NREUP 2015

National Research Experience for Undergraduates Program (NREUP) at University of Michigan-Dearborn: May 11, 2015 - July 3, 2015.

Acknowledgement

This program is an MAA activity funded by NSF grant DMS-1359016 and NSA grant H98230-15-1-0020.

Summary

General idea of decomposing a generic function (or a signal) into a superposition of symmetric pieces is a powerful tool in mathematics and science. A few specific areas it is used frequently include differential equations, signal processing, image coding and information theory. In the proposed summer program, we plan to present this general idea using two classical tools of decomposition, Fourier series and refinable functions, with applications in different areas of science. Many students learn these or similar tools in different courses but a coherent explanation with real world applications may get lost in the exposition. Our main goal is to get students familiar with this ubiquitous idea and prepare them to apply it to different problems.

In the program, students will obtain an in-depth understanding of Fourier series and refinable functions. They will learn about mathematical issues arising in the process of superposing pieces, like different convergence and divergence problems. They will also practice how to overcome these issues using new techniques. 

We believe this program will provide a valuable opportunity for students to learn more about how to use tools of mathematical analysis to resolve issues arising in real world applications. In addition, students will practice on scientific writing, oral presentations, and team work.

Faculty mentors: Hyejin Kim and Yunus Zeytuncu

Students: Mathew Gomez (University of North Texas), Jose Chavez (University of North Texas), Angel Castillo (Texas A&M), Jamie Jeffries (UMD), and Brandi Wingate (UMD), 

Guest students (UMD REU students) : David Wendl (UMD), Joshua Postel (UMD), and Joe Gaber (UMD).

Outcomes: Students worked in pairs on four different projects, 

- Angel Castillo and Jose Chavez, "Convergence Preserving Permutations and Divergent Fourier Series". 

- Jamie Jeffries and Brandi Wingate, "Isoperimetric Inequality in the Third Dimension". 

- Mathew Gomez and David Wendl, "How to refine multivariate polynomials and transformations of refinement masks". 

- Joe Gaber and Joshua Postel, "Investigation $\lambda$--permutations". 

1. Joe Gaber and Joshua Postel gave a 15 minutes presentation, "Investigation $\lambda$--permutations", at the MathFest 2015, Washington D.C. on August 7, 2015 and they received MAA student travel grant.

2. Angel Castillo and Jose Chavez were invited for twenty--minute oral presentation, "Convergence Preserving Permutations and Divergent Fourier Series", at the Young Mathematicians Conference which was held at the Ohio State University from August 21 to August 23 with full travel support.

3. Jamie Jefferies and Brandi Wingate were invited for a poster presentation, "Isoperimetric Inequality in the Third Dimension", at the Young Mathematicians Conference which was held at the Ohio State University from August 21 to August 23 with full travel support.

4. Jose Chavez and Angel Castillo gave a poster presentation, "Convergence Preserving Permutations and Divergent Fourier Series", at 2016 Joint Mathematics Meetings in Seattle, Washington on January 8, 2016. Their poster was selected as one of outstanding posters, that is, in the top 50 projects out of more than 300 projects.

5. Jamie Jeffries and Brandi Wingate gave a poster presentation, "Isoperimetric Inequality in the Third Dimension", at the 6th Annual Undergraduate Research Showcase on March 25, 2016.

   1. Joshua Postel gave a poster presentation, "Investigation $\lambda$--permutations", at the 6th Annual Undergraduate Research Showcase on March 25, 2016.