Should I Become a Mathematician?

[atyy]

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.

1) all the mathematicians I personally know are turkeys.

What about Bourbaki1123's analysis? I don't understand how (ii) is linked to (i) and (iii).

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

Beautiful!

 

[atyy]

mathwonk and uman, thanks! I have just learned what a contrapositive is

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

I saw the movie and loved it. Unfortunately, I don't remember details except what you are supplying, so I am only going on them. I think Cruise needed a confession, or at least something beyond reasoning from the axioms ("that your men always do exactly as you tell them"), since he had not established that the axioms were (experimentally?) true.

 

[uman]

A defense attorney doesn't have to prove innocence, just that there is a reasonably believable possibility if innocence. Thus, unlike in mathematics, an argument may rely on hypotheses that aren't certainly true but that may be.

These were more interesting than the UGa application essays. You should talk to the admissions people... ;-)

 

[mathwonk]

Thank you, I live for the feedback from intelligent readers.

 

[arshavin]

hi mathwonk or anyone,

Are gauss's books, in particular "General Investigations of Curved Surfaces" accessible to someone with only calculus and linear algebra knowledge?

 

[mathwonk]

I have not read this book, but I recall that Michael Spivak had his differential geometry class read this work as part of the course he taught that led to (especially the second volume of) his own book on differential geometry.

I believe he said his class enjoyed, or perhaps was impressed by, Gauss's book. So just plunge in and see for yourself. Or get hold of Spivak's second volume of his opus on diferential geometry.

 

[Bourbaki1123]

Mathwonk,

Have you read Hausdorff's Set Theory? If so, do you feel it gives one a good sense of set theory?

Also, what are the usual prerequisites for learning Category Theory? It seems that the actual material is not too far removed from just understanding homomorphisms, but it also speaks of metric spaces, isometries ect. Since the goal is to generalize mathematical structures, it seems that the need for broad knowledge is substantial.

 

[mathwonk]

i have not read all of hausdorff's set theory but have read some and was quite impressed by its high quality. i recommend reading as much as one finds appealing. in general i recommend reading also einstein, euclid, archimedes, riemann, and other brilliant geniuses.

as to category theory, it is a useful language for most people, and a specialty for a few people. i myself enjoyed reading the book abelian categories by peter freyd, and never read further. as to prerequisites, there are not too many, and abstract algebra should suffice.

 

[Wretchosoft]

Would you recommend taking graduate level classes in mathematics even if you don't plan to go to graduate school? I will be in a position to do so by my senior or possibly junior year, if everything goes well.

 

[mathwonk]

if you do not plan on grad school in math, but love math, then grad courses are still fine if you have completed all relevant undergrad courses.

in general, take what you enjoy. courses are just courses. there is no firewall between grad and undergrad.

 

[Bourbaki1123]

Mathwonk,

Would you say that it would be a good idea for an undergraduate to join the AMS and attend the meetings/talks?

 

[mathwonk]

I don't know what to advise on this. I have not myself been very active in the AMS. They do have a lot of meetings, and there can be some very good talks at them on up to the minute research work. These general talks might be useful for undergrads.

maybe someone knows more abut this than I do. have any undergrads here been to an AMS meeting and enjoyed it, or have any had other positive experiences with the AMS?

 

[qntty]

Mathwonk, after taking Calculus with one and many variables, which university class(es) will typically give you a taste of what Mathematicians do? (I am in High school so I can only take two or three classes per year and have one year to taken them)

the classes that I can take are
Linear Algebra, Diff Eq, More Calculus, Geometry, Probability, Proofs/Logic, possibly Abstract Algebra

 

[mathwonk]

that is hard to answer. college classes are typically presented in a way that does not give the feel for what mathematicians do. basically what they do is discover the stuff you are learning there.

so in a way, what we do is solve problems like the ones in your courses, only harder. or prove theorems like the ones in your courses. to get a feel for what we do, you could take any good course and try to prove the theorems yourself.

there are special seminars and courses some places that are meant to introduce students to research, but not everywhere.

as to specific courses, abstract algebra might give the best idea. or you could try reading a good creative geometry book like thurston's.

