Other Should I Become a Mathematician?

[QuantumP7]

It's not going to impress Harvard

I'm curious. What WOULD impress Harvard or MIT or the other top math programs?

 

[Calabi_Yau]

This is an interesting thread. I'm a freshamn in college, studying Physics but right now I'm seriously pondering about switching to a maths degree. I have always been good at math, and every math teacher I had, told me I was talented at it. However, I got into physics mainly because I read 3 years ago Kaku's Parallel Worlds, and having watched many science tv programmes about the marvels and excentricities of the cutting edge theories in theoretical physics I decided that it was that I wanted to do.

Recently, I have read the book "The Man Who Loved Only Numbers" which portraits the life of the great mathematician Paul Erdös, and my attentions shifted to math again. Basically, when I read about maths I want to become a mathematician and when Iread the lectures of Feynman I want to become a phycist again. So I guess I'll be working on something related with mathematical physics.

The problem is that I don't know whether I should better major in physics and minor in math, or do the opposite instead, since in my country it's impossible to double major at once. Porbably I'm majoring in Physics, with a minor in maths concerning some topics about abstarct algebra, differential geometry and galois theory. But I really don't know. That's my story so far lol, I'd like to read about those who are passing through the same, or already have. It seems I will only get an answer through personal experience.

 

[Mandelbroth]

This is an interesting thread. I'm a freshamn in college, studying Physics but right now I'm seriously pondering about switching to a maths degree. I have always been good at math, and every math teacher I had, told me I was talented at it. However, I got into physics mainly because I read 3 years ago Kaku's Parallel Worlds, and having watched many science tv programmes about the marvels and excentricities of the cutting edge theories in theoretical physics I decided that it was that I wanted to do.

Recently, I have read the book "The Man Who Loved Only Numbers" which portraits the life of the great mathematician Paul Erdös, and my attentions shifted to math again. Basically, when I read about maths I want to become a mathematician and when Iread the lectures of Feynman I want to become a phycist again. So I guess I'll be working on something related with mathematical physics.

The problem is that I don't know whether I should better major in physics and minor in math, or do the opposite instead, since in my country it's impossible to double major at once. Porbably I'm majoring in Physics, with a minor in maths concerning some topics about abstarct algebra, differential geometry and galois theory. But I really don't know. That's my story so far lol, I'd like to read about those who are passing through the same, or already have. It seems I will only get an answer through personal experience.

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory. Differential geometry, on the other hand, is my candidate for a foundation for modern physics. Manifolds are an important part of contemporary studies of physics, so you will definitely want to take that. All three of them are beautiful subjects with many aesthetically pleasing results, though, so if you really like mathematics I would definitely advise taking all three.

I used to think I wanted to be a doctor of medicine. Then, I figured out that the real world is kind of boring to study. Math is where it's at. If you are really considering going into mathematics, I think you should go the distance.

 

[R136a1]

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory.

One thing you can study is coding theory. You'll see how things like finite fields and ideals are applicable to generate good codes.
For (finite) groups, they are very applicable in chemistry. Just google it and you'll find a lot of hits.

 

[George Jones]

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory. Differential geometry, on the other hand, is my candidate for a foundation for modern physics. Manifolds are an important part of contemporary studies of physics, so you will definitely want to take that. All three of them are beautiful subjects with many aesthetically pleasing results, though, so if you really like mathematics I would definitely advise taking all three.

The combination of abstract algebra and differential geometry is extremely important in theoretic physics. Continuous symmetries (both spacetime and "internal" symmetric in quantum field theory) are modeled by representations of Lie groups, which are groups that are both groups and differentiable manifolds, with the group operations being differentiable.

In fact, right now, I am reviewing the relationship between the spacetime Poincare group, its Lie algebra, and relativistic wave equations.

 

[IGU]

One consideration for you might be that you can't do physics without math, but you can do math without physics.

 

[Calabi_Yau]

One consideration for you might be that you can't do physics without math, but you can do math without physics.

That is correct, but I think those who start in physics can change to maths easier than those who start in maths can change to physics. That is, in my opinion, because during a physics course you acquire the basics and the the skills necessary to do maths (although with less rigour). But if you finish maths and want to pursue physics, you'll have a greater deal to catch up, you may be an ace in mathematics but know nothing about the underlying principles of mechanics or electromagnetism for example.

 

[deluks917]

I'm curious. What WOULD impress Harvard or MIT or the other top math programs?

