I've seen in other popular books

The Compleat Strategist by Williams -- elementary game theory

Books by Keith Devlin: The Numbers behind Numb3rs TV show, The Millenial Problems

Coincidence, Chaos and All That Math Jazz by Starbird and Burger -- many cool paradoxes and mysteries

Number Freak by Niederman -- arcane facts about each number from 1 to 200 great as conversation starters for shy mathematicians

Sander Bais books on Special Relativity and The Equations -- extremely introductory treatments of the most famous equations of math and scienceMore advanced reading:

The Princeton Companion to Mathematics edited by Gowers -- covers many topics of math written by major players in each field

Mathematical Methods for Physicists by Arfken and Webber -- applied math topics suitable for mathematicians with a physical sense of reality

The Road to Reality by Penrose -- physics, math and history with a Penrose twist

Space Through the Ages by Lanczos -- covers the development of geometry up to the present ie Gauss, Riemann, Einstein GR, Abstract Spaces and Projective Geometry

Differential Forms by Flanders -- go beyond Vector Analysis to Differential Forms

Origami Design Secrets by Lang -- very interesting treatment of origami covering a lot of topics, fertile ground for new mathematical ideas

An Introduction to Computer Simulation Methods by Gould, Tobochnik and Christian -- a great way to learn applied math via physics and computer modeling with Java

Principles of Quantum Mechanics by Dirac -- rigorous treatment of Quantum Mechanical principles

The Dreams That Stuff Are Made Of edited by Hawking -- foundational papers of Quantum Mechanics all in one volume

Einstein Gravity in a Nutshell by Zee -- latest developments in General Relativity a kind of update to the Wheeler classic on Gravitation (see next)

Gravitation by Misner, Thorne and Wheeler -- everything about General Relativity in multiple tracks of learning, great drawings too...

That's all for now folks. I suppose you could file this post under books for the armchair mathematician who likes to dream about Mathematical things.

 

[Cod]

Here's an interesting math / programming blog I came across a few months back: http://jeremykun.com/

 

[David Carroll]

The Mathematical Mechanic by Levi -- uses physics to demonstrate the truth of mathematics

But what if we lived in a world where everytime you picked two apples off a tree, they immediately became 3 apples?

 

[jedishrfu]

I'm sure you would enjoy the book regardless of the world you live on.

 

[waddlingnarwhal]

Hi Y'All. I have two questions, one specific and one general.

(1) Why is the Residue Theorem often stated specifically for meromorphic functions. In the statements I've read or Googled, it is specified that the singularities are isolated poles. Wikipedia mentions that the theorem also holds for essential singularities, but that the latter result is something "more general". However, it seems that one of the common proofs often given for the meromorphic case works perfectly well even if the singularity is essential.

For simplicity, assume we have just one isolated essential singularity "z" and a simple closed curve C that winds around it once. Suppose f is holomorphic in a bigger disc containing C and its interior. Then is it true that the integral of f around C is equal to 2(pi)(i)*Res(f,z)?

It seems to me that we can punch a keyhole in C to get a contour K. The integral around this contour is 0. Then by shrinking the width of the keyhole but keeping the size of the hole H the same, we equate the integral around C with the integral around the hole H. H is contained in a small annulus centered at "z". We expand f in a Laurent series in this annulus. The principal part converges absolutely outside the inner hole of the annulus, and thus it converges absolutely on H. Thus we can integrate term-by-term, leaving us with the residue.

(2) I am looking for a math book to read over the winter holidays. I'd like a textbook rather than casual expository material, so for example Ian Stewart's Galois Theory would be preferred over his Concepts of Mathematics, but I would also like something short enough to finish during break :). Does anyone have suggestions?

 

[jedishrfu]

What about Penrose's Road to Reality, it's perhaps more physics than math but you could think of it as applied math.

https://www.amazon.com/dp/0679776311/?tag=pfamazon01-20

 

[waddlingnarwhal]

What about Penrose's Road to Reality, it's perhaps more physics than math but you could think of it as applied math.

Thanks for the recommendation! I don't think I'll be able to finish that during winter break though...

 

[jedishrfu]

Thanks for the recommendation! I don't think I'll be able to finish that during winter break though...

