A Eulogy Of Lipschitz Maps

A Lipschitz map (/function) is one that does not extend distances by more than a pre-assigned factor: $f: X \longrightarrow Y$ is Lipschitz if there exists an $L \in \mathbb{R}$ such that

$$ \forall x, \ \  \forall y \ \ \ d(f(x),f(y)) \leq L d(x,y) \ .$$

The definition makes sense as long as a distance is defined on the spaces. This makes Lipschitz maps highly versatile. Almost every space you deal with daily is either a priori a metric space (a set plus a distance function) or can be made one by endowing it with a distance. (Check for instance the word metric on a group that builds metric spaces out of groups, opening the doors to the beautiful topic of Geometric Group Theory.)

We will, nonetheless, limit ourselves to studying (some of) the properties of Lipschitz maps between Euclidean spaces. Properties?! Look at the definition again, what else could we expect of a Lipschitz map? What else could that condition possibly impose on $f$? Continue reading

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Introduction to Ideal Class Groups

Algebraic number theory is a really interesting subject, but unlike some other subjects, it’s not 100% clear what objects people study. This post provides an introduction to the class group of a finite dimensional field extension of $\mathbb{Q}$, an object often used in modern number theory.

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Towards Embracing Diverse Mathematical Communities

As graduate students, we interact with a wide variety of people: local communities,  students, peers, and professors. Within these interactions, there is great diversity: different backgrounds, experiences, and cultures. I believe this is something that makes our mathematical community vibrant if we choose to embrace it.

However, through our training as mathematicians, are we asking ourselves:  How are we contributing to this diversity?  How do we create environments that embrace the identities of those who “do” mathematics? Are we making mathematics accessible and inclusive? These are questions that I ask myself, but would be interested in making them part of a larger narrative.

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Mathematics Students and Legitimate Peripheral Participation

One of the things that mathematics educators often talk about is the idea of teaching the norms of the discipline of mathematics to students, starting at a fairly young age.  In Jo Boaler and Cathlee Humphreys’ book Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning, the authors have compiled several examples of teaching the basics of mathematical argumentation to middle school students.  Students are told that when making a mathematical claim, they need to first convince themselves of the claim, then convince a friend of the claim, and then convince an enemy of the claim.  The idea is that students can engage in mathematical practices in  the classroom which approximate the kind of practices that mathematicians use in their work.  There have been many attempts to systematize these practices; most recently, with the publication of the Common Core State Standards for Mathematics and its eight mathematical practices.

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Freedom in Failure

“Preliminary examination results: Fail. Please meet with your committee members to discuss your exam.”

These are the words I have seen five out of the six times I’ve opened an envelope after pouring my soul into studying for a prelim exam. That’s right – my prelim pass rate is 0.1667. That’s not even good in baseball, where the standard batting average is somewhere around a 0.300. Were I a baseball player, the coach would have benched me a long time ago (probably after my first three prelims, on which I went “oh for three”, as they say). The fact of the matter is that I’ve benched myself several times. Too many times. But I learned a lot sitting on that bench.

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