This blog post corresponds to my newest olympiad handout on mixtilinear incircles. My favorite circle associated to a triangle is the $latex {A}&fg=000000$-mixtilinear incircle. While it rarely shows up on olympiads, it is one of the richest configurations I have seen, with many unexpected coincidences showing up, and I would be overjoyed if they become… Continue reading The Mixtilinear Incircle
Category: Mathematics
Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
\title{A Variant of Linnik for Elliptic Curves} \maketitle Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields. 1. Motivation: Arithemtic Progressions Given a property $latex {P}&fg=000000$ about primes, there's two questions we can ask: How many primes $latex {\le x}&fg=000000$ are there with this… Continue reading Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
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
Zeros and Primes
Prerequisites for this post: previous post, and complex analysis. For this entire post, $latex {s}&fg=000000$ is a complex variable with $latex {s = \sigma + it}&fg=000000$. 1. The $latex {\Gamma}&fg=000000$ function So there's this thing called the Gamma function. Denoted $latex {\Gamma(s)}&fg=000000$, it is defined by $latex \displaystyle \Gamma(s) = \int_0^{\infty} x^{s-1} e^{-x} \; dx… Continue reading Zeros and Primes
von Mangoldt and Zeta
Prerequisites for this post: definition of Dirichlet convolution, and big $latex {O}&fg=000000$-notation. Normally I don't like to blog about something until I'm pretty confident that I have a reasonably good understanding of what's happening, but I desperately need to sort out my thoughts, so here I go\dots 1. Primes One day, an alien explorer lands… Continue reading von Mangoldt and Zeta
On Problem Sets
(It appears to be May 7 -- good luck to all the national MathCounts competitors tomorrow!) 1. An 8.044 Problem Recently I saw a 8.044 physics problem set which contained the problem Consider a system of $latex {N}&fg=000000$ almost independent harmonic oscillators whose energy in a microcanonical ensemble is given by $latex {E = \frac… Continue reading On Problem Sets
Cauchy’s Functional Equation and Zorn’s Lemma
This is a draft of an appendix chapter for my Napkin project. In the world of olympiad math, there's a famous functional equation that goes as follows: $latex \displaystyle f : {\mathbb R} \rightarrow {\mathbb R} \qquad f(x+y) = f(x) + f(y). &fg=000000$ Everyone knows what its solutions are! There's an obvious family of solutions… Continue reading Cauchy’s Functional Equation and Zorn’s Lemma
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
Power Overwhelming 