Exploring Advanced Topics in Mathematics: Fractals and Non-Euclidean Geometry

[PiAreSquared]

Hello,

I was wondering if anyone had any good ideas on a topic or course that I could take next Fall as an independent study? I will be taking Number Theory this summer and Real Analysis, Abstract Algebra and either Numerical Analysis or Complex Analysis next Fall.I spoke to one of my professors today and he said that he would do an independent study with me, but I am unsure of what course or topic to do it on. I plan on attending graduate school in mathematics upon graduating, but I still am unsure whether I want to do applied or pure math.

Thanks in advance.

 

[micromass]

Some more information would be welcome, since there are so many things out there. What courses did you find interesting? What are you aiming to do research in? Do you have a vague clue what you would like to do? What's the speciality of the professor? What courses did you complete already?

 

[PiAreSquared]

Lets see, I have taken the normal calculus sequences(through Calc III), ODE's, Linear Algebra, Probability, Statistics, and an intro. to proofs course. I have really enjoyed all of the maths course, save for Statistics. Linear Algebra piques my interest, but I am going to take a second course in that in next spring so it is out of the running. My professor's specialty is in analysis(I'll be taking the real analysis sequence with him next year). Just looking through the mathematics course list of my school, it is easy enough to narrow it down( I was thinking maybe advanced calculus, mathematical modeling, or maybe graph theory), but I was unaware that I could deviate from actual courses altogether, and do it on other topics. I realize that I really need to do some research to find out what topics sound most interesting to me, but I just talked to my professor today, and was just seeing if anyone had any interesting ideas. My interest in mathematics is very broad, and I find most topics that I come across very interesting.

 

[homeomorphic]

Complex Analysis is nice to know and it's a great subject. Differential geometry of surfaces would be good at that stage. Graph theory is awesome.

Kind of hard to answer, though, without more direction.

 

[micromass]

OK, that helps.

Here are some suggestions:

Calculus on manifolds (Spivak's book is excellent).
This is basically a bit of a continuation/rigorous version of calc III. The main result is the general Stokes theorem. This would be interesting because there are many useful techniques in the course: differential forms, tensors, covectors, etc. You can even venture a bit in De Rham cohomology!
Since your professor in an analyst, he should be able to help with this.

Differential geometry (this book is good: https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 )
This is a very interesting course as it is very geometrical. Basically, you will investigate geometric phenomena using analysis techniques. All of this is very useful in physics.

Fourier analysis
The topics of Fourier series and Fourier transforms are very useful both in applied as in pure mathematics. They really have tons of applications. One such applications is in PDE's. You can also use it to give a very elegant proof of the isoperimetric inequality (this basically says that of any curve with fixed perimeter, the circle has the largest surface area). If you like pure, then you can go on and investigate convergence issues of the series and you can look at Hilbert spaces (which are very important in physics). If you like applications, then a study of PDE's is possible.

Functional analysis (easy introduction https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 )
This usually needs real analysis, but Kreyszig wrote a nice little book that just needs calculus prerequisites. Functional analysis generalizes and extends results from linear algebra to infinite dimensions. It has many applications including the solution of ODE's and PDE's.

Calculus of variations
This is a very applied topic. You can use this to answer questions like "if I take a rope and I tie one end in one placee and the other end in another place, what shape will the rope take on".

Discrete mathematics
Your professor might not like this as an analyst, but you might. There are many exciting topics here: graph theory, combinatorial designs, generating functions
This book is very popular: https://www.amazon.com/dp/0201558025/?tag=pfamazon01-20
It doesn't contain graph theory though (I think), but it's a pretty read.

 

[PiAreSquared]

Thanks for your input! I have done a little research on your suggestions, but I am a little confused on what the difference between Differential Geometry and Calculus on Manifolds exactly is. I have looked at a few textbooks in the library regarding Differential Geometry before and I looked up Calculus on Manifolds on the internet as well as the book by Spivak last night but they seem to be very similar topics. So I was wondering if you could provide a bit of clarification as to what the differences of these two subjects are.

Thanks.

 

[homeomorphic]

I would recommend doing differential geometry of surfaces to start out with. More down to Earth subject.

 

[Broccoli21]

All the choices the micromass listed are good. I found Kreyszig very easy to read as well. For a good book on differential geometry, this book is good:
https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20
although De Carmo is great as well.

Concrete math doesn't have graph theory, so here is the best graph theory book ever:
https://www.amazon.com/dp/0486247759/?tag=pfamazon01-20

 

[transphenomen]

Since your teacher specializes in real analysis, you could learn more about fractals. I, myself, am going to do a reading course on fractals at some point, as well as non-euclidean geometry since they are not offered at my school and they have interested me before I even knew what a derivative was.