True, but think of the fun you'll have just trying... :-)

 

[mathwonk]

narwhal, maybe the residue theorem is often stated for meromorphic functions because those are easier to compute residues for. i.e. there is perhaps essentially no way to compete them for essential singularities, so although the theorem is true there, it is not of much use?

 

[waddlingnarwhal]

mathwonk - That makes a lot of sense. I guess I was only thinking of "very nice" essential singularities like e^(1/z).

Complex analysis is such a great class. I feel that I've internalized the theorems and their proofs, at least those we covered in class, a lot better than those from real analysis. It may be because the book we used is much better for the former class.

 

[mathwonk]

you make a good point. any time you actually have your hands on the laurent series, of course you know the residue. that could be useful.

for some odd reason, almost all books on complex analysis tend to be good, while the opposite seems true often for reals. maybe it's just inherent in the subjects, or maybe the early workers in complex already wrote it up so well we just repeat what they said.

or maybe it's because complex analysis deals with almost the absolutely nicest functions, those which are analytic, and reals deals with the worst ones, those which are merely measurable. The worst behaved phenomena in complex by the way seem to me to be essential singularities.

 

[Mathmaybe]

What is your Recommendation for books on real analysis with an applied bent? Folland is unpleasant to read, Roydens is pretty good but it has a massive list of errors.

I want something slightly chatty, emphasizing institution as well as of course, formal proofs.

 

[waddlingnarwhal]

Hi Mathmaybe. From the books you listed, it sounds like you are looking for a second course in analysis. I found the real analysis volume of the Stein and Shakarchi series to be good. It is clearly written and provides historical motivation. The pace of the exposition is fairly slow.

 

[mathwonk]

i am myself not at all expert in real analysis and have trouble recommending books on it. i did like the chapter on integration in the book calculus of several variables, by wendell fleming, and i like most books by sterling berberian, including his fundamentals of real analysis. i also like looking into the classic functional analysis by riesz-nagy. i like volume one of dieudonne's classic foundations of modern analysis very much but dislike his later volume on integration theory. i think royden is a good place to learn somethings, especially measure and integration on R^1, and i would not worry about errors. if there is a long list that is a good thing as it means they have been pointed out. a book with no such list probably has as many unidentified errors. experts i know like wheeden and zygmund, but unfortunately i do not own it.

the book by stein and shakarchi does look quite clear on the concept of measurability. in general i caution against recommending a jointly written book just because one of the authors is a famous and brilliant mathematician, since he probably did not write it.

https://www.amazon.com/dp/0824764994/?tag=pfamazon01-20

 

[Mathmaybe]

Thank you for the recommendations. You are correct that it would be a second(or really a third) course, there seems to be some inconsistencies in how people refer to a subject.

Anyway about the books, it seems like a lot of high level books are purposefully obtuse/difficult in both the presentation and the problems. There is value in having a variety of exercises but some books feel like the author is purposefully being a jerk.

 

[waddlingnarwhal]

If you want something very down-to-earth, there's a "Schaum's Outline of Real Variables and Lebesgue Measure". It's out of print, but I've seen copies in my school's library, and there are used copies online. It seems like most of the Schaum's Outlines aimed at upper-division undergraduate classes have gone out of print :(.

 

[CosmicCube]

I have a somewhat tangential but related question vis-a-vis the title of this thread. Is there an age limit when it comes to learning math. For instance, is the brain less mathematical, say after 40 or 50, such that mathematical concepts and computations become much harder and perhaps higher math becomes almost impossible to learn?

 

[mathwonk]

It may get harder to learn math as we age, but so what? I'm going to age anyway, and I just have to choose whether to keep learning or give up. Even tying my shoes is getting harder, but I still need to wear them to get around. Learning math is fairly hard for pretty much everyone, but also satisfying.

I think in my own case, learning math is harder now that I am isolated from a group of mathematicians, learning together and teaching each other by talking and presenting their work. So the main factor affecting learning math for me is immersion in an active learning environment. I learned most and fastest the year I was on leave as a research fellow at Harvard, at age 38, surrounded by some of the most active and knowledgeable experts in my subject.

 

[waddlingnarwhal]

mathwonk - It's very interesting to read about your personal experiences in this thread. Anecdotes like the one you just shared are one reason I enjoy learning in school or from forums more than just working by myself.