For Grad school doing well on the Putnam is considered very impressive by about half the professors at top schools. The other half think it looks good but is somewhat overrated.

However impressing half the professors at these schools is pretty likely to do a lot for your admissions chances. However doing well on the Putnam is exceedingly difficult.

 

[Seydlitz]

This is perhaps already asked before, so excuse me if I have not conducted a search beforehand in the thread, but my question is this:

How can one prepare for international sort of competition like Putnam, and IMO? In this case, I don't even dare to think to solve the majority of the problem, I just want to know what topics or what one should learn in order to be able to solve at least one or two questions in the competition, considering that their level are significantly higher in comparison to ordinary math problems given in textbook and day to day activity?

To deliver the point further, I don't even understand what is being asked by the problems (I just skimmed through one Putnam past paper.) I've never learned formal math so to say beyond application of calculus in high school and A-Level, but when I read through physics olympiad question I know at least what the question means even though I don't know the answer to it.

Can these advance problem-solving skills be learned? Again, I don't even think of participating in those competitions, but I'm hoping to learn some of the skills that could be eventually useful in my university study.

 

[QuantumP7]

I asked a Putnam Fellow this question. He said that the best way to do really well on the Putnam is to practice. Go over the old questions, and practice a lot! I'm going to do this all of 2014, and take the Putnam in December 2014. I'll let everyone know how it turns out.

 

[Cod]

Not sure if its been posted, but here is a link a lot of math and computer science book reviews (more in-depth than an "everyday book" review) done by multiple university professors from around the globe: http://www.cs.umd.edu/~gasarch/bookrev/bookrev.html

The focus is more on CS, but there are a good bit of math books.

 

[superduck]

about studying mathematics: questions

Hello,
I am a Japanese student of university. I am a philosophy of science major. But, to tell the truth, I really want to be atheoretical physicist. Unfortunately I have big lack of mathematics and everything academic skills because of I have got a mental illness sinse I was a high school student. But, I'll never give up my deam to be a theoretical physicist. Then, I am studying mathe matics by myself ( I am in correspondence course). I have to start from high school level math. You recommended several books. It is very helpul. But, I want to ask you about geometory textbook. Are thete any good books? At the moment, I am thinking to use "Foundation Mathematics" by K Stroud. Do you know this book? If you know this, is this book useful for studying high school math? I have another question. Which is better way to study mathematics, to use a thick multiple textbook like which is carried algebra and trigonomketory and geometory etc,or separated books which is carried one topic specially?
I'll be grad if you answer me.

 

[matqkks]

Use Engineering Mathematics Through Applications by Singh because it has complete solutions online to all the questions.
For Linear Algebra use 'Linear Algebra Step by Step' by Singh. Again it has complete solutions to all the problems in the book so ideal for distance learning.

 

[TheKracken]

Hello, currently I am at a community college and after tons of reading and thinking I have decided I want to be a math major.
Anyways, I also want to join the military for one term (usually 4 years), this is a something I want to complete for many reasons including the honor, the family tradition and just in general feeling responsible for serving my country.

Would it be best to join now that I have 15 college credits and would go up a rank or would it be better to join after college when I would be an officer. My goal would be to go back to academics and possibly get a Phd in pure mathematics, but I feel like a 4 year term in the service would cause me to forget most of the material.

Does anyone have anything to say about this topic? I have also considered going the NSA route to serve my country, but it just isn't the same.

Thank you everyone.

 

[homeomorphic]

Would it be best to join now that I have 15 college credits and would go up a rank or would it be better to join after college when I would be an officer. My goal would be to go back to academics and possibly get a Phd in pure mathematics, but I feel like a 4 year term in the service would cause me to forget most of the material.

I'm not sure, but throwing 4 years of military service would be making an already extremely difficult path even more difficult. You need recommendation letters to get into grad school. That could be tricky if the last time you took a math class was 4 years ago.

I'm not sure you forget all the material, though, if you know what you are doing when you learned it. I can't comment much on math, since I never stopped doing it, but I'm working on programming a game right now, and I basically can still program, even though I didn't really do any programming for the last 8 or 9 years. So far, I've barely had to look anything up. That's from taking 3 programming classes. Sure, I'm a bit rusty on some stuff that I haven't had to use yet, but I'm sure it will come right back. Plus, programming is not one of the subjects I did the best job of learning--most of the stuff I've forgotten could probably be attributed to lack of understanding of the motivation (i.e. what's the point of object-oriented programming, and how does it help you in concrete situations?). With the basic stuff like iteration and functions, it's easier to remember because you see why it's useful and as soon as you think about writing a program that does this or that, the need for them is obvious--that, and because it's simpler, and you use it over and over again if you take the next couple CS classes. I actually think taking a break from programming after the first two classes and then having to remember it later when I took data structures is one reason why I still remember a lot of it now. It almost seems like I know it better than when I was taking that data structures class, having to remember back to the previous class a couple years earlier. When you have to work to remember, that's one of the things that implants things in long term memory more firmly.

You just have to have a good strategy for making it stick. Learn how long term memory works. If you really know how to learn, the knowledge lasts a lot longer. So, that could be a possible solution, if you can figure out that puzzle of how to make the best use your own mind.

 

[drewb]

Wow! This thread is really comprehensive... and humbling. I have a long way to go if I want to become a mathematician.

I'm just finishing up my BS in Astrophysics. I'm thinking about making a thread asking for advice on what to do next. :P

Thanks for all this!

 

[mathwonk]

bless you. and good luck!

 

[MarneMath]

Hello, currently I am at a community college and after tons of reading and thinking I have decided I want to be a math major.
Anyways, I also want to join the military for one term (usually 4 years), this is a something I want to complete for many reasons including the honor, the family tradition and just in general feeling responsible for serving my country.

Would it be best to join now that I have 15 college credits and would go up a rank or would it be better to join after college when I would be an officer. My goal would be to go back to academics and possibly get a Phd in pure mathematics, but I feel like a 4 year term in the service would cause me to forget most of the material.

Does anyone have anything to say about this topic? I have also considered going the NSA route to serve my country, but it just isn't the same.

Thank you everyone.

I just saw this and figured I would comment on this. I did nearly the same thing. I had 15+ college credits prior to college and after being thrown out of my first college I ended up in the military for 7 wonderful years. No sarcasm in that statement. I loved my job and would've stayed in longer if I was physically able too. First, I would heavily advise against joining after college and between a PhD. You want your recommendation writers to actually remember you and not struggle to recall what they liked about you four years ago. Secondly, it's much much much more difficult to recall four years of undergraduate mathematics than perhaps a semester or two of calculus. Thirdly, the longer you delay a PhD, the more life throws at you. A PhD becomes less and less attractive (at least for me) after you spent x amount of years working making money and living on your own. The prospect of giving that up and to struggle in subject you barely remember isn't very enticing. Lastly, it makes no sense to go to college, get a degree, do a job a high school student can do and then go to a graduate school in a weaker position than before. If you really have intention to serve in the military after college, do it as an officer or at worse in the national guard. I have met quite a few engineers, one lawyer, and one PhD Chemistry student who are in the guard for one reason or another, so it is do-able, but it will eat up your time on certain years.

 

[dkotschessaa]

I think I have been reading this thread since I went back to school in 2011. I am getting my B.A. in May and going to Grad school next fall. Thanks Mathwonk for your encouragement and advice. I will still need it!

-Dave K

 

[mathwonk]

you will be fine dave k. they would not have taken you unless they had confidence in you. besides we know you better from your history here, and you are a proven quantity.

 

[Supremo]

@Mathwork,

I know you are familiar with Spivak Calculus, how about N.Piskunov Calculus. I have read somewhere in Physics Forum that Piskunov Calculus is a great book that it has chapters uncommon to other calculus book, and do not cover topics just to cover them just like some author.

Sir can you give a detailed difference (advantage) of using one over the other between Piskunov and Spivak's Calculus book for college freshman. How about the topic presentation (discussion), is it ideal for young student of average level in math. Which one is the better choice between the two.

 

[mathwonk]

i am sorry, i have not seen piskunov. is it online? but in general, it is not important which of several great books you use, just get hold of one and start thinking and working.

 

[x BlueRobot]

You mentioned there is three branches of Mathematics, what about the other areas such as Set Theory, Number Theory and Graph Theory etc.?

Graph Theory is my favorite area of Mathematics, I find Analysis and Calculus rather boring and dull.

 

[homeomorphic]

You mentioned there is three branches of Mathematics, what about the other areas such as Set Theory, Number Theory and Graph Theory etc.?

I'm guessing mathwonk probably said topology/geometry, analysis, and algebra. Those were just broad areas. Number theory could be considered part of algebra, and you'd see why if you studied enough algebra and number theory.

I think you might have to add combinatorics (including graph theory) and logic/foundations (including set theory) to the list. But with those 5, I think you could probably cover everything in broad strokes. But names and compartmentalization aren't that important, anyway. Who cares? They are just names.

