Harvard College Mathematics Review

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Motivated by a remarkable 18th-century result about polyhedra known as Euler's formula, we will develop the notion of the Euler characteristic in the more modern context of CW complexes. The fact that is a homotopy invariant gives an easy (perhaps trivializing) proof of Euler's formula. We then develop two non-elementary methods of computing in specific cases: Morse theory and the Poincaré-Hopf Index Theorem. Both will be used to compute the Euler characteristic of closed orientable surfaces, using culinary analogies. In an appendix, the former will also be used to compute the Euler characteristic of real projective space.

The focus of this paper is the famous theorem on primes in arithmetic progressions due to Dirichlet: if a and m > 0 are relatively prime integers, then there exist infinitely many primes of the form a+km with k a positive integer. The proof of this theorem in the general case uses analytic techniques, and in fact some key statements heavily rely on complex analysis. The case a = 1, however, can be handled by purely algebraic methods as we will show in Section 2.1 following suggestions given in [La]. In Section 2.2, we will outline the idea of the proof of Dirichlet's theorem as it is presented in [IR] and [Kn] for the case m = 4. Finally, in Section 2.3, after a brief discussion of characters of finite abelian groups following [Se], we will present the proof of Dirichlet's theorem (cf. [IR,Kn,Se]).

This project is based on the study of two kinds of representation theory: quiver representation theory and Lie algebra representation theory. By looking at some simple examples, we'll show how the two are connected. Indeed, we'll identify the isomophism classes of simple and indecomposable representations of a particular quiver with relation with the equivalence classes of simple and indecomposable representations of .

Complex networks such as the World Wide Web and social relationship networks are prevalent in the real world, and many exhibit similar structural properties. In this paper, a fitness-based model is developed for these complex networks. This model employs a purely "better-get-richer" method of network construction that is believed to realistically simulate the growth process of most real-world networks. Both computer-simulated results and theoretical analysis show that the degree distribution of networks created with this model depends on the distribution of vertex fitnesses; a power-law fitness distribution results in the commonly observed scale-free network structure. In addition, results indicate a small average path length and large clustering coefficient, in accordance with real-world phenomena. It is proposed that this model may serve as a possible explanation of the prevalence of scale-free networks in the real world.

The practice of neglecting small terms of an equation is analyzed in the case of polynomial root approximations. Our discussion centers on the following new result: The roots of a polynomial can be approximated self-consistently by roots of much simpler equations consisting of pairs of terms from the polynomial.

The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if A, B, C are relatively prime integers such that A + B = C then A, B, C cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat's Last Theorem to the formulation of the ABC conjecture by Masser and Oesterlé. We then state the conjecture and give a sample of its many consequences and the few very partial results available. Next we recite Mason's proof of an analogous assertion for polynomials A(t), B(t), C(t) that implies, among other things, that one cannot hope to disprove the ABC conjecture using a polynomial identity such as the one that solves the Diophantine equation x² + y² = z². We conclude by solving a Putnam problem that predates Mason's theorem but is solved using the same method, and outlining some further open questions and fragmentary results beyond the ABC conjecture.

Unlike any other article in this journal, this one begins with a warning: Categories, beautiful and powerful as they may be, are not panacea and should be used with great prudence. This short note presents a fun, but silly use of categories.