Three-Dimensional Geometry and Topology: Volume 1 (Hardcover)
by William P. Thurston (Author), Silvio Levy (Editor)

 

[abelian jeff]

I don't know what to advise on this. I have not myself been very active in the AMS. They do have a lot of meetings, and there can be some very good talks at them on up to the minute research work. These general talks might be useful for undergrads.

maybe someone knows more abut this than I do. have any undergrads here been to an AMS meeting and enjoyed it, or have any had other positive experiences with the AMS?

I went to MathFest two years ago, and it was amazing. I'll be attending the JMM next week, and if I remember, I'll let you know how it was, but my guess is that it, too, will be amazing. (I'm currently a senior undergraduate.)

 

[samspotting]

In my university the pure math degree does not require any applied math courses like ode's or pde's. Are understandings of these fields that you get in applied math courses good for a pure mathematician?

 

[Bourbaki1123]

I don't think so, unless you want to do analysis, then maybe a rigorous treatment of them, but ode's are pretty much strictly application. I have found that the only reason to take these courses if you are not going into applied math is that many grad programs want you to have taken them.

 

[mathwonk]

i would say that ode and pde are not at all applied courses, just courses with important applications. some of those applications are in pure math subjects like topology and differential and algebraic geometry.

wouldn't you say differential equations were the key to perelmans recent solution of the poincare conjecture?

i know the complex heat equation is just crucial in much beautiful work on moduli of abelian varieties.

 

[samspotting]

The problem is that I don't have enough time to take everything, with the pure math major I can either do a load of combinatorics classes (graph theory and enumeration) which look interesting, or a few courses in the applied math department:

Calculus 4
Vector integral calculus-line integrals, surface integrals and vector fields, Green's theorem, the Divergence theorem, and Stokes' theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction to Fourier analysis. Fourier series and the Fourier transform. Parseval's formula. Frequency analysis of signals. Discrete and continuous spectra. [Offered: F,W,S]

Introduction to Differential Equations
Physical systems which lead to differential equations (examples include mechanical vibrations, population dynamics, and mixing processes). Dimensional analysis and dimensionless variables. Solving linear differential equations: first- and second-order scalar equations and first -order vector equations. Laplace transform methods of solving differential equations. [Offered: F,W,S]

Ordinary Differential Equations 2
Second order linear differential equations with non-constant coefficients, Sturm comparison, oscillation and separation theorems, series solutions and special functions. Linear vector differential equations in Rn, an introduction to dynamical systems. Laplace transforms applied to linear vector differential equations, transfer functions, the convolution theorem. Perturbation methods for differential equations. Numerical methods for differential equations. Applications are discussed throughout. [Offered: F,S]

Partial Differential Equations 1
Second order linear partial differential equations - the diffusion equation, wave equation, and Laplace's equation. Methods of solution - separation of variables and eigenfunction expansions, the Fourier transform. Physical interpretation of solutions in terms of diffusion, waves and steady states. First order non-linear partial differential equations and the method of characteristics. Applications are emphasized throughout. [Offered: W,S]

Partial Differential Equations 2
A thorough discussion of the class of second-order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations -- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations.

 

[mathwonk]

calc 4 maybe?

 

[samspotting]

I'll talk to the profs about it, I don't really know much about differential equations except the part in calculus 2 where you solve these really simple first order linear equations that you had to build from confusing and complex word problems.

Differential equations deal with change, I have heard from one professor that pure mathematics does not care so much about solutions to differential equations as to whether solutions exist.

 

[mathwonk]

here is a statement: a compact manifold on which there exists a smooth function with exactly two critical points, which may be assumed non degenerate, is a sphere.

how would you go about proving this? consider the flow given by the gradient of the function, and use the solutions structure theorem for odes.

 

[samspotting]

Ill check it out once I take calculus 3 (this semester). Thanks.

Oh btw, I'm entering a version of calculus 3 where multivariable calculus is taught from a more rigorous perspective. I never took the more rigorous version of the single variable calculus, is there a quick primer somewhere?

I'm only done up to sequences in rudin's book, I feel like I am entering this course really unprepared.

 

[altcmdesc]

samspotting, I'm currently taking a class similar to that. If yours will be any thing like mine, learn some linear algebra while you're at it (be sure you're comfortable working with matrices and linear mappings, even at the basic levels).

Also, if you haven't covered basic point-set topology (interior, boundary, limit points, open/closed balls) or vector geometry cover that too.

