The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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Some quadratic reciprocity

In the previous post we showed that the splitting behavior of a rational prime in the ring of cyclotomic integers depends only on the residue class of . This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof. The key result is the following fundamental observation. [...]

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Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when [...]

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Extracting the diagonal

Needless to say, I have been very, very busy. But enough about me. Suppose you are given a bivariate generating function in “closed form,” where I’ll be vague about what that means. Such a generating function may arise, for example, from counting lattice paths in ; then might count the number of paths from to [...]

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Some examples of graded algebras

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds [...]

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