Modular index theory and spectral flow

“线性运算符”是一个数学对象，它（在许多情况下）可以像电子表格一样用数字的正方形数组（或“矩阵”）表示。通过称为“矩阵乘法”的过程，这样的运算符对数字的垂直列表（称为“向量”）进行“操作”以产生新的垂直列表（新向量）。如果A是一个操作符的符号，X是一个垂直列表（向量）的符号，那么我们用AX表示新转换的列表（向量）。特别重要的是找到运算符A的特定向量，比如Y，以及特定的数（取决于A和Y），比如2，使得AY是2乘以Y，符号为AY=2Y。在这种情况下，我们可以说2在A的“谱”中。也就是说，A的谱是所有这样的数n的集合，对于这些数n，有某个向量，比如Z，使得AZ=nZ。当然，不同的运营商通常会有不同的频谱。 这种算子及其频谱的例子经常出现在科学和经济应用中，以至于学生们在大学一年级（甚至高中）经常以“矩阵理论”或“线性代数”的名义教授这门学科的基础知识。在更深的层次上，这些算子对于理解量子力学和该领域的所有无数应用至关重要。 我计划继续详细研究这些运营商的频谱如何随着运营商本身的不断变化而不断变化。这就是“流动频谱”或“频谱流”的概念。特别重要的是，能够计算频谱从负到正的变化量，而不必经历在流动的每个时刻计算所有频谱的非常耗时的过程。我和我的合著者一直在开发在许多不同情况下都能做到这一点的公式。 注：上述描述是高度简化的说明的目的。

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.