Grad Appreciation Week Poster Session Winner, Sarah Wolff
Congratulations to Sarah Wolff, graduate student in the Department of Mathematics, who was one of four winners of the Graduate Poster Session held recently in Alumni Hall! (Below is a summary of Wolff’s poster.)
Poster Title: Separation of Variables and the Computation of Fourier Transforms on Finite Groups
Fourier transforms are functions with applications in audio signal processing, medical imaging, image processing, pattern recognition, and much more! A Fourier transform allows us to view data (for instance, a signal or sound wave) in a setting where it is easier to analyze. For example, suppose you are listening to an orchestra and recording the sound wave as a function of time. We call this viewing the sound wave in the “time domain.” The graph you get is useful for telling us what the orchestra sounds like at each moment, but what if we wanted to be able to figure out what instruments are being played by just looking at the graph? One way to find this out would be to analyze the frequencies of the sound wave. However, that can be hard to determine at first glance! This is where the Fourier transform is useful; it gives us a way to move from the “time domain” to the “frequency domain”—a setting in which the sound wave is now viewed as a function of frequency instead of time. This provides us with all sorts of new information about the sound wave that would have been much harder to figure out with the original graph.
In its many applications, the Fourier transform is the key to analyzing and processing signal data, and the fact that we can do a Fourier transform very quickly (first discovered by Carl Friedrich Gauss, a German mathematician, but rediscovered by James Cooley and John Tukey, American mathematicians, in 1965) has given us much of the digital technology that we take for granted today.
I work with generalizations of Fourier transforms. One such generalization is to consider functions on permutations— the different ways of ordering a bunch of letters. For example all the permutations of the letters A, B, and C are: ABC, ACB, BCA, BAC, CBA, CAB. Permutations are useful in voting theory when we consider elections where voters rank the candidates. For example, suppose I want to rank Candidate B first, Candidate C second, and Candidate A third. This can be represented by the permutation BCA. With this viewpoint, a function on permutations would tell us how many votes each possible ranking received. We call this a function in the “voting domain.” Here is where the Fourier transform is useful again! It allows us to move from the “voting domain” to the “outcome domain”—a setting where it is easier to analyze the outcome of the election and determine which rankings had the biggest impact on the outcome.
These and other generalizations make Fourier transforms and efficient algorithms to compute them an exciting topic of study!
poster summary by Sarah Wolff
Wolff, along with the other poster winners, explains her research in this video: