Modular index theory and spectral flow

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A "linear operator" is a mathematical object which (in many cases) can be represented by a square array (or"matrix") of numbers just like a spreadsheet. By a process called "matrix multiplication", such operators"operate" on vertical lists of numbers (called "vectors") to produce new vertical lists (new vectors). If A is the symbol for an operator and X is the symbol for a vertical list (vector), then we denote the newly transformedlist (vector) by AX. It is especially important to find particular vectors for the operator A, say Y, and particularnumbers (depending on A and Y) say 2 for example so that AY is 2 times Y, in symbols AY=2Y. In this case, we would say that 2 is in the "spectrum" of A. That is, the spectrum of A is the collection of all such numbers n for whichthere is some vector, say Z so that AZ=nZ. Of course different operators will usually have different spectrums.   Examples of such operators and their spectrums occur so often in scientific and economic applications thatstudents are routinely taught the rudiments of the subject in first year university (or even high school) in theguise of "Matrix Theory" or "Linear Algebra". At a deeper level, these operators are crucial to the understanding of quantum mechanics and all the myriad applications of that field.   I plan to continue my detailed study of how the spectrums of such operators change continuously as the operators themselves change continuously. This is the notion of "flowing spectrum" or "Spectral Flow". In particular, it is important to be able to calculate how much of the spectrum changes from negative to positive,without having to go through the hugely time-consuming process of calculating all the spectrums at each instant of the flow. My co-authors and I have been developing formulas which do this in many diverse cases.   WARNING: The above descriptions are highly over-simplified for the purposes of exposition.