These notes are from the February 23, 2016 lecture of 18.757, Representations of Lie Algebras, taught by Laura Rider. Fix a field $latex {k}&fg=000000$ and let $latex {G}&fg=000000$ be a finite group. In this post we will show that one can reconstruct the group $latex {G}&fg=000000$ from the monoidal category of $latex {k[G]}&fg=000000$-modules (i.e. its… Continue reading Tannakian Reconstruction
Category: Mathematics
Some Advice for Olympiad Geometry
I know some friends who are fantastic at synthetic geometry. I can give them any problem and they'll come up with an incredibly impressive synthetic solution. I also have some friends who are very bad at synthetic geometry, but have such good fortitude at computations that they can get away with using Cartesian coordinates for… Continue reading Some Advice for Olympiad Geometry
Rant: Matrices Are Not Arrays of Numbers
The following is an excerpt from a current work of mine. I thought I'd share it here, as some people have told me they enjoyed it. As I'll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an $latex {m \times n}&fg=000000$ matrix is merely a… Continue reading Rant: Matrices Are Not Arrays of Numbers
Writing Olympiad Geometry Problems
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
Uniqueness of Solutions for DiffEq’s
Let $latex {V}&fg=000000$ be a normed finite-dimensional real vector space and let $latex {U \subseteq V}&fg=000000$ be an open set. A vector field on $latex {U}&fg=000000$ is a function $latex {\xi : U \rightarrow V}&fg=000000$. (In the words of Gaitsgory: ``you should imagine a vector field as a domain, and at every point there is… Continue reading Uniqueness of Solutions for DiffEq’s
Putnam 2015 Aftermath
(EDIT: These solutions earned me a slot in N1, top 16.) I solved eight problems on the Putnam last Saturday. Here were the solutions I found during the exam, plus my repaired solution to B3 (the solution to B3 I submitted originally had a mistake). Some comments about the test. I thought that the A… Continue reading Putnam 2015 Aftermath
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
Constructing the Tangent and Cotangent Space
This one confused me for a long time, so I figured I should write this down before I forgot again. Let $latex {M}&fg=000000$ be an abstract smooth manifold. We want to define the notion of a tangent vector to $latex {M}&fg=000000$ at a point $latex {p \in M}&fg=000000$. With that, we can define the tangent… Continue reading Constructing the Tangent and Cotangent Space
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
Power Overwhelming 