Harmonic Analysis and Representation Commons^™

The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal

Honors Theses

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Parametric Polynomials For Small Galois Groups, Claire Huang

Honors Theses

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field. …