Undergraduate research topics in topology?

[rtista]

TL;DR Summary: In search of a suitable topic for an interesting undergraduate dissertation.

I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential fields and forms, integration on chains, and a little bit of Theory of manifolds (mainly building to Stokes' theorem following the pathway in Calculus on Manifolds, Michael Spivak). I am comfortable with concepts in point set topology and have a limited knowledge in Algebraic Topology (Topology by James Munkres) (having finished courses on abstract algebraic structures beforehand), and plan to expand it within the next 3-4 weeks. I am in search of a topic suitable for an interesting undergraduate dissertation culminating the above mentioned subjects. Thanks to anyone who takes the time to read and write me some input.

 

[fresh_42]

TL;DR Summary: In search of a suitable topic for an interesting undergraduate dissertation.

I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential fields and forms, integration on chains, and a little bit of Theory of manifolds (mainly building to Stokes' theorem following the pathway in Calculus on Manifolds, Michael Spivak). I am comfortable with concepts in point set topology and have a limited knowledge in Algebraic Topology (Topology by James Munkres) (having finished courses on abstract algebraic structures beforehand), and plan to expand it within the next 3-4 weeks. I am in search of a topic suitable for an interesting undergraduate dissertation culminating the above mentioned subjects. Thanks to anyone who takes the time to read and write me some input.

Hello again!

I have a few ideas but I don't know whether they fit your profile.

1) Elaborate the proof and discuss what is the crucial point of the Banach-Tarski paradox, in particular concerning the question of whether it is the axiom of choice or our lack of understanding the concept of points that makes it so surprising.

2) $G=\left\{\left.\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}\, , \,\begin{pmatrix}1&x\\0&1\end{pmatrix}\right| \;t,x\in \mathbb{R}\right\}$ defines an analytical (topological) group (manifold). Elaborate the various terms of differential geometry like atlas, geodesics, Levi-Civita connection, etc. on this example.

3) You could use the same example for algebraic topology and calculate its properties: compact or not, (path-) connected or not, homology groups, (co-)chain complexes, its Lie algebra, and the Cartan-Eilenberg complex.

4) Or you could discuss the various definitions of dimensions of fractals.

If you want to have something new, I can only hand you an algebraic topic.

 

[CrysPhys]

OP: Any suggestions from your advisor, who should best know the level expected and what is viable within the allotted time?

 

[FactChecker]

You list a broad range of subjects that the topic can "culminate in" but the title specifies topology. Are you restricting your request to topology?

 

[fresh_42]

If you want to do something on topological fields, you could

5) investigate the topology on p-adic numbers and their approximation theorem.
or
6) function theory on p-adic numbers,
or
7) any other (or all) approximation theorems.

 

[gmax137]

p-adic numbers

I stumbled upon this Veritasium video, which is about p-adic numbers. I'm not a mathematician by any means but this really blew my mind. Fascinating.

Something Strange Happens When You Keep Squaring

 

[mathwonk]

Late to the discussion, but with your background, a topic in differential topology seems appropriate, such as expositing some of Milnor's Topology from the differentiable viewpoint, maybe even the final Hopf theorem on framed cobordism, or part of chapter 1 of Bott-Tu, covering deRham cohomology, Mayer Vietoris, and Poincare duality.