Quick Thoughts on DataFrame.stack & DataFrame.unstack | 99 Problems but pandas ain't 1 Part 4

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pandas, NumPy—how hard can they really be?

In this series, we will address the following pandas and NumPy questions, identifying common concepts, mental models, and approaches necessary to solve each one of them effectively and efficiently.

We will look to answer the following questions:

Given a pandas.Series containing snapshots of some evolving process, how do we determine the state at any given point in time (such that we can index align this result against other structures)? How do we derive state at any given point in time given a pandas.Series containing deltas?
Given a numpy.ndarray or pandas.array containing numeric values, how do we identify the N most closely grouped values?
How do we use Einstein summation notation via numpy.einsum without getting confused? Why might we choose to use this notation?
How do we turn one data set in pandas into multiple? When do we want to do this and why? What do pandas.DataFrame.stack and pandas.DataFrame.unstack actually mean (and how can we generalize these operations to NumPy, and why is this conceptually distinct from xarray.DataArray.stack and xarray.DataArray.unstack?)?
Given a numpy.ndarray containing the position of entities in 3D space, how do efficiently compute collisions using spatial subdivision tools provided by SciPy?
Given multiple pandas.Series and pandas.DataFrame with MultiIndexes with overlapping levels how do we perform index aligned operations to solve complex nested problems without explicit Python-level loops?
How do we perform minimizations, maximizations, and sorting on alternate predicates in pandas (equivalent to the use of the key= argument in the Python __builtins__.max, __builtins__min, and __builtins__.sorted)?
How do we use numpy.swapaxes, numpy.moveaxis, and numpy.rollaxis without getting confused? How do we use pandas.Index.swaplevel, and pandas.Index.reorder_levels?