Columbia Journal of Undergraduate Mathematics

Front matter: Letter from the Editors, Masthead, and Contents

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Front matter for the Columbia Journal of Undegraduate Mathematics Volume 1 Issue 1. Contains the Letter from the Editors, Masthead, and Contents.

A topological proof of the Riemann–Hurwitz formula

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The Riemann–Hurwitz formula is generally given as a result from algebraic geometry that provides a means of constraining branched covers of surfaces via their Euler characteristic. By restricting to the special case of compact Riemann surfaces, we develop an alternative proof of the formula that draws on topology and manifold theory as opposed to more advanced algebraic machinery. We first discuss the foundation in manifold theory, defining Riemann surfaces and providing an example of the complex projective line. We then discuss the local topological structure of holomorphic maps between Riemann surfaces, introducing the notion of a branched cover and of branch points. Next, we discuss triangulations of a topological space and use this to intro- duce the Euler characteristic of Riemann surfaces. Using these definitions, we explicate and prove the Riemann–Hurwitz formula on compact Riemann surfaces. To conclude, we discuss consequences of this formula for adjacent fields such as algebraic topology. We provide visual intuition and examples throughout, drawing primarily on Szameuly’s Galois Groups and Fundamental Groups (2009), as well as Forster’s Lectures on Riemann Surfaces (1981), Guillemin and Pollack’s Differential Topology (1974), and a few other supplementary sources. The main prerequisite for this paper is a background in topology and covering spaces.

Representations of complex tori and GL(2, C)

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Groups and their representations have been studied for a long time. One can extend the notion of a group by asking the group axioms to hold in other categories. A group in the category of smooth manifolds is a Lie group, and a group in the category of algebraic varieties is an algebraic group. In this paper, we discuss the representation theory of algebraic groups, in particular complex tori and GL(2,C).

The Gauss–Bonnet theorem

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The Gauss–Bonnet theorem is a crowning result of surface theory that gives a fundamental connection between geometry and topology. Roughly speaking, geometry refers to the “local” properties—lengths, angles, curvature— of some fixed object, while topology seeks to identify the “global” properties that are unchanged by a continuous deformation, such as stretching or twisting. The theorem formalizes an intuitive idea: continuous changes to curvature on one region of a surface will be balanced out elsewhere, so the total curvature of the surface stays the same.

Explicitly, the Gauss–Bonnet theorem says that a surface’s total curvature, defined using its local Gaussian curvature, is directly proportional to the number of holes in the surface, which comes from an invariant quantity called its Euler characteristic. The Euler characteristic is a way of classifying which surfaces can be continuously deformed into one another; as an informal example, the classic joke that “a topologist is a person who cannot tell the difference between a coffee mug and a doughnut” comes from the fact that the objects each have one hole. Even though a coffee mug and a doughnut have visibly different geometric shapes, according to the Gauss–Bonnet theorem, both objects will have the same total curvature. The proof itself is delightfully systematic: we first find the total curvature of a curve on a plane, extend that result to curves on three-dimensional surfaces, extend that result to “polygons” on surfaces, and finally the entire surface.

In Section 2, we prove Hopf’s Umlaufsatz for the total curvature of a simple closed curve in R². Sections 3, 4, and 5 introduce concepts from differential geometry to define Gaussian curvature. In Section 6, we prove the local Gauss–Bonnet theorem for the total curvature of a surface polygon. At last, in Section 7, we prove the global Gauss–Bonnet theorem for compact surfaces by covering the surface with polygons and applying the local Gauss–Bonnet theorem to each one.

Our discussion focuses on exposition, and references will be given in place of tedious computations when reasonable. This paper assumes a somewhat rigorous understanding of multivariable calculus and linear algebra, as well as some elementary group theory.

The Peter–Weyl theorem & harmonic analysis on S^n

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For finite groups, the Artin–Wedderburn theorem gives a precise decomposition of the algebra of all C-valued functions into matrix algebras. Specialized to the case of cyclic groups, this produces the classical discrete Fourier transform. In this paper, we endeavor to extend these techniques to compact topological groups, proving similar harmonic decompositions on S¹, S², and S³.

Elliptic bootstrapping and the nonlinear Cauchy–Riemann equation

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The goal of this paper is to deduce a nonlinear elliptic regularity result from a linear one. In particular, elliptic bootstrapping is a powerful method to determine the regularity of a solution to a partial differential equation. We apply elliptic bootstrapping and linear elliptic regularity to the nonlinear Cauchy–Riemann equation. In doing so, we generalize the fundamental analytic result that holomorphic functions are automatically smooth. In particular, we show that, under certain conditions, the same is true for so-called J-holomorphic functions. We conclude by discussing how this nonlinear regularity result relates to ideas in symplectic geometry.