 

[Linear Space]

Mathwonk,

Would you say that Spivak's Calculus fully prepares you for his Calculus on Manifolds?...or What else do you need?

Also, after both of these books I plan on tackling Spivak's Differential Geometry series. I noticed that you said in the prereq to diff geometry thread that any amount of algebra and topology will only broaden your knowledge. Where within this sequence of books would you recommend picking this stuff up?

Thank you.

 

[arshavin]

What is the typical session length(in terms of weeks) at a standard university in the US or UK?


In Australia we have 2 semesters (12 weeks each) and we have a summer break of almost 4 months (during which no maths courses are offered).

This just seems completely ridiculous to me.

Is it the same everywhere else?

 

[snits]

I imagine most universities in the US offer math courses during the summer session.

 

[mathwonk]

yes spivak's pwn calculus is adequate preparationm for his calc on manifolds.

then volume one of his diff geom is chock full of great topology and basic manifolds.

a little heavy going in the basic theory of manifolds.

the problems are also wonderful, and the extra chapters on lie groups and de rham cohomology are terrific.

just volume one is very valuable information on manifolds and cohomology.

then volume 2 is the world's best historically oriented but modern version of an analysis of the most important gadget in diff geom, the curvature tensor.

those are the only two volumes i own. i am tempted by volume 5 i think it is, on the chern version of the gauss bonnet theorem?

but i myself have little time for learning now. maybe soon.

 

[uman]

Mathwonk or others,

Long ago I was reading both an Algebra book (Artin) and a Mathematical Analysis one (Apostol). I had to take the Algebra book back to the library, so I've since been doing only Apostol's book. I'd like to get back into Algebra to give me a change of pace, but I don't have access to Artin's book; my local library now only has Dummit and Foote (long story). Is it worth buying the Artin book (which I really liked) or will D&F suffice? Is it accessible to someone who doesn't know much about Algebra?

 

[desti]

Dummit and Foote is very easy to follow. It has loads of examples and the exercises vary from trivial to moderately hard, so it's easy to find toy problems to test understanding of definitions. I think its biggest downside is that it doesn't have any harder exercises. There is some stuff in Artin that you won't find in Dummit and Foote though. There's a very brief intro to algebraic number theory in Artin and then there is the section on wallpaper symmetries. It's hard to say which is better though. Artin also focuses a lot more on connections with linear algebra by using matrices as examples for almost anything.

If you consider your "mathematical maturity" as ok (i.e. you're comfortable writing proofs), I would recommend reading Herstein's, Topics in Algebra, for a quick intro to group theory and then go straight into e.g. Lang's Algebra and use Dummit and Foote mainly for examples and easy exercises if you get stuck. This gets you quicker to advanced material. If you read Lang, I would recommend that you teach yourself LaTeX and write your own account of any major theorems after you've read about them in Lang. It's slow and takes time, but when you go back to the stuff later, you'll notice that having your own notes makes you recall the stuff a lot faster.

 

[abelian jeff]

I went to MathFest two years ago, and it was amazing. I'll be attending the JMM next week, and if I remember, I'll let you know how it was, but my guess is that it, too, will be amazing. (I'm currently a senior undergraduate.)

So, the Joint Math Meetings were awesome. I highly advise any undergraduates who can get funding (or pay their own way) to attend MathFest and JMM. Both are great experiences.

 

[rodigee]

I (undergrad) went to the JMM this year as well to present a poster. I'll echo what Jeff said. Overall, it was quite fun. No matter your interests, you should be able to find a session of talks about something intriguing. However, understanding the talks may be another matter all together.

I really enjoyed seeing D.C. as well.

 

[uman]

desti: thanks. I'm not sure how "mathematically mature" you would consider me: I got through five chapters or so of Artin without too much trouble before I give it back. I guess I'll look into Herstein if you think D&F is too trivial... In any case having multiple resources can't hurt.

Anyone else have an opinion on this?

 

[jacksonpeeble]

Hell no. Maths and economics majors know jack about maths either pure or applied. Economists struggle to add up, never mind do maths properly (including applied maths).

I find this entertaining (because it's absolutely true); my economics teacher seems clueless about math in general, much less the complicated things.

Besides, economists mess up all the time. Mathematicians don't seem particularly apt or happy to make errors.

 

[PhysicalAnomaly]

Does that mean that if I study some stochastic and financial maths units, they'll hire me to do their sums? =P