Harvard College Mathematics Review

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The study of arrangements of non-overlapping spheres in space, known as sphere packings, has given rise to numerous questions such as finding the densest sphere packing and the kissing number problem. Aside from these theoretical considerations, sphere packings is also closely related to the theory of codes. This paper introduces the basic problem of finding efficient error-correcting codes and discusses its geometric interpretation as a sphere packing problem. The paper focuses on certain families of codes with special properties and explores these properties in connection with two famous codes.

This paper rephrases Kummer's proof of many cases of Fermat's Last Theorem in contemporary notation that was in fact derived from his work. Additionally, this paper develops a reformulation of the proof using class field theory from a modern perspective in a manner similar to the tactics used for the complete proof, and describes how Kummer's proof strategy can generalize to solve the theorem for a broader set of primes.

Inspired by geometric intuition, the Braid Group admits a neat algebraic structure with complexity sufficient to suggest cryptographic applications. We investigate these applications and present an algorithm for normalizing words in the braid group. Such normalization is critical to a realistic implementation of a computational system based on braids. We also present a simple cryptographic protocol based on the braid group.

We explore an intimate connection between Young tableaux and representations of the symmetric group. We describe the construction of Specht modules which are irreducible representations of S_n, and also highlight some interesting results such as the branching rule and Young's rule.
Some knowledge of basic representation theory is assumed.

In this paper we provide two simple new versions of Arrow's impossibility theorem, in a model with only one preference profile. Both versions are transparent, requiring minimal mathematical sophistication. The first version assumes there are only two people in society, whose preferences are being aggregated; the second version assumes two or more people. Both theorems rely on assumptions about diversity of preferences, and we explore alternative notions of diversity at some length. Our first theorem also uses a neutrality assumption, commonly used in the literature; our second theorem uses a neutrality/monotonicity assumption, which is stronger and less commonly used. We provide examples to illustrate our points.

We discuss a famous problem about right triangles with rational side lengths. This elementary-sounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis).