Visualizing combinatorics with trees
In all counting problems, we have some $n$ number of items and we want to choose some $k$ of them. There are two fundamental tools for computing this - permu...
Sarah's Blog
About me
Thanks for visiting my blog! I am in my third year at Columbia University, studying CS and math. Hence, most of my posts are related to the math and CS topics I’ve been learning through my studies and research.
I’m currently involved in two research projects.
- MOSES (Columbia Gang Lab) - A DNA origami bond coloring algorithm which uses group theory and symmetry mapping to reduce the number of unique DNA sequences to enable the self-assembly of a nanoparticle superlattice. This blog post talks about it more in depth.
- ??? (NYU Dennis Shasha Group) - A practice tool for musicians which compares a users’ music performances to a MIDI score, plotting their pitch, volume, rhythm over time, and giving feedback. (Name TBD)
For fun, I also play violin in the Barrio Tango Orchestra with some really amazing people. My friend and I have both started learning tango as well. 👯♀️
Recent posts
Darboux integration
Darboux integration (or Riemann integration) is similar to what we learned in Calc. In Analysis, we broke down how the integral sign is composed of these ser...
Derivatives
Finally - differentiation!!
Continuity
Continuity is one of the core concepts in Analysis. Now that I have a good handle on compactness, series, and sequences, I feel equipped to talk about contin...
Sequences, series, convergence
The variables Sequences A sequence is precisely defined as the map $f:\mathbb{N}\rightarrow X$ from the natural numbers to some infinite cardinality set.
Compactness
A compact set is one where every open cover has a finite subcover. WTF?
Balls and sets
While my course is primarily an Analysis course, we went over a significant amount of basic topology. Ultimately it gave us a richer understanding the differ...
The art of naming variables
The helix problem Just as I thought I had finally debugged all of the edge cases of the algorithm, last week I went into the lab and was asked to try out the...
Research, ego, and PSETs
The perovskite problem Last I had a meeting with Jason debugging the coloring algorithm - where I realized that the colortree post-processing script may not ...
Presentations; Group problem solving
Group meetings: Overview Every Friday, Gang lab holds a group meeting from 10am-12pm where 1-2 lab members present on the progress they’ve made on their proj...
Metric spaces
The metric space bridges real analysis with topology by introducing the notion of a distance onto a set of elements. They are a general setting for studying ...
The real numbers are an ordered field
Real numbers The set of real numbers $\mathbb{R}$ is an ordered field, meaning it’s defined as being a “tuple” of four elements, which includes An ordered...
My monitor problem; an algorithm
The monitor problem I ordered a new monitor for my dorm last week, and it arrived yesterday! But it’s 16 inches large, and I’m quite lazy to retrieve it from...
Game trees
It turns out I was wrong - the algorithm is not just trying to minimize the number of colors, but more specifically it must minimize the unique voxel+bond co...
Ganglab; MOSES Overview
Ganglab - Fall 2024 This past summer, I started working with the Gang Lab at Columbia University as a chemical engineering research assistant. I received a g...