Columbia Journal of Undergraduate Mathematics

Demystifying Compactness

Emory Sun — 2025-09-22

We explain how to think about and use the open cover definition of compactness through several examples. In particular, we argue that compactness should be thought of as a finiteness condition which is algorithmic in nature.

Multiplicative Weights: An Elegant Application to Maximum Flow

Chon Hou (Jophy) Ye — 2025-09-22

Multiplicative weights is a class of meta-algorithms commonly found in learning theory. It typically carries out several rounds of queries to oracles/agents, each time learning weights in an online manner given feedback from the current system to capture proficiency of them. It has found its use in game theory, machine learning, fast algorithms for optimization, etc. We begin with the classical example of multiplicative weights weighted majority. In the second part, we will see a delicate usage of multiplicative weights for approximating the maximum network flow, a well-known problem in theoretical computer science with many practical usages, accompanied by visualization from our simulation. The algorithm approximates maximum flow by repeatedly solving a related, computationally easier problem, the electrical flow of a circuit, whose parameters are derived from multiplicative weights. Multiplicative weights come in to adjust the resistances of the circuit online so that edge capacities are gradually obeyed. It is an elegant piece of work drawing insights
from learning theory, physics, numerical methods, and theoretical computer science.
This journal is intended for undergraduate readers broadly interested in mathematics and theoretical computer science, who have developed some mathematical maturity and are familiar with basic algorithms.

Exploring Parking Functions: Poset and Polytope Perspectives

Yan Liu — 2025-09-22

This paper provides an exploration of parking functions, a classical combinatorial object. We present two viewpoints on their structure and properties through poset of noncrossing partitions and polytopes.

Elliptic Curve Cryptography: Secure Digital Communication

Kian Kyars — 2025-09-22

The goal of this article is to explain the basic operations of elliptic curves in modular arithmetic. Knowing the principles behind elliptic curves is important for understanding the cryptographic security they provide. The widespread use of elliptic curves in cryptocurrencies and devices of the Fourth Industrial Revolution, both defining movements of our day, demonstrates its real-world significance. The target audience is early-year undergraduate students. We include appropriate formulas and figures to present elliptic curves both mathematically and geometrically. We conclude by illustrating elliptic curve applications in cryptography.