As I begin thinking about whether I want to continue my mathematical studies in graduate school, I'm finding it difficult to sift through all the advice I receive since people whom I greatly respect often have different and conflicting opinions. Is it true that all roads lead to Rome?

 

[mathwonk]

absolutely.

 

[Hercuflea]

Would it be possible to get a professorship in Mathematics with a Master's in Mathematics and a PhD in an engineering discipline? Especially if your research is all applied math?

 

[mathwonk]

I don't know how likely this is. Of course anything is possible. E.g. Edward Witten has a PhD in physics but won the Fields medal in mathematics and hence would surely be welcome in virtually any math department.

 

[sunny79]

Mathwonk! Can you please give tips on studying habits for aspiring undergrads who wish to excel at Math and physics. Perhaps could you or someone could redirect me to the posts if this has been covered earlier by you. I would really appreciate it.

 

[waddlingnarwhal]

sunny79 - On page 2 of this thread, mathwonk gives some advice for undergraduate students. His suggestions and book recommendations seem spot-on to me. I also noticed that he linked this article in another thread - http://www.xavier.edu/diversity/documents/1Studying_Students_Studying_Calculus_A_Look_at_the_lives_of_minority_mathematics_students_in_college.pdf . I think it gives some good advice about study habits. I particularly agree with the author regarding the positive value of studying with a group of dedicated math students.

 

[waddlingnarwhal]

anybody taking group theory? here is a very basic question a student just asked me today:
suppose you have a group G and a subgroup H of index n. Prove there must be a normal subgroup K contained in H, such that #(G/K) divides n!
think "group actions".

I was reading through the older posts in the thread and came across this question. I decided to take a stab at it since I just finished my group theory course. G acts on the cosets of H by left multiplication, and this gives a homomorphism of G into S_n. The image of this homomorphism is isomorphic to the quotient of G by the kernel of the homomorphism. This kernel is the K we are looking for.

I myself would like to pose a question. This question comes from an exam I took, and I could not solve it during the exam :(. Let G be a simple group of order 168. Prove that G has no element of order 21.

I was wondering what classes other people will be taking after the winter holidays :). I thought it would be fun to discuss.

 

[sunny79]

Waddlingnarwhal! Thanks a lot...There is a wealth of information in the previous threads including Mathwonk's notes, which are outstanding. Tons of great advice. Due to GED background, my math skills were lackluster as I proceeded towards college so there were quite a few discrepancies which showed from time to time manifesting in the form of errors, costing me points. Need to get back to the drawing board and fix the problems... :)

 

[lavinia]

I am not a mathematician so take these thoughts as completely personal, my opinion based only on my experience.
I have been learning mathematics on my own and recently have begun attending seminars and sitting in on graduate courses.

- While course work either in school or from a book is essential more important perhaps is learning to think mathematically. Mathematics is not only a body of knowledge but also a mental discipline. This discipline needs learning as much as any subject. The way I did it was to try to figure problems out on my own without asking questions and without even reading the book first. I took each problem as completely new and unknown, as a research question rather than as an exercise. When I reached the point where I clearly understood what facts were missing, I looked for them in the book. I think that a great subject for doing this is point set topology. It is conceptual, proof based - as opposed to calculation based - elementary, and not very hard relatively speaking.

Don't give up too quickly. That aha moment will come unexpectedly. Take time.
I find that on Physics Forums people ask questions too soon. They have not struggled with the problem enough. Getting answers is not the same as understanding.

- Work through examples. Theorems are nice, but examples bring them to life Get your examples from anywhere you can find them, engineering books, physics books, math books, finance books, whatever. For instance, when learning Differential Geometry - a subject that can get amazingly abstract - I worked through Struik's book on the classical differential geometry of curves and surfaces. The book is chock full of wonderful examples. When learning vector calculus I worked though Feynmann's Lecture on Electricity and Magnetism. Maxwell's Equations make vector calculus so real you can taste it. i also worked through an old engineering book on the planar motion of rigid bodies. Great book.

- If you want to learn a subject, learn something that uses it as a tool. This is how Physics teaches mathematics and it works. It gives the mathematics meaning. So if you want to learn linear algebra, learn multivariate calculus and tensor analysis, or group representation theory. If you want to learn point set topology, learn complex analysis. If you want to learn complex analysis, learn about Riemann surfaces.

