The Jacobson radical

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one [...]

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Schanuel’s conjecture and the Mandelbrot Competition

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition: Let be positive integers such that . Find the minimum value of . After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: should be replaced by [...]

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The Yoneda lemma I

For two categories let denote the functor category, whose objects are functors and whose morphisms are natural transformations. For a locally small category, the Yoneda embedding is the functor sending an object to the contravariant functor and sending a morphism to the natural transformation given by composition. The goal of the next few posts is [...]

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Morita equivalence and the bicategory of bimodules

In the previous post we learned that it is possible to recover the center of a ring from its category of left modules (as an -enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring). For [...]

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Centers, 2-categories, and the Eckmann-Hilton argument

The center of a group is an interesting construction: it associates to every group an abelian group in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor (unlike the abelianization ). We might wonder, then, exactly what kind of [...]

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A first blog post on noncommutative rings

In this post, I’d like to record a few basic definitions and results regarding noncommutative rings. This is a subject clearly of great importance and generality, but I haven’t had much exposure to it, and I’m trying to fix that. I am working mostly from Lam’s A first course in noncommutative rings.

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A less biased definition of a group

Here’s what seems like a silly question: what’s the universal group? That is, what’s the universal example of a set together with maps satisfying the identities , , ? A moment’s reflection shows that there isn’t such a group; the existence of the groups , where is an arbitrary set, shows that there exist groups [...]

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Estimating roots

In lieu of a real blog post, which will have to wait for at least another two weeks, let me offer an estimation exercise: bound, as best you can, the unique positive real root of the polynomial . The intermediate value theorem shows that , which was the subject of a recent math.SE question that [...]

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Morality

Apologies for the lack of updates; I’ve been attempting to apply to graduate school. In the meantime, I want to link to a fantastic paper I just heard about by Eugenia Cheng on moral truth in mathematics. In private (or for me, on MathOverflow), mathematicians often say things like “well, morally, this should be true [...]

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Constructing Poisson algebras

(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that is a Poisson algebra with bracket uniquely defined by ; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra). [...]

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