George Coote's Blog

Welcome

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I’m George, I am a second year CMI student supervised by Matthew Colbrook and Anders Hansen. My Part III essay (which will appear here at some point – I am now more likely to rewrite its content on here) was on computational impossibility results for the spectral problem for self-adjoint operators and relatively compact perturbations thereof. Prior to coming to Cambridge I did a BSc in Maths at Warwick. A copy of my CV can be found here.

My interests are fairly broad, but my research at the moment is on the interface between spectral theory and computability theory. In layman’s terms, we might have some physical system we can take measurements from or simulate, and someone might ask whether it is possible to calculate a quantity associated with the system based off a certain set of measurements. The answer in many cases is no even with theoretically perfect measurements, and this can be proven mathematically. One might ask if there is other information that could be collected from the system to make the calculations possible, and the answer is often yes.

In more mathematical terms, I look at what information about a closed linear operator is needed to compute its spectrum (or other spectral sets such as the absolutely continuous spectrum), and quantify the failure of spectral sets to be computable using certain information within the Solvability Complexity Index hierarchy. This is essentially a generalisation of (and has strong connection with) the arithmetical hierarchy: the arithmetical hierarchy is equivalent to the SCI hierarchy in the case of decision problems on the natural numbers, and the same notation is used. Most recently I have been looking at the (lack of) computability of spectral type of self-adjoint Schrodinger operators. Said results were previously only known in the discretized (tridiagonal) case.

Content on this blog will fall in three broad categories:

  • Compilations of results that are hard to find with proof in literature, (or hard to find in one place, etc.) or cool results that are rarely covered in structured university study of a subject.
  • Expositions about my own research, though original work will only appear after it is on arXiv.
  • Tripos past paper solutions: I decided to start this blog the day before my Part III Functional Analysis exam in order to procrastinate deposit typed solutions for most of Zsak’s past papers. I am open to writing solutions for other courses as well. Please drop me a line if this is something you and ideally some others are interested in before I sink a lot of time into it.

I can be reached by email at gc602 at cam dot ac dot uk.

Research

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