Math on rkempe’s blog

What Doing My First (Short) Math Lecture Taught Me

Sun, 29 Jun 2025 13:02:46 -0400

For context, these are some things I learned in the process of putting together and delivering a guest lecture to a first-year discrete math course last summer. The talk was about the research I was doing at the time, and I was allotted about half an hour for the presentation. Again, I meant to write and post this last year, but clearly that didn’t happen.

Why Does Theory Matter in Computer Science? (Part 5)

Thu, 22 May 2025 21:34:31 -0400

Here is a long list of sources I consulted at various points while doing research for this project. As it might have become clear through reading the previous installments of this series, this was more a talk about the densest subgraph problem than it was about theory being useful in computer science. There are a few reasons for that, which I might explain in more detail at some point in the future. This means that I presented a lot of information that I borrowed from other people! Here’s an annotated bibliography of sorts, in case you were interested in going deeper, for whatever reason.

Why Does Theory Matter in Computer Science? (Part 4)

Thu, 22 May 2025 21:28:43 -0400

In Part 3, we discussed the densest subgraph problem (DSP) and some algorithms for solving it. In this section, we’ll be looking at how we can generalize this problem and those algorithms to solve some other problems. This is the part of the talk where I will introduce some fancy new math I had to learn in order to understand how one might generalize iterative peeling to solve adjacent problems.

Why Does Theory Matter in Computer Science? (Part 3)

Thu, 13 Feb 2025 23:17:01 -0500

In Part 2 of this talk, we gave a crash course to graph theory and showed how we can use it to view some real-world problems as instances of the densest subgraph problem (DSP). But what exactly is the DSP? If you’ve studied graph theory, you may have heard of something called the Maximum Clique Problem. The goal of the max clique problem is to find the largest complete subgraph in a graph. If we consider our vertices to be people, and edges to represent a friendship relationship between two people, in the max clique problem we are trying to find the largest friend group in a community.

Which Math Courses Should I Take in First Year?

Sun, 09 Feb 2025 00:00:01 -0500

This page was last updated on February 9, 2025.

What is COMP 1805? Is it math or programming?

COMP 1805 involves no programming, though at times reading pseudocode may be involved. It is very explicitly a math course, though it’s probably not the math you are used to doing in high school. Whereas high school math courses teach you techniques for doing different sorts of calculations and algebraic manipulations, there are close to no calculations in COMP 1805. Instead, the course is about learning the language used by computer scientists and mathematicians to describe problems more formally (which in turn, makes it easier to decide whether or not a problem has been “solved”). Topics covered in COMP 1805 include “foundations of math and reasoning” such as propositional logic, basic set theory, functions and relations on sets, and proof techniques, as well as topics intended for computer scientists, such as an introduction to graphs and networks, Big O notation, and asymptotic analysis of algorithms.

Do You Need to Understand the Math Behind a System to Implement It?

Sun, 26 Jan 2025 14:17:20 -0500

A while ago, someone in a Discord server I’m in asked how much of the math behind a system you need to know to implement it. I thought it was an interesting question, and I felt qualified to answer it, so I ended up writing quite a lengthy response. It just occurred to me that it might also be useful to other people, so I thought I would clean it up a little bit and archive it here.

Why Does Theory Matter in Computer Science? (Part 2)

Fri, 24 Jan 2025 12:47:46 -0500

In the first part of this talk, I made the case that theory is useful because it allows us to find (or at the very least, have the correct toolkit and language to explore) solutions to real-world problems. In this part, we are going to look at some examples of such problems and develop mathematical language to be able to discuss them more abstractly. I’ve put the term “real-world” in quotes in the title, because I’m going to be talking about these problems in a lot of generality. However, I want to stress that specific instances of these problems are actually relevant in industry, and I think it’ll be easy to see why once I start talking about them.

Research Reflections: On Reading (Math and Math-Adjacent) Academic Papers

Sun, 29 Dec 2024 23:51:14 -0500

Last summer, I spent a good chunk of my time doing an undergraduate research project during which I worked on a project largely by myself, under the supervision of a math professor. I then took a graduate level course in a related area this fall, where I investigated the theoretical underpinning behind my summer project. I had no idea what I was doing or what I had gotten myself into. As a result, I learned a lot, and tried to keep note of the various things I had learned. This is the first installment, on what I learned from trying to read math and theoretical CS papers.

Why Does Theory Matter in Computer Science? (Part 1)

Sun, 29 Dec 2024 00:38:01 -0500

If you’re a computer science student, you probably had to take an introductory discrete math course at some point. Did you enjoy it? If so, this talk probably isn’t for you, so you can feel free to skip the rest. (Or not – hopefully you feel like you can still learn something from me!) Jokes aside, it’s actually okay not to enjoy your intro to discrete math course: like, personally, I loved mine, but I also completely hated my discrete probability course and would prefer never to see it again. But I pick on discrete math because I feel like if it’s taught well, it can be a turning point for many people, and it certainly was for me.

Heuristics, Approximation Algorithms, and Relaxations: An Introduction

Sun, 24 Nov 2024 00:03:31 -0500

While all NP-hard optimization problems are identical in terms of exact solvability, they may differ wildly from the approximative point of view. If the goal is to obtain an answer that is “good enough”, some problems become much easier (such as KNAPSACK), while others (such as CLIQUE) remain extremely hard.

Hypergraph Theory Basics

Fri, 22 Nov 2024 22:35:01 -0500

Graphs can be seen as a way to represent pairwise relationships between objects. With graphs, we have one object type and one relationship type. In one of the most common canonical applications or graph theory, social networking, we are trying to understand and represent social groups using graphs. In that case, our object is people, our pairwise relationship is friendship, and two people have a relationship between them if they are friends.

Some Advice for Taking Your First Proof-Based Math Course

Sat, 10 Feb 2024 00:00:01 -0400

If you are like me, you did not particularly enjoy math in high school, so the idea of learning an entirely new type of math in university might be terrifying to you. I know this can be scary and overwhelming (it was for me!), so here are some things I learned when taking my first proof-based mathematics course. Hopefully you find this helpful.