https://journals.library.columbia.edu/index.php/cjum <p><span style="font-weight: 400;">The primary goal of the Columbia Journal of Undergraduate Mathematics is to provide undergraduate readers with high-quality, accessible articles on challenging topics, or novel approaches to teaching more familiar concepts. Articles published are purely expository; we do not accept research papers. Most range from 5 to 15 pages in length, with the primary exceptions being senior theses written by students at Columbia and other universities alike. The journal also accepts and publishes mathematical artwork with clear pedagogical value.</span></p> Columbia University Libraries en-US Columbia Journal of Undergraduate Mathematics 3065-1670 <p><span style="font-weight: 400;">All content is subject to a Creative Commons </span><a href="https://creativecommons.org/licenses/by-nc-nd/4.0/" target="_blank" rel="noopener">Attribution-NonCommercial-NoDerivs 4.0 International</a> License.</p> https://journals.library.columbia.edu/index.php/cjum/article/view/14896 <p>This paper gives a detailed proof of the boundedness of the scalar wave equation on a Schwarzschild spacetime using modern energy estimates. The first proof of this statement was given by Kay and Wald in by using Killing vector fields and discrete isometries of the spacetime. However, since then, new techniques to analyze vector fields near the event horizon have been developed by Dafermos and Rodnianski in which has given rise to a new proof strategy. This new proof, using the red shift effect, has been sketched in many lecture notes, but many details are left out. This paper fills in those details and presents the theorem in a self-contained manner.</p> Katherine Mekechuk Copyright (c) 2026 Katherine Mekechuk https://creativecommons.org/licenses/by-nc-nd/4.0 2026-06-29 2026-06-29 3 1 10.52214/cjum.v3i1.14896 https://journals.library.columbia.edu/index.php/cjum/article/view/14895 <p>In this exposition, we will focus on John Nash's Isometric Embedding Problem: What is the lowest dimension d(n) of Euclidean space that a compact Riemannian manifold of dimension n can be isometrically embedded into? First, we set up the underlying perturbation problem and introduce the loss of differentiability issue that arises. Then, we discuss an elegant solution to the loss of differentiability by Matthias Günther discovered in 1987: His solution involves a nice property of the Laplace operator--in particular, that Δ-1 has a well-defined bounded inverse on Hölder spaces. Günther was able to use this to avoid the monstrosities of proving the inverse function theorem of Nash--Moser and to improve Nash's original upper bound of n(3n+11)/2 for the required dimension of the ambient Euclidean space to max{n(n+5)/2, n(n+3)/2+5}.</p> Shiv Yajnik Copyright (c) 2026 Shiv Yajnik https://creativecommons.org/licenses/by-nc-nd/4.0 2026-06-29 2026-06-29 3 1 10.52214/cjum.v3i1.14895