# For all the smart Mathematicians

1. Aug 1, 2004

### woodysooner

For all of you with broad knowledge help me out a little here.

I was given a list of math's to know before i could get a good understanding of Quantum and GR. I have had already some of them.

Here they are,

Can some of you out there tell me about what some of these are, I have never heard of them, and how I should go about learning them, like what book to buy, or books if its really gonna be hard.

Thanx a million for you all the help.

cheers woody

2. Aug 1, 2004

### Janitor

Discrete Group Theory-- I can give you a sketch of that one. Group theory is a branch of abstract algebra. A group is a set plus a rule for mapping pairs of elements (order often counts!) to a unique element within the same set, i.e. a group has closure under binary composition. There are some additional properties that the composition must satisfy. A group can be continuous or discrete. If it is discrete, it either has a finite number of elements, or a countably infinite number of elements.

(If I goofed up anywhere, it would be in my final sentence. Maybe I did not state it carefully enough.)

Last edited: Aug 1, 2004
3. Aug 1, 2004

### Zone Ranger

Advanced Probability theory requires a lot more than calculus. It requires abstract measure theory, real and complex analysis, and sometimes group and ring theory.

4. Aug 1, 2004

### mathwonk

I do not know quite how to take your request, as I am a 60 year old PhD pure mathematician and college professor, and although I specialize in one of the areas you name, I do not know all those topics by any means.

Thus obviously it would take you years and years to "know" all those things, if ever, especially if you are now at the precalculus stage!

If you do not know calculus yet, try Silvanus P. Thompson's Calculus Made Easy. My classmate at Harvard introduced me to that classic in 1960, and I recommend it to any beginner.

[where did you get that list? "global topology: very easy"!!, Oh? Have you tried proving that any simply connected three manifold is a 3-sphere lately?]

The easiest introd. to algebraic geometry is probably the one by Miles Reid. And for lie theory, try the appropriate chapters in the book of Michael Artin on Algebra.

But really, you are not ready for most of that stuff, are you? Just start somewhere, and see what you can learn. trying to absorb a list like that all at once is unrealistic.

5. Aug 1, 2004

### Janitor

I recognized it as being a post at this website, circa May or June of this year.

6. Aug 1, 2004

### woodysooner

no i have had all of calc already and fixin to be in DiffE

7. Aug 1, 2004

### woodysooner

mathwonk

thanx for the help, i was more less asking what to do first, where do i start and what book with. I have had pre calc, calc I II III and startin diffE. studied alil statisitics and prob.

8. Aug 1, 2004

### mathwonk

well forgive my discouraging you. obviously one can learn some of those topics without learning all of them. besides you apparently really want to learn quantum physics and general relativity rather than mathematics, so you might do well to read a book on physics that introduces those topics as needed. the great work on GR when I was a young person was the tome by John Wheeler, Kip Thorne, et al.

As to global topology, cohomology, homology, differential forms, etc.... one reason the assignments of different degrees of difficulty to these puzzles me, is that they are in some sense all the same. I.e. the basic mathematical tools for studying global topology are homology and cohomology, so they cannot be hard and the subject itself be easy.

On another count, the subject of cohomology is actually an abstract version of the theory of differential forms, so if they are easy so is cohomology. In fact the famous theorem of deRham says that on a compact differentiable manifold (curved space) that every cohomology group with real or complex coefficients is computable as a quotient space of differential forms. forms with compact support are also used to define cohomology on non compact spaces.

Homology is the sudy of the connectivity of space- it measures connectivity by asking when a loop or closed surface, or higher dimensional object in that space separates the space, and this is equivalent to asking when the loop say is itself the boundary of another piece of the space.

Riemann invented homology as follows to differentiate between planar and non planar surfaces such as the surface of a doughnut. I.e. he asked about a surface, how many closed loop cuts need to be made before the surface becomes planar?

On the surface of a doughnut one closed loop cut suffices to reduce it to a plane annular region, i.e. the region between two concentric circles. On the other hand it takes two closed loop cuts on the surface of a doughnut to reduce it to a rectangle.

The skin of a doughnut with two holes requires two loop cuts to reduce it to a planar region (like the boundary of a pair of elton john or dame edna eyeglasses) and 4 cuts to reduce it to a region equivalent to a disc such as a rectangle. The homology group of a surface is the group formed by those closed loops that are not boundaries of two dimensional regions in the space. Thus a single holed doughnut has 2 dimensional homology, a 2 holed doughnut has 4 dimensional homology etc...

There are analogous concepts in higher dimensions but lets stick to this case for now.

a cohomology elelement on the other hand is an object that spits out a number when it sees a homology element i.e. a loop. This is exactly what a differential form does when integrated over the loop. So a cohomology element is nothing but a generalization of a differential form.

Thus to understand these concepts you need to study path and surface integration, and especially greens and stokes theorems. There is a simple formal calculation with partial derivatives (equality of mixed partials) that shows that if you start with a 1 form which happens to be a gradient, then the curl of this 1 form is zero, i.,e. the curl of every gradient is zero. A basic differential equation is to consider a 1 form whose curl is zero and ask the opposite question, i.e. is it actually a gradient?

Now a 1 form will in fact be a gradient if and only if it has integral zero over every path. But Greens theorem says that the path integral of a 1 form over a path whoich happens to boiund a region equals the integral of the curl of the 1 form over the region. Hence if the curl is zero, then at elast the 1 form will have zero integral over every path which bounds, i.e. which is zero in hopmology.

So homology is an equaivalence relation on paths wherein two loops or paths are equaivalent if all closed 1 forms have the same integral oievr them. hence the homology of a space is equivalently either the number of loops in the space that do not bound subregions of thbat space, or dually it is the number of closed 1 forms that are not gradients.

The second dual point of view is called cohomology and the first is called homology.

One dimension up, we know the divergence of a curl is always zero, so the analogous question is which 2 forms with divergence = zero are curls? This is related to how many closed surfaces in the space do not bound three dimensional regions, etc...

I recommend since you have studied calculus, to read the book differential topology by Guillemin and Pollack, where your knowledge of calculus will be extended to a grasp of deRham cohomology and differential manifolds.

Another more high powered book after that, would be Differential Forms in algebraic topology, by Bott and Tu, where the de Rham cohomology, i.e. closed differential forms, will be used to study global topology in a very powerful way.

But look also at Kip Thorne, J.A. Wheeler, et al, since it is a physics book.

these are my personal recommendations, but remember I am not a physicist, but a geometer.

Oh, the book considered by my engineering and physics friends as the best introd to quantum mechanics in the 60's was the third lecture of Feynman on introductory physics at Cal Tech.

best wishes,

mathwonk

9. Aug 1, 2004

### mathwonk

oh, and as to algebraic geometry, i suspect this is related to gossip I have ehard tha modern physicists think space is not just old fashioned 4 diemnsional space time, but also has a 6 dimensional factor which is a tiny copy of a "Calabi Yau" complex 3 fold.

Well I believe Calabi Yau 3 folds are (possibly simply connected, i.e. sort of like saying all closed 1 forms are gradients) manifolds on which there is a never vanishing complex 3 form, so it again comes back to some property of (complex) differential forms. The basic example is the hypersurface defined by a degree 5 homogeneous polynomial in 4 dimensional complex projective space, and analogous objects in lower dimensions are the surfaces of degree 4 in 3 space, and the plane curves of degree 3.

There is also a survey of K theory, which is a generalized cohomology theory, by Michael Atiyah on the web somewhere, but it may be a little off the deep end. Basically K theory replaces loops and differential forms by vector bundles, and more general objects called "coherent sheaves" which are quotients of vector bundles by subbundles. I know very little about this.

10. Aug 1, 2004

### mathwonk

oh, and non commutative geometry seems to be derived from operator theory, a branch of real analysis, (higher dimensional linear algebra plus integration theory).

Atiyah also has a little book on a survey of these topics including knot theory, but it is (necessarily) pretty sketchy.

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11. Aug 1, 2004

### woodysooner

thanx mathwonk

Thanx alot for you analysis of those subjects. I would love to be you with all the math knowledge. I like math more than physics, but my father will not let me be a math major, he wants me to be a chem Eng, so i have let's say begged for a double major in Eng Phys. so i can do something more that I want, but it is the MATH the drives the physics that drives all, it's so elegant in form and so beautiful. I envy you.

Thanx for the help.
I may be searching you out in the future for help.
I hope that's ok.

Cheers Woody

12. Aug 1, 2004

### mathwonk

Dear Woody, I am sure your father wants what is best for you, and may be thinking of job opportunities, but if you tell him what you love, I believe he will be flexible. Give him a chance.

I was also very, very set on my oldest son becoming a math major but he stood his ground and found his own way, and we are the best of friends now.

best regards,

roy

PS. it is my pleasure to advise in ny capacity I can, you made my day!

13. Aug 1, 2004

### woodysooner

I will do that with dad but I am gonna ease into it, he was asking me tonight about it actually that's funny. Because it's summer and I am in the kitchen trying to learn complex anyalsis lol. Thanx a whole lot, I'm sure I'll be looking you up soon.

14. Aug 2, 2004

### mathwonk

As to complex analysis there are lots of good books I think. One of my favorites is the one by Henri Cartan (analytic functions of one and several complex variables?), now avaliable in paperback for a very small price. Very elegant.

15. Aug 2, 2004

### shmoe

Hi Woody,

When I started life my dad wanted me to be a banker like he was. When I started university, I went into Engineering and he was satisfied with that, given the job prospects. Mid first year I jumped ship and went into math. He wasn't terribly pleased, but since I was paying for university myself (scholarships+summer jobs) he didn't really get a say. If your dad is worried about job prospects, maybe you could look into what sort of employment people with BSc in math get to try to convince him. It really is an employable degree judging from where my friends have gone after their undergrads. It's also a great stepping stone for graduate school in just about any field. I know one math BSc and one PhD in math who went on to med school and said their analytical skills picked up from their degrees went a long way on getting accepted.

ps. it's summer, you shouldn't be studying complex analysis in the kitchen. You should be studying complex analysis under a shady tree in a quiet park!

16. Aug 2, 2004

### woodysooner

Shmoe, I like you, we could be pals lol.

Thanx for the advice, I have all my school paid for the a big university plus alot so maybe i'll use that against him....

17. Aug 2, 2004

### pnaj

Hey schmoe, my best friend went straight into banking (mergers and acquisitions) after getting his bsc, and is having a very successful career. Your dad would love him, I'm sure!