Graph Theory is my favorite area of Mathematics, I find Analysis and Calculus rather boring and dull.

If you find calculus/analysis boring and dull, that hints to me that you might have a less than complete understanding of it. I wouldn't blame you for thinking certain aspects of calculus are dull, but if you look at the bigger picture and some of its applications, it's pretty nice. We shouldn't be so spoiled and judge a subject so harshly because of a few silly trigonometric integrals that we might have had to put up with. Although I'm a topologist, or maybe I should say, a failed topologist, I might have to say graph theory is arguably my favorite subject, too.

Of course, I think almost every subject in math is too complicated for my tastes once you get to research-level stuff because I don't have time to understand everything clearly and understanding clearly is exactly what I like about math (which is why I like elementary analysis, at least). The struggle for understanding is good, too, but it gets to be too much like pitting an ant against the US army at some point, in terms of that battle for understanding at the level that I desired. I at least have to have a fighting chance for it to be enjoyable. This is coming from someone with a PhD in math, so I don't think I'm the only ant out there--I think all mathematicians have become ants with respect to the entire subject. A few Fields medalists might be like big queen ants or something, but they are still awfully puny and pitiful compared to the whole subject.

 

[x BlueRobot]

I understand what you mean about Number Theory, and I do like Series within Calculus/Analysis and some other little bits and pieces of the area, but overall Analysis doesn't really appeal to me as much as other areas like Graph Theory.

 

[mathwonk]

go with what you enjoy! the fun is the main motivation in math.

 

[dkotschessaa]

Man, feel like I'm doing a 180. (Colloquially speaking). I was going down the "pure math" road. Now I am doing a project in mathematical oncology and it's really fascinating. http://moffitt.org/research--clinical-trials/research-disciplines/departments/integrated-mathematical-oncology

I'm working on differential equations (I did not enjoy this subject when I first took it) modeling of tumor growth (I have no biology background).

You just never know what's going to happen do you?

-Dave K

 

[mathwonk]

I also disliked diff eq until I read the books of martin braun , hurewicz, guterman and nitecki, and especially arnol'd.

 

[dkotschessaa]

I also disliked diff eq until I read the books of martin braun , hurewicz, guterman and nitecki, and especially arnol'd.

Thanks. I actually hated the subject so much I got rid of all materials relating to it (some good ones too) saying "I think I'll just avoid that subject for the rest of my math career." I feel very silly about that now. I just found the guterman one for 20 cents on amazon and purchased. ($4.00 shipping of course).

Regards,

Dave K

 

[dkotschessaa]

Just got Gutermen and Nitecki. If this book had been my introduction to Differential Equations, my perspective would have been much different.

I am finding all too often that my experience of a particular area of mathematics has to do with how I am introduced to it. Kind of a shame in a way.

 

[dkotschessaa]

Most spot-on quote from my cousin, a geophysicist.

"Anything worth doing research-wise, requires you to learn a bunch of stuff that you've never thought about before just to get to the point where you understand the problem you're trying to solve."

I keep reading it over and over...

 

[x BlueRobot]

Most spot-on quote from my cousin, a geophysicist.

"Anything worth doing research-wise, requires you to learn a bunch of stuff that you've never thought about before just to get to the point where you understand the problem you're trying to solve."

I keep reading it over and over...

Science and Mathematics is like solving lots of little problems, which eventually lead to a solution to a much larger problem.

 

[dkotschessaa]

I'm not sure if it's kosher to post in a thread asking for another thread to be answered. But darn the torpedoes, if anyone, including mathwonk, could look at my recent thread First Year Math graduate school - Full of possibility!. I post a lot of threads that don't get a reply, so I'd like if this is not one of them.

Thanks

Dave K

 

[x BlueRobot]

I never used to like Pure Mathematics, but when I started to think of it as more of a puzzle, then I started to enjoy the subject much more.

 

[homeomorphic]

I never used to like Pure Mathematics, but when I started to think of it as more of a puzzle, then I started to enjoy the subject much more.

I would agree with that. I don't mind my math being pure if it's something I can sit down and solve in a couple hours, or even a few days, or up to a week or two. As a brain-teaser it's great. But if it's something I have to devote my life to and read hundreds and hundreds of pages of stuff (plus, trying to get by with a minimal conceptual understanding of it, since there isn't time to really process it all) and work on stuff that takes several years to solve, that's where the problem comes. At that point, I have realized, I need external motivation beyond it being having a brain-teaser to solve. It took me a PhD in pure math to realize that.

Also, a lot of the point isn't just solving puzzles, but wrapping your head around cool ideas.

 

[x BlueRobot]

I given myself a little research project on Bull Graphs, I've been doing labeling problems for the last few days. I'm going to check my conjecture that Bull Graphs can be prime labeled tomorrow.

What area of Mathematics do you specialize in?

 

[mathwonk]

remember, EVERYTHING is interesting if seen in the right perspective, which usually comes from an introduction at the hands of someone who loves and understands it.

 

[homeomorphic]

What area of Mathematics do you specialize in?

Not sure if you mean me or mathwonk. Mathwonk does algebraic geometry. I wrote my dissertation in topological quantum field theory, but I don't really do much math any more. I'm just trying to become a programmer, now. To the extent that I still do math, it's mostly probability, statistics, financial math, theoretical computer science, and a tiny bit of graph theory, and very much on the applied side of it all (and nowhere near research level). I've been meaning to learn more about numerical methods but haven't gotten around to it, yet. So much to do, so little time. The idea of specializing in something is sort of repugnant to me.

 

[Philip]

Hypertorus Exploration and Mathematics

Well, I'm certainly not sure what I want to do. I've been a professional bike mechanic for 14 years ( the kind you pedal ), and I've made some interesting mathematical discoveries. Lately, in the last 7 months, it seems that I have acquired knowledge of far-reaching things, into unexplored territory. These things are known about theoretically, but I'm not sure to what extent.

And, it seemed by accident, or pure chance that I would be able to learn it. On a quiet forum, somewhere on the internet, is a funny-looking notation system, called toratopic notation. It was made by others, before I joined. It just so happens to stand for the equations of multidimensional toroids. By reducing the implicit surface equation for, say a circle, sphere, and torus, we can get something like:

circle : (II) : x^2 + y^2 - R1^2 = 0

sphere : (III) : x^2 + y^2 + z^2 - R1^2 = 0

torus : ((II)I) : (sqrt(x^2 + y^2) - R1)^2 + z^2 - R2^2 = 0

The end result is a combinatoric sequence of surfaces of revolution. By repeating these rotations into various hyperplanes, one can build a notation sequence for a shape, which converts into an implicit surface equation.

Going beyond 3D, we have many more possible shapes, per dimension:

4D:
(IIII) - x^2 + y^2 + z^2 + w^2 - R1^2 = 0
((II)II) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
((II)(II)) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0
((III)I) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
(((II)I)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0

5D:
(IIIII) - x^2 + y^2 + z^2 + w^2 + v^2 - R1^2 = 0
((II)III) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 - R2^2 = 0
((II)(II)I) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 - R2^2 = 0
((III)II) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 - R2^2 = 0
(((II)I)II) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2 - R3^2 = 0
((III)(II)) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2^2 = 0
(((II)I)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3^2 = 0
((IIII)I) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 - R2^2 = 0
(((II)II)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2 - R3^2 = 0
(((II)(II))I) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 - R3^2 = 0
(((III)I)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 - R3^2 = 0
((((II)I)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 - R4^2= 0

6D:
(IIIIII) - x^2 + y^2 + z^2 + w^2 + v^2 + u^2 - R1^2 = 0
((II)IIII) - (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2 + u^2 - R2^2 = 0
((II)(II)II) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 + u^2 - R2^2 = 0
((II)(II)(II)) - (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2^2 = 0
((III)III) - (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2 + u^2 - R2^2 = 0
(((II)I)III) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 + v^2 + u^2 - R3^2 = 0
((III)(II)I) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 - R2^2 = 0
(((II)I)(II)I) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2 - R3^2 = 0
((III)(III)) - (sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 - R2^2 = 0
(((II)I)(III)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2 + u^2) - R1b)^2 - R3^2 = 0
(((II)I)((II)I)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3^2 = 0
((IIII)II) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2 + u^2 - R2^2 = 0
(((II)II)II) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2 + u^2 - R3^2 = 0
(((II)(II))II) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 + u^2 - R3^2 = 0
(((III)I)II) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2 + u^2 - R3^2 = 0
((((II)I)I)II) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 + u^2 - R4^2= 0
((IIII)(II)) - (sqrt(x^2 + y^2 + z^2 + w^2) - R1a)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R2^2 = 0
(((II)II)(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3^2 = 0
(((II)(II))(II)) - (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3^2 = 0
(((III)I)(II)) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1a)^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R3^2 = 0
((((II)I)I)(II)) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + w^2) - R3)^2 + (sqrt(v^2 + u^2) - R1b)^2 - R4^2 = 0
((IIIII)I) - (sqrt(x^2 + y^2 + z^2 + w^2 + v^2) - R1)^2 + u^2 - R2^2 = 0
(((II)III)I) - (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
(((II)(II)I)I) - ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + v^2 - R2)^2 + u^2 - R3^2 = 0
(((III)II)I) - (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
((((II)I)II)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2) - R3)^2 + u^2 - R4^2= 0
(((III)(II))I) - ((sqrt(x^2 + y^2 + z^2) - R1a)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R2)^2 + u^2 - R3^2 = 0
((((II)I)(II))I) - ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 - R3)^2 + u^2 - R4^2 = 0
(((IIII)I)I) - (sqrt((sqrt(x^2 + y^2 + z^2 + w^2) - R1)^2 + v^2) - R2)^2 + u^2 - R3^2 = 0
((((II)II)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2= 0
((((II)(II))I)I) - (sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0
((((III)I)I)I) - (sqrt((sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 ) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2= 0
(((((II)I)I)I)I) - (sqrt((sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3)^2 + v^2) - R4)^2 + u^2 - R5^2 = 0

As you can see, these are very large and complex surfaces of revolution, the basic concept behind a hypetorus. The number of hypertori in each dimension is 1,1,2,5,12,33,90,261, etc, which is the A000669 integer sequence on the OEIS. These equations and notations are defining discrete hypershapes in an n-dimensional Euclidean plane.

Once the equation is derived, one can reduce it to a 3D equation, as a cross section of the hypertorus. Then add rotate and translate parameters to move the slice around. I put these enormous functions into a great program, CalcPlot3D. It handles 3D implicit graphing quite well. While exploring the various functions for a shape, I'll see fascinating things happening all the time. That notation system can be used to interpret cross sections abstractly, too. By removing the uppercase " I " you make a cut, by setting that dimension to zero. Take 6D hypertorus (((II)I)((II)I)) for example:Dimensional Map of (((II)I)((II)I)) Hyperplane Intercepts

XYZWVU 6D Hyperplane
(((II)I)((II)I)) - 1x tiger duotorus
-------------------------------------
XYZWV 5D Intercepts
(((II)I)((I)I)) - 2x tigritoruses (((II)I)(II)) in 1x1x1x2x1 vert column
(((II)I)((II))) - 2x tigritoruses (((II)I)(II)) in major1 concentric pairs
--------------------------------------
XYZW 4D Intercepts
(((I)I)((I)I)) - 4x tigers ((II)(II)) in 2x1x2x1 vert square
(((II)I)((I))) - 4x ditoruses (((II)I)I) in 1x1x1x4 vert column
(((II))((II))) - 4x tigers ((II)(II)) in concentric maj1/maj2 pairs
(((I)I)((II))) - 4x tigers ((II)(II)) in maj2 concentric pairs in 2x1x1x1 line
(((II)I)(()I)) - empty
--------------------------------------
XYZ 3D Intercepts
(((I)I)((I))) - 8x torii ((II)I) in 4x1x2 vertical rectangle
(((II))((I))) - 8x torii ((II)I) in 2 concentric maj pairs along 1x1x4 vertical column
(((I)I)(()I)) - empty
(((II)I)(())) - empty
((()I)((II))) - empty
----------------------------------------
XY 2D Intercepts
(((I))((I))) - 16 circles in 4x4 squareUsing the notation sequence (((II)I)((II)I)), the implicit equation can be derived like this:

(((II)I)((II)I)) = 0
((II)I)((II)I) = 0
((xy)z)((wv)u) = 0
((x + y) +z) + ((w + v) +u) = 0
((x + y -R1a) +z -R2a) + ((w + v -R1b) +u- R2b) -R3 = 0
((x + y -R1a)^2 +z -R2a)^2 + ((w + v -R1b)^2 +u- R2b)^2 -R3^2 = 0
((sqrt(x + y) -R1a)^2 +z -R2a)^2 + ((w + v -R1b)^2 +u- R2b)^2 -R3^2 = 0
((sqrt(x + y) -R1a)^2 +z -R2a)^2 + ((sqrt(w + v) -R1b)^2 +u- R2b)^2 -R3^2 = 0
(sqrt((sqrt(x + y) -R1a)^2 +z) -R2a)^2 + ((sqrt(w + v) -R1b)^2 +u- R2b)^2 -R3^2 = 0
(sqrt((sqrt(x + y) -R1a)^2 +z) -R2a)^2 + (sqrt((sqrt(w + v) -R1b)^2 +u) -R2b)^2 -R3^2 = 0
(sqrt((sqrt(x^2 + y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) -R1b)^2 +u^2) -R2b)^2 -R3^2 = 0

Establish diameter values for non-intersection, and make 3D equation from cut (((II))((I))) :

(sqrt((sqrt(x^2 + y^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2)^2 + 0^2) -1)^2 = 0.4^2

Establish rotate + translate parameters:

(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2 - 0.3^2 = 0

 

[homeomorphic]

So, what kind of future can this hold? Are there any other groups investigating this field of discrete high dimension geometry? I know Algebraic Topology is a related field, and I would love to be able to write the language for these shapes and concepts. It's pretty wild stuff!

I'm not sure, but it seems like a bit of a stretch to say algebraic topology is a related field. You could certainly apply algebraic topology techniques to the figures to tell them apart topologically (or homotopically), but that seems pretty tangential to what you're doing, and modern algebraic topologists generally seem to work on very algebraic stuff, like cobordism spectra, algebraic K-theory, or stable homotopy theory, which is pretty far removed from making computer plots of algebraic equations, and are very difficult to describe to the layman.

You might want to check out this subject, though:

http://en.wikipedia.org/wiki/Computational_geometry

There are probably other branches of math that could be relevant, but I'm not exactly sure where you want to go with it, and I'm not an expert on all nooks and crannies of math. I can tell you, it's probably a nook or a cranny you'll have to find if you want to just continue on that theme alone.

 

[Philip]

Yeah, you're probably right. It's a very small and specific thing, but who knows what the outcome may be. I've come across some interesting papers on the subject of " Hypertoric Varieties" : http://pages.uoregon.edu/njp/su.pdf . It describes the combinatorial nature in polynomials, which solve into exact roots as the intercepts. It's much more in-depth than I understand, but speaks of some familiar things. For discrete hypersurfaces, the topic itself seems so new, since there's no +5D hypertorus discussion or visuals anywhere else, other than one place: http://hddb.teamikaria.com/forum/viewforum.php?f=24 . Youtube has probably 5 people making hypertorus vids.

I can't find much more research on them other than what Mr. Proudfoot writes about. The animations are just a recreational math thing, in artistic form. It's the deeper understanding that I'm after now. I'd like to explore the maths that define them, and other related methods. It was quite an effort in learning the notation, and the shapes/concepts it stands for. Where, there seems to be so many high-level maths that barely get into their particulars. It makes me think if it's something novel and unexplored.

 

[sunny79]

Man, feel like I'm doing a 180. (Colloquially speaking). I was going down the "pure math" road. Now I am doing a project in mathematical oncology and it's really fascinating. http://moffitt.org/research--clinical-trials/research-disciplines/departments/integrated-mathematical-oncology

I'm working on differential equations (I did not enjoy this subject when I first took it) modeling of tumor growth (I have no biology background).

You just never know what's going to happen do you?

-Dave K

Dave! For an undergrad who is a math major wanting to pursue this path how would one go about it?

Here is another link...
http://mathematicalneurooncology.org

 

[sunny79]

Dave! My plan is to pursue a pure math track in undegrad while taking a good bunch of science classes and then either go to grad school for applied mathematics and follow it up with phd or medical school ( interested in radiation oncology a very technical specialty)
https://sites.google.com/site/jacobgscott/theoretical-biology

What were your experiences as a math major in undergraduate?What advice would you give to an upcoming math major? I am not a math whizz but I love math with a passion and am a hard worker. Also what do the top graduate schools look at when they accept you in their program?

 

[dkotschessaa]

Dave! For an undergrad who is a math major wanting to pursue this path how would one go about it?

Here is another link...
http://mathematicalneurooncology.org

I got kind of lucky because we have Moffitt here, which is the only hospital that has a dedicated mathematical oncology department. Of course it wasn't all luck. I kept emailing them about internship possibilities until they gave me one.

For you, I'd say try to get some background in biology (maybe double major) and learn some "scientific" programming, like matlab. Mathematically, differential equations (both partial and ordinary) are the big thing.

There are people doing mathematical oncology in other places, just not with a dedicated department. See if you can find these people and tell them you'd like to help. They might have some sort of project sitting around that is not high priority but that you can work on.

-Dave K

 

[dkotschessaa]

Dave! My plan is to pursue a pure math track in undegrad while taking a good bunch of science classes and then either go to grad school for applied mathematics and follow it up with phd or medical school ( interested in radiation oncology a very technical specialty)
https://sites.google.com/site/jacobgscott/theoretical-biology

What were your experiences as a math major in undergraduate?What advice would you give to an upcoming math major? I am not a math whizz but I love math with a passion and am a hard worker. Also what do the top graduate schools look at when they accept you in their program?

Not much time to answer this now (Grad school!) but I see nothing wrong with your plan, though you might also look at bioinformatics for grad school. (maybe).

 

[sunny79]

Dave! Currently I am in sophomore year. While pursuing a pure math and statistics track. I am also fulfilling all the science courses requirement for medical school including upper division courses, namely biology. Initially, I had considered majoring in biomathematics but felt that the amount of math which the degree offers isn't enough. I would love to get more tips from a fellow math major like yourself. My current goal is to ace all my math classes, take grad level courses in undergrad, try getting in research, perhaps attend Budapest semesters of mathematics and maybe, just maybe compete in Putnam and place well.

 

[David Carroll]

Re: The general "Should I become a Mathematician?" question.

I prefer pure math to any of the hard sciences for several reasons.

1) Math is morally neutral.

It is impossible to get angry doing math. You can sometimes get very frustrated but you can't get angry. Philosophy used to be my bag, but I found myself getting increasingly angry at the world because there exist too many people in the world who adhere to the "wrong" philosophy. Or who adhere to the right philosophy but for wrong reasons. You can't get angry at someone who believes that there exist zero's of the Riemann Zeta function that have a non-trivial real part of some value other than 1/2. But you can get VERY angry at someone who believes that Israel should be annihilated as a state.

If you hold to a mathematical position that is wrong, you are subject to the epithet "Idiot". But if you hold a philosophical/scientific/historical position that is wrong, you are subject to the epithets "Bigot", "Commie", "Terrorist", "Sexist", "Homophobe", AND to the epithet "Idiot".

Save yourself, O people, from this!

Furthermore, millions of people have been imprisoned, killed, persecuted, tortured, and defrauded for the sake of scientific, historical, or economic theories (see Hitler, KKK, Pol Pot, etc.). No one has ever been imprisoned, killed, persecuted, tortured, or defrauded for the sake of a Math theorem.

2) There are no such thing as opinionated mathematicians. Only wrong ones.

3) Math is the most democratic of all the branches of knowledge.

If you want to prove a math theorem, all you need is a pen and a piece of paper. You can be a beggar on the streets of Kansas City or Calcutta and if you submit such a theorem to the right journal, you are famous overnight. Try submitting a theory of physics, economics, or microbiology when you are a beggar on the streets of Kansas City or Calcutta. You won't even have the benefit of hearing the laughs aimed at you.

No stuffy philanthropist is needed to fund your equations written on notebooks.

I could go on and on, but I'll leave it at that.

 

[mathwonk]

The original title of this thread, chosen by its starter, was "Who wants to be a mathematician?"

 

[homeomorphic]

1) Math is morally neutral.
2) There are no such thing as opinionated mathematicians. Only wrong ones.
3) Math is the most democratic of all the branches of knowledge.

1) Math can't really be cut off from the sciences because there are extremely rare instances in which things trickle down from math and eventually become "useful". Not to mention there are evil mathematicians like me who actually care primarily about contributing something useful to society, rather than being paid to sit around and play abstract games.

2) Go read Doron Zeilberger's blog for a counter-example. There's more to math than raw facts and theorems.

3) What are the odds that a beggar would be able to acquire anything even remotely approaching the ultra-specialized knowledge required to publish in today's top math journals? There are rumors that certain branches, like combinatorics might not require as much background knowledge. I have a PhD from a highly ranked program, left whole classes of math majors in the dust as an undergrad, and I'm pretty sure my odds would have been slim to zero to contribute anything meaningful if I had stayed in academia. Also, aside from people like Wiles or Perelman who get media coverage, no one really gets famous in math anymore.

These kinds of things actually could have a great impact on whether someone wants to be a mathematician.

 

[David Carroll]

I'll only briefly reply since the powers-that-be might consider this thread to have taken a philosophical tangent (which is, ironically, breaking the very rules I myself set out to lay down in the previous post).

A wrench is morally neutral. The fact that I can take this wrench and bop somebody upside the head with it doesn't negate the wrench's moral neutrality.