REU 2018
2018 REU Site in Mathematical Analysis, Algebraic Music Theory, and their Applications
Dates: Tuesday, May 29--Friday, July 20.
This REU program is supported by the National Science Foundation (DMS-1659203), the National Security Agency, the College of Arts, Sciences, and Letters, and the Department of Mathematics and Statistics at the University of Michigan-Dearborn.
Outcomes:
All groups presented at the 2018 Summer Undergraduate Michigan Mathematics Research Conference at Michigan State University on July 19, 2018.
Ahn, Bansil, Brown, Cardin were invited for twenty-minute oral presentation, "Spectra of Kohn Laplacians on Spheres", at the Young Mathematicians Conference with travel support.
Ahn, Bansil, Brown, Cardin, and Zeytuncu have uploaded to ArXiv their paper "Spectra of Kohn Laplacians on Spheres", and submitted to a journal for publication.
Balukonis, Bowden, Hourani, Lochner were invited for twenty-minute oral presentation, "The Numerical Range of a Composition Operator on the Hardy Space on the Unit Ball", at the Young Mathematicians Conference with travel support.
DeBrito, Keiser, Younes were invited for twenty-minute oral presentation, "Efficiency of a Moving Mesh System with a Curvature-type Monitor and an Application to Burger's Equation", at the Young Mathematicians Conference with travel support.
2018 Projects:
Spectral Theory and Complex Analysis (Advisor: Yunus E. Zeytuncu):
Students: John Ahn (Bowdoin College), Mohit Bansil (Michigan State University), Garrett Brown (Harvard University), Emilee Cardin (College of William and Mary)
The spectrum of an operator contains a lot of information about the operator and also the domain/manifold it is defined on. A famous example of this is the phenomena of hearing the shape of a drum as explained in the first two articles below. Domains and manifolds in several complex variables come with some canonical operators defined on them, and spectral properties of these operators can be used to answer many geometric questions. One of the particular geometric questions is the embeddability of abstract manifolds in the complex Euclidean space. This question can be answered by studying the eigenvalues of the d-bar Neumann problem. In this project, we will learn more about spectral theory, complex analysis, CR manifolds, and complex geometry. Prerequisites include a course in linear algebra, some familiarity with complex analysis, and basic real analysis/advanced calculus.
Selected References:
Daniel M. Burns and Charles L. Epstein. Embeddability for three-dimensional CR-manifolds. J. Amer. Math. Soc., 3(4):809–841, 1990.
A. Boggess. CR Manifolds and the Tangential Cauchy Riemann Complex. Studies in Advanced Mathematics. Taylor & Francis, 1991.
Siqi Fu. Hearing pseudoconvexity with the Kohn Laplacian. Math. Ann., 331(2):475–485, 2005.
Mathematical Music Theory: An Algebraic Analysis of Schubert's Piano Sonata in B-flat Major, D960 (Advisor: Thomas Fiore):
Students: Emma Bidwell (George Fox University), Patrick DeBonis (University of New Mexico), Siri Mellem (St. Olaf College)
The goal of this project is a global analysis of Schubert's Piano Sonata in B-flat Major, D960, using the group-theoretical framework of David Lewin. We will first highlight some of the key research in the extensive music-theoretical literature on this sonata (and sonata form in general), and study the recent group-theoretical results of Fiore, Noll, and Satyendra. Then we will revisit the Schubert literature from within this algebraic framework, and finally make our own analytical discoveries in the sonata. In the process we will formulate and prove theorems tailored to the musical phenomena of the sonata. We will also use the HexaChord software of Bigo to explore the piece topologically. Participants will use the Open Music software to typeset the musical illustrations in the write-up. Another approach to the sonata, using discrete Fourier transformations as developed by Amiot, Yust, and others, might also be explored. Prerequisites: A course in abstract algebra and knowledge of music theory would be very beneficial. Applicants should mention any previous experience with either of these fields in their personal statement.
Selected References:
Richard Cohn, “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert,” Nineteenth-Century Music, Volume 22, Issue 3, pages 213-232, 1999.
David Lewin. Generalized musical intervals and transformations. Oxford: Oxford University Press, 2011.
Thomas Noll. "Der Halbton im Zweilicht von Leitton- und Moduswechsel: Zum Stellenwert von Enharmonik in Schuberts B-Dur Sonata (D960)" in Musiktheorie an ihren Grenzen: Neue und Alte Musik, edited by Angelika Moths, Markus Jans, John MacKeown, and Balz Trümpy, Bern: Peter Lang, 2009.
Charles Rosen. Sonata Forms. New York: W.W. Norton & Co, 2006.
Thomas M. Fiore, Thomas Noll, and Ramon Satyendra. “Morphisms of Generalized Interval Systems and PR-groups,” Journal of Mathematics and Music, Volume 7, Number 1, pages 3–27, 2013.
Alissa Crans, Thomas M. Fiore, and Ramon Satyendra. “Musical Actions of Dihedral Groups,” American Mathematical Monthly, Volume 116, Number 6, June–July 2009, pages 479–495.
Thomas M. Fiore and Thomas Noll. “Voicing Transformations of Triads,” Preprint, 2017.
Louis Bigo, Moreno Andreatta, Jean-Louis Giavitto, Olivier Michel, and Antoine Spicher (2013), "Computation and Visualization of Musical Structures in Chord-based Simplicial Complexes, "Proceedings of the Conference MCM 2013. Lecture Notes in Computer Science / LNAI, Springer, pages 38-51.
Louis Bigo, Daniele Ghisi, Antoine Spicher, and Moreno Andreatta (2014), "Spatial Transformations in Simplicial Chord Spaces", in Anastasia Georgaki and Georgios Kouroupetroglou (Eds.), Proceedings ICMC-SMC 2014, pages 1112-1119.
Louis Bigo and Moreno Andreatta (2015), "Topological Structures in Computer-Aided Music Analysis," in David Meredith (ed.), Computational Musicology, Springer, p. 57-80.
Linear Fractional Transformations (Advisor: John Clifford):
Students: Julia Balukonis (Providence College), Laney Bowden (Colorado State University), Fatme Hourani (University of Michigan-Dearborn), Ellie Lochner (University of Wisconsin-Eau Clare)
The focus of the summer REU project will be the study the numerical range of a composition operator on the Hardy space. The three main ideas are numerical range, composition operator, and Hardy space. We will start by studying the basic properties of the numerical range; the numerical range of a 2 by 2 matrix is an ellipse, the numerical range is invariant under unitary conjugation, the numerical range is convex and the numerical range of a normal matrix is the convex hull of the eigenvalues. The goal of the project will be to