Prof. Edward Aboufadel's Applied Mathematics Projects
This page provides an overview of applied mathematics projects, many that I have done with students, during my career. A significant portion of this work has been done at the Mathematics REU at Grand Valley State University. Starting in 2014, my projects are part of the Center for Applied Mathematics Projects (CAMP) at GVSU. Current areas of focus for CAMP are 3D printing, data analytics and visualization, wavelets, and applied mathematics competitions. Watch this page for updates.
I enjoy working on interesting applications of mathematics to various areas of science, medicine, and engineering. Many of the projects I have pursued utilize wavelets. As I learned about wavelets in the 1990's, I came across how the FBI uses wavelets, and an application of wavelets to denoising musical recordings. Since 2000, I have been working with students on other applications of wavelets. Wavelet-based filters are applied for basically two reasons: information rearrangement and information reduction. In each of these projects, the use of wavelets is a key step in the method.
This is a list of the projects and the people involved, along with links to publications or videos, when available:
2015: Visualizing Water Pollution Data Using Beck-Style Flow Path Maps. This entry (found at the bottom of the linked web page) was a Runner Up in the Visualizing Nutrients Challenge. Collaborator: Daniel P. Huffman.
2013-16: 3D Printing. Student Researchers: Melissa Sherman-Bennett and Sylvanna Krawczyk (2013), Lindsay Czap (2014), Samantha Law (2016), along with further work done in 2014-16 by EFA.
2014: NASA-Nex Challenge. This entry was a winner of the "Ideation" stage of the Challenge. (No collaborators.)
2014: Ordinary Generating Functions of Context-Free Grammars. Student Researcher: Tanner Swett
2009: Classification of Diabetic Patients Using Wavelets. Student Researchers: Robert Castellano and Derek Olsen. We also have their award-winning MathFest talk available.
2008: Creating Chuck Close-like Portraits Using Wavelets. Student Researchers: Clara Madsen and Sarah Boyenger Kaindl.
2004: Identifying Airplanes in Aerial Photographs Using Wavelets. Student Researchers: Kevin Brink and Drew Colthorp.
1997: Qualitative Analysis of a Singularly-Perturbed System of Differential Equations Related to the van der Pol Equations. (No collaborators, but thank you to my thesis advisor, Jane Smiley Cronin Scanlon.)
1996: Research in Differential Games. Student Researcher: David Szurley
Books: Discovering Wavelets, with Prof. Steven Schlicker (1999). Advanced Engineering Mathematics with Profs. Merle Potter and Jack Goldberg (2005).
About Wavelets ...
Figure 2 is a graph of the scaling function known as D4. Scaling functions play a role similar to the sine and cosine functions of Fourier analysis in that other wavelets in a wavelet family are generated from the scaling function. The scaling function is also called the father wavelet.
Some of this material is based upon work supported by the National Science Foundation under Grants No. DMS-0451254, DMS-0137264, DMS-9820221, DMS-1003993, and DMS-1262342. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).