Visualizing Discrete Fourier Coefficients

Over the course of the 2010 University of Georgia REU, we spent an extensive amount of time understanding and applying properties of the discrete Fourier transform to topics in arithmetic combinatorics. In order to get an intuitive idea of what we were really looking at, I wrote a GUI program in Python to visualize the . . . [Read More]

On Fourier Pseudorandomness

As a final project at the 2010 University of Georgia mathematics REU, I wrote a manuscript discussing the properties and meaning of a mathematical notion called Fourier Pseudorandomness, which provides a quantitative measure of the randomness (in some sense) of a finite set of integers. Some of the more involved proofs require a first course . . . [Read More]

A Statement of the Riemann Hypothesis

The Riemann Hypothesis, a longstanding unsolved conjecture in analytic number theory, is considered by many mathematicians to be one of the most important unsolved problems in theoretical mathematics. To understand the statement takes only a typical undergraduate mathematics education, but to find a proof would be the capstone of a mathematical career.

The Riemann zeta function . . . [Read More]

Concerning Uncountable Sums

Sums are a beautiful notion which are sometimes taken for granted. 2 + 2 = 4, the sum from 1 to infinity of 1/2n is equal to 1, and so forth. In the case of countably infinite sums, questions of convergence become important. But what happens when you have a sum over uncountably many summands? It . . . [Read More]

A Small Subset of the Reals

Monet Let {qn} be an enumeration of the rationals, and for each n, let Un be an open interval of length 1/2n centered at qn. Denote the union of all Un by U. Then U is a dense open . . . [Read More]