People often ask "What is mathematics used for?" Well, a lot of mathematics is used for other mathematics.

- Don't try to memorize. Rather try to understand the ideas. Know what things are supposed to be like then derive the equations on your own. Don't think you know something just because you have memorized how to do a calculation.

- If you can, get a mentor. One on one guidance in my mind is the best environment. Large lecture courses often fail the students and discourage them.
At some point down the road,hang out with mathematicians. I have found them open and accepting even of relative beginners like myself.

- Don't try to be a whiz. Don't measure yourself by exam scores. Measure yourself by that inner sense of truth. Don't compete.

- Don't overdo rigor. While rigorous proofs are indispensible, as is knowing how to do them, ideas are more important because they are the material for which proofs are fashioned. I sat in on a basic algebraic topology course with a renown mathematician and every time a student tried to answer a question with a rigorous demonstration, he got angry and said "that's not a proof." To him, the proof was the idea, or if you like, the geometric insight.

In some sense, rigorous fully elaborated proofs seem to be the bookkeeping for the business, not the business of mathematics itself.

- Here are some book that I have used.

Struik, Lectures on Classical Differential Geometry - great examples. Also good for learning Multivariate Calculus and basic Linear Algebra

Hurewicz, Lectures on Ordinary Differential Equations. - amazingly clear exposition. Good for learning basic Linear Algebra and Point Set Topology

Milnor, Topology from the Differentiable Viewpoint This book shows the geometric insights behind multivariate calculus. It s more advanced but still elementary.

Feynmann, Lectures on Physics The more physics you know, the better mathematician you will likely be.

Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry - This is an undergraduate text that introduces modern mathematical ideas. Its geometry section is priceless. It develops the differential geometry of surfaces, the simplest case after smooth curves, from the modern point of view of connections on fibers bundles. With this book together with Struik's book, life is good. Also a good book for learning integration theory on manifolds.

Bott and Tu, Differential Forms in Algebraic Topology This is a difficult book - at least for me - but has an integrated view of the use of calculus in topology. Well worth the pain, at least so far. I am only half way through it. It does not have enough examples or exercises so you have to go even slower and supplement the book with other sources.

Milner, Characteristic Classes This is a great companion to Bott and Tu. It is an advanced book but beautifully written and completely fascinating.

Weeks, The Shape of Space Another book that introduces modern topology and geometry to the uninitiated. Extremely conceptual and elementary. A friend who can't even add two fractions worked through it with no trouble ( except a lot of work).

Baxter and Renni, Financial Calculus This is a book on Derivatives for practitioners. It introduces the ideas of Stochastic Calculus clearly without plunging into the technicalities of the Ito calculus. One learns the math from the "real world" problem of pricing derivatives. Read it if you think you might want to be a math quant on a trading desk.

Klein, On Riemann's Theory of Algebraic functions and their Integrals A short book on Riemann's theory of complex functions. A classic from one of the greats. Importantly, it gives insight into the thought process that led to the theory of Riemann surfaces.

Feller, An Introduction to Probability Theory and Its Applications I can't recommend volume 1 enough. What a joy! Book 2 less so and it also omits Martingales. You could think of this as a back door into real analysis.

Taylor and Wheeler, Exploring Black Holes: An Introduction to General Relativity This book gets your hands dirty right away with the Schwarzschild Metric. It is mathematically simple, but rich in insight. and examples.

Greenberg, Lectures on Algebraic Topology This is an old probably outdated book that I picked up at random. It is very well written and sticks to the basics.

Rudin, Real and Complex Analysis This book is way too hard to use as a text but has great exercises.

 

[sunny79]

Lavinia! Appreciate the input. :)

 

[mathwonk]

Just want to thank Lavinia for the great post!

 

[mathwonk]

@narwhal: see if this flies:

if a simple group of order 168 had an element of order 21, then it would generate a cyclic subgroup of order 21 of course, and that subgroup would be abelian, so would be contained in the normalizer subgroup of every one of its elements. Thus the conjugacy class of each of its elements would have order a factor of 8. But G acts transitively on that conjugacy class, so the order must equal 8, or else we would get a homomorphism from G to a small permutation group, and the kernel would contradict simplicity.

So every non trivial element of the group of order 21is conjugate to exactly 8 elements. But look at an element of order 3 in there. It generates a Sylow 3 subgroup, of which there are either 1,4,7,10,... such subgroups. But there can't be < 7 or we get an action on a small permutation group by conjugation. And there can't be 10 or more since all such subgroups are conjugate, and our element is only conjugate to 8 elements.

I think it follows there cannot be 7 such subgroups, so there must be only 4 such subgroups, a contradiction. I.e. either our element is conjugate to its square or it isn't, hence the number of conjugates should equal either the number of Sylow 3 subgroups, or twice that number. Either way 8 doesn't work.I hope this makes sense. Even if this is right, I would not have solved that during the time given for most tests though, unless I was really in good form.

The moral though is again, it has to follow from Sylow and group actions, usually by conjugation.

 

[waddlingnarwhal]

Hi mathwonk. I've read your solution, and I think everything you said is correct. I will look at it again tomorrow to see if anything sticks out. Thanks for the reply! For now, I would like to offer two short, alternative solutions.

(1) This was how I eventually did it. The cyclic group of order 21 contains one of the Sylow-3 subgroups (call it P) of G and is contained in the normalizer of P. By Sylow's Theorem, n3 = 1, 4, 7, or 28. By orbit stabilizer, n3 = 168 / |N(P)|. Since N(P) contains an element of order 7, we have already divided the 7 out of 168, so n3 = 1 or 4. This contradicts that G is simple.
(2) This is a solution my T.A. proposed. I think it is very nice. G acts on the 8 Sylow-7 subgroups by conjugation. Since the group is simple, this gives an embedding of the group into S8, which contains no element of order 21.

lavina - Your post was great! I am so impressed by the level you've reached studying by yourself. It is very inspirational.

 

[mathwonk]

yes yours is much neater than mine, using one extra fact about n3 i ignored. your TA's solution uses a fact i did not know about S8. I tried a similar proof and got that G embeds in both A8 and A7, but did not know how to finish.

 

[waddlingnarwhal]

I believe that S8 has no element of order 21 because the order of a permutation is the least common multiple of the lengths of the disjoint cycles in the disjoint cycle decomposition.

If anyone is interested, here is another interesting problem I did this term. It is a probability problem.

You have a box of n toothpicks. Each time you pick a whole toothpick, you break it in half, put half back in the box, and throw the other half out. Each time you pick a half toothpick, you throw it out. Thus the box will be empty after 2n steps. At a given step, each whole toothpick has the same chance of being chosen as any other whole toothpick, and each half toothpick has the same chance of being chosen as any other half toothpick. Each whole toothpick is twice as likely to be chosen as each half toothpick. Let H be the number of half toothpicks remaining after the last whole toothpick is chosen. Give a closed-form expression (i.e. involving only factorials, fractions, exponents, binomial coefficients, or products) for P(H = k).

 

[mathwonk]

narwhal: thinking about your group theory problem again, I think your solution is the most natural one. And we really could have solved it quickly by applying the basic principles.

I.e. the fundamental fact that in a transitive group action, the order of the group always equals the product of the orders of the orbit and one isotropy subgroup, implies the most basic fact of all for conjugation actions: the order of the conjugacy class (orbit), equals the index of the normalizer (isotropy subgroup).

Since the most fundamental objects to let G act on are the sylow subgroups we have three cases, those for the primes 2,3, and 7.

Since the normalizers for these have orders which are multiples of 8, 21, 21, respectively, from what is given, the conjugacy classes have orders

dividing 21, 8, 8 (and greater than 1, since G is simple). But they also have (by sylow 3), orders congruent to 1, mod 2, mod 3, and mod 7. thus those orders are factors of 21, 4, 8.

This gives us actions on the conjugacy classes of those orders. But a transitive action on a set of (more than 1 and) less than 7 elements

implies G is not simple.This is your proof, but I see now it follows from pursuing the most basic facts: sylow theorems plus the mantra: "order of conjugacy class equals index of normalizer".

I'm a little out of practice after 4 years of retirement and maybe 20 since I taught the course. But I'm just trying to emphasize that these problems seem to usually follow from the same basic principles.