You can use a wide range of wild, cultivated or supermarket greens in this recipe. Consider nettles, beet tops, turnip tops, spinach, or watercress in place of chard. The combination is also up to you so choose the ones you like most. --- Y. Ottolenghi. Plenty More In this post I'll describe how I come… Continue reading Writing Olympiad Geometry Problems
Tag: math
Models of ZFC
Model theory is really meta, so you will have to pay attention here. Roughly, a ``model of $latex {\mathsf{ZFC}}&fg=000000$'' is a set with a binary relation that satisfies the $latex {\mathsf{ZFC}}&fg=000000$ axioms, just as a group is a set with a binary operation that satisfies the group axioms. Unfortunately, unlike with groups, it is very… Continue reading Models of ZFC
Cardinals
(Standard post on cardinals, as a prerequisite for forthcoming theory model post.) An ordinal measures a total ordering. However, it does not do a fantastic job at measuring size. For example, there is a bijection between the elements of $latex {\omega}&fg=000000$ and $latex {\omega+1}&fg=000000$: $latex \displaystyle \begin{array}{rccccccc} \omega+1 = & \{ & \omega & 0… Continue reading Cardinals
Some Notes on Valuations
There are some notes on valuations from the first lecture of Math 223a at Harvard. 1. Valuations Let $latex {k}&fg=000000$ be a field. Definition 1 A valuation $latex \displaystyle \left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0} &fg=000000$ is a function obeying the axioms $latex {\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0}&fg=000000$.… Continue reading Some Notes on Valuations
Proof of Dirichlet’s Theorem on Arithmetic Progressions
In this post I will sketch a proof Dirichlet Theorem's in the following form: Theorem 1 (Dirichlet's Theorem on Arithmetic Progression) Let $latex \displaystyle \psi(x;q,a) = \sum_{\substack{n \le x \\ n \equiv a \mod q}} \Lambda(n). &fg=000000$ Let $latex {N}&fg=000000$ be a positive constant. Then for some constant $latex {C(N) > 0}&fg=000000$ depending on $latex… Continue reading Proof of Dirichlet’s Theorem on Arithmetic Progressions
Teaching A* USAMO Camp
In the last week of December I got a position as the morning instructor for the A* USAMO winter camp. Having long lost interest in coaching for short-answer contests, I'd been looking forward to an opportunity to teach an olympiad class for ages, and so I was absolutely psyched for that week. In this post… Continue reading Teaching A* USAMO Camp
Representation Theory, Part 4: The Finite Regular Representation
Good luck to everyone taking the January TST for the IMO 2015 tomorrow! Now that we have products of irreducibles under our belt, I'll talk about the finite regular representation and use it to derive the following two results about irreducibles. The number of (isomorphsim classes) of irreducibles $latex {\rho_\alpha}&fg=000000$ is equal to the number… Continue reading Representation Theory, Part 4: The Finite Regular Representation
Represenation Theory, Part 3: Products of Representations
Happy New Year to all! A quick reminder that $latex {2015 = 5 \cdot 13 \cdot 31}&fg=000000$. This post will set the stage by examining products of two representations. In particular, I'll characterize all the irreducibles of $latex {G_1 \times G_2}&fg=000000$ in terms of those for $latex {G_1}&fg=000000$ and $latex {G_2}&fg=000000$. This will set the… Continue reading Represenation Theory, Part 3: Products of Representations
Representation Theory, Part 2: Schur’s Lemma
Merry Christmas! In the previous post I introduced the idea of an irreducible representation and showed that except in fields of low characteristic, these representations decompose completely. In this post I'll present Schur's Lemma at talk about what Schur and Maschke tell us about homomorphisms of representations. 1. Motivation Fix a group $latex {G}&fg=000000$ now,… Continue reading Representation Theory, Part 2: Schur’s Lemma
Representation Theory, Part 1: Irreducibles and Maschke’s Theorem
Good luck to everyone taking the December TST tomorrow! The goal of this post is to give the reader a taste of representation theory, a la Math 55a. In theory, this post should be accessible to anyone with a knowledge of group actions and abstract vector spaces. Fix a ground field $latex {k}&fg=000000$ (for all… Continue reading Representation Theory, Part 1: Irreducibles and Maschke’s Theorem
Power Overwhelming 