Path - Geometry

Question

I am currently completing the end of a Bachelor degree in pure mathematics. I would like to work on an interesting project (by myself) this summer in the field of spectral geometry. Does someone could essentially offer me some books on which I could work? Better yet, is that someone could describe me a personal project (detailed) that I could take to the summer in the latter area of research?

To describe me roughly, I succeed to obtain $21$ for the PUTNAM contest $2014$. I have experience in differential topology and geometry. I read the books " Differential Topology" of Guillemin and Pollack, "Differential geometry of curves and surfaces" of Do Carmo, a bit of "Morse theory" by John Milnor and finally, "Topology : Point-set and geometric" of Paul L. Shick.

To be a bit more precise, I need someone to present me a project that a professor had given one of his students. I do not wish you to build a new project I could work. From projects that offer me, I will analyze each of them and take what's best for me.

A project for example only be able to prove a certain theorem, and describe me what books I could pass to get there.

Thanks!

Answer 1

Disclaimer : I am not an expert in this area.

From what I have understood by some self-reading is that given a Riemannian manifold $M$ we have the Laplace-Beltrami operator $\Delta$ acting on $C^{\infty}(M)$. We can look at the spectrum of this operator. The main idea is that how much of geometry of $M$ can be recovered from analysisng the spectrum of $\Delta$.

As a starting point you can look at these notes.

The following webpage is a graduate course offered by Masoud Khalkhali at the University of Western Ontario. It has some very useful references.

The following notes by Shing-Tung Yau discuss a lot of open problems in Riemannian Geometry, Geometric Analysis and related areas including Spectral Geometry. I don't know whether problems from these notes will be suitable for a Bachelor's project, but atleast these will give you an idea about the questions in which people are interested.

Hope it helps.

Answer 2

Here's a couple key books. I took a course on spectral geometry a few years ago, but I can't find my notes, which had a list of classical papers like Kac 1966 "Can one hear the shape of a drum?" (you should read this one regardless - it's well written and considered a landmark).

• "Eigenvalues in Riemannian Geometry" by Chavel
• "Old and New Aspects in Spectral Geometry" Craioveanu