New Math Website

Hello everyone,

Sorry I have not been around in a while- I have a qualifying exam in 2 weeks that I have been cramming for. To say the least, I have been dreaming about cosets and quotent groups.

I wanted to let everyone know that I am starting a new website dedicated to Math. The specific purpose of the site is to make abstract forms of math more approachable to non-mathematicians, possibly attract High Schoolers and Community College Students to pursue Mathematics in College, and to provide a resource for University students to develope intuition about abstract mathematical concepts. (I plan to use a lot of pictures and animations to accomplish this).

Since this website will be in the service of the community, I was wondering what types of features you would like to see in such a site. How would such material be best represented to you?

For now, I do plan to focus on Topology and Abstract Algebra (and perhaps a sprinkling of Combinatorics), but would like to incorporate other disciplines in as well if I can get the nessasary support from friends.

Any ideas, requests?

Some number theory, real analysis as well. Graph theory as examples of combinatorics, because very accessible (if sometimes difficult).

Graph theory... good idea. I could probably look through old putnam exams for good stuff to post.

1. Turing Machines, ex.

2. Complex Analysis

3. Elliptical eqn's and L functions, try introducing the concepts geometrically.

HEY!!! I didn't even think of Turning Machines!!! One of my favorite topics!

L-functions!? On a page that tries to encourage people to take up math at a University level.. Wouldn't that do the opposite? I know stuff like that is what drew me into mathematics while I was in high school, but most people are afraid of math because it makes them feel stupid..

I am all for the Turing machines and complex analysis (as long as the CA is kept simple).. But how do you explain CA intuitively?

Btw, what about some dynamical systems analysis from a geometrical point of view? Topology also plays a big part in this field.. I could send you (ColdCoffee) my explanation of why linearization works. It is from my masters thesis, and I have tried to explain it intuitively, so that new-comers to the field of dynamical systems would be able to read my thesis as well..

Maybe you should do a sort of rating system, so that visitors will know what level the mathematics they are reading corresponds to. =) Then I would have no problem with L-functions..

I have considered that rating system. I have a SQL server set up with my webhost so I don't think that would be unreasonable (assuming that a criteria can be established). I would appreciate any contribution you can make certainly.

As far as the CA idea. I would probably be inclined to used things like Phasors from electrical engineering to motivate why CA matters and then rely on the Riemann Sphere to help develop intuition about some basic CA function. (Like Mobius Transformations).

BTW, by L-Functions you guys are referring to elements of L-Space from measure theory right?

I meant the l functions of elliptical equations. Ex: the Birch and Swinerton-Dyer Conjecture. The rating system sounds like a great idea.

Ah, ok. I am not yet familiar with elliptical equations in a formal sense. I am sure by the end of it all I will be- as I believe elliptical curves are studied in Low Dimensional Topology.

I have a friend that it into that stuff, I wonder if he has any ideas...

hmmm... correct me if I am wrong but now that I think about it, I this Andrew Wiles used this theory in proving Fermat's last theorem.

Yes you are right. You reference the Taniyama Shamura Conjecture and the dualities between the modular forms and elliptic curves...you should read "Rational Points on Elliptic Curves" by Joseph Silverman (?). Some undergrads are introduced to it.

That sounds very advanced to me! Perhaps undergraduate syllabuses have changed emphasis since Andrew Wiles work. I recall studying Rings and Modules, Group theory, then Galois theory. I passed algebraic geometry because I was an analyst, not an algebraist, but even if I had studied that (and every other relevant course), I think I would still been a fair way short of the basic theory needed for Wiles work.

Riemann sphere! Excellent idea. Also tying in to the depth of modular functions etc, consider some algebraic number theory - lots of depth and interest, just dealing with subrings of the complex numbers.

I dont think most Undergrads see a lot of that material. I am sure many universities have courses in elliptical curves but I would be surprised to find a program which includes a course as part of its core curriculum.

I've seen undergrads study that for independent study sometimes.

Elroch:

Ha! An Analyst! Now we know you're weird for sure

poor analysts, they have the best reputation for being the "weird" mathematicians.

I like it. Admittedly not my area, but I am sure eventually I can dig something up. I plan to dive into more CA this year- call it a new years resolution. I am reading a book right now that has a section dedicated to using complex analysis in Surface Topology. CA is very good stuff. People who don't take it miss out on a very beautiful number system!

Not here, analyists have the reputation of being really nitpickey!! Topologists and Combinatorialists are considered the weird ones!!!

Let's face it. All mathematicians are weird, and trying to pick the weirdest subset is splitting hairs.

On the fascinating subject of the choice of topics, the things that have fascinated me most long since having left research are what might call the nexuses of mathematics, where several subjects collide and give an impression of deep truth. For a popular presentation of one of the most fascinating (in my opinion) collisions, see John Baez on the Monster (the article has something entirely unconnected at the start). Complex analysis is a subject which appears in most of these connections, and is one of the most fascinating and rich areas, without doubt. The subject has a sort of rigidity, where results rarely seem accidental or arbitrary. Simple but powerful facts like one time differentiability implies infinite differentiability and knowledge of the derivatives at a point give the values of the function everywhere are part of the "rigidity". Adding the point at infinity and showing it can be dealt with exactly the same as others is a fascinating simple extension, leading to Riemann Surfaces (hinted at by ColdCoffee), but to take this further is not so suitable, as defining differentiable manifolds properly requires a fair amount of preparation, and is found to be difficult by less experienced students, I believe.

Oh believe me!! I intend to get into manifolds!! That is the good stuff. Actually, I plan to kick it up another notch and get into orbifolds at some point (these articles will not be for high schoolers of course). Orbifolds are a hot research topic in the Differential/Riemannian Geometry realms right now, and for good reason- they are dang cool!

FYI, for all of the audience members listening that are not familiar with orbifolds:

An n-manifold is basically a space in which every point has a neighborhood that is homeomorphic (or diffeomorphic in the case of differential manifolds), to R^n. So if you are a bug living on a manifold and you look all around you, everything looks like R^n, even if the overall space does not look like R^n.

Oh yeah, homeomorphic means "topologically equivalent" it means that there is some mapping out there that preserves open sets, diffeomorphic means that there exists such a mapping which is smooth (infinitly differentiable).

Now! An n-orbifold is basically an n-manifold that has locally been modded out by a finite group (in the case of the trivial group, we just have the plain old manifold, so manifolds are orbifolds). So every point looks like (R^n)/G where G is some finite group.

Very cool stuff.

Dang, I am getting all excited now!