We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the ...

for all infinite sets X X and Y Y. Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn ...

Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are ...

Aug 2, 2008 In this new version of our paper, we systematically explain how n-dimensional field theories give n-plectic manifolds. We also say how a B field affects the 2-plectic structure for a ...

Apr 19, 2017 Conditional entropy, joint entropy and mutual information all have something important to say about the biological diversity of a metacommunity. Pre- and Postdictions of the NCG Standard ...

every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”; ...

Previously: Part 6. Next: Part 8. As the course continues, the axioms fade into the background. They rarely get mentioned these days. Much more often, the facts we’re leaning on are theorems that were ...

Thurston gave a concrete procedure to construct triangulations of the 2-sphere where 5 or 6 triangles meet at each vertex. How can you get the icosahedron using this procedure? Gerard Westendorp has a ...

Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint ...

Thurston’s paper Shapes of polyhedra and triangulations of the sphere is really remarkable. I’m writing about it in my next column for the Notices of the American Mathematical Society. Here’s a draft ...