Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms? Lingua Franca links relativity and quantum theories with spectral geometry
Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics community? Is this work ground breaking or just the start of something that might be?
From what I can understand this physicist related two types of maths that we use to model the world around us. Which has been hard to do, because the types of maths are incompatible with each other. 
But what is this spectral geometry? How does it relate the two types of maths in relativity and quantum theory? What might those maths be? What about the two has made it hard to unite them?
 A: Spectral geometry is one of the many ways mathematicians think about geometry.  The general idea is that if you have some manifold equipped with a metric, you can cook up some canonical differential operators.  These operators can be thought of as linear operators, acting on (infinite-dimensional) vector spaces of functions, tensors, spinors, and the like.  Each such linear operator will have a set of eigenvalues.  Spectral geometry is concerned with relationships between these eigenvalues and the geometry of the manifold you started with
The most obvious linear operator to associate to a metric is the Laplacian, which is a linear operator on the space of functions on the manifold.  In the early/middle part of the 20th century, mathematicians started wondering "If you know the set of eigenvalues of the Laplacian, can you reconstruct the manifold?", or as Mark Kac famously put it:  "Can one hear the shape of a drum?"
The answer is no; the set of eigenvalues alone doesn't let you reconstruct the manifold and its metric.  The map which sends a manifold with metric to the set of eigenvalues of the Laplacian is not invertible.  
But it's such a pretty idea that people haven't given up on it.  Alain Connes, for example, figured out that the answer to a slightly different question is "yes".  If you have a commutative spectral triple (basically, the Dirac operator on a compact spin manifold), you can reconstruct the metric from this data.
The physicists interviewed in the linked article are trying a slightly different variation on the spectral geometry problem.  They're considering systematic finite-dimensional approximations to the "derivative" of the map $F$ which sends a manifold to its set of eigenvalues of its Laplacian acting on tensors of low-degree, and trying to show that these approximations are invertible.  This should let them show that this map $F$ is invertible in small regions of the space of manifolds.
It's a nice idea, and looks like some fun experimental mathematics.  Trying to write down a theory of gravity in explicitly gauge invariant terms is also a good idea.  But I'd be quite surprised if this line of thinking bears any fruit.  It seems more likely to me that we need some essentially new physical ideas than a clever way of rewriting what we already have.  
A: What I will comment here is supplementary to user1504's answer and what is not clear about iso-spectrality (iso for same and spectral for eîgenvalues) and spectral triple. 
I have been working on a different topic but astonishingly found connections between spectral geometry and it. I have been working on Non-Hermitian Quantum Mechanics and a recent result suggests that Non-Hermitian operators having Real Eigenvalues 'always' have a sort of brother, a Hermitian operator with same set of eigenvalues ( these kind of operators are called iso-spectral)! A somewhat deeper thought suggests correctly that, they should be related by Similarity Transformation. Moreover, their state spaces form Hilbert space (A complete orthogonal vector space) with different metrics though (a metric is something similar to a length scale).
You can check all that User1504 suggested by taking a complex 2*2 matrix instead of using a differential equations and doing hardcore math. Just take a 2*2 general matrix make it Hermitian, perform a Non-Unitary Similarity transform on it and you will see that the new matix has same eigenvalues ( i.e. it is isospectral) but it has now become Non-Hermitian ! ( Why does this happen ?, well because similarity transformation always preserve eigenvalues, a small google search will give a lot of links to it's proof)
This directly suggests what Alain Connes tries to say about spectral triple, A complete information is embedded not only in spectrum but your measurement scale that is Metric which is usually ignored and the algebra you are using for example C* in my case. 
You can refer to Appendix of my paper if you want. Note just skip to the Appendix and everything will be clear. 
http://vixra.org/abs/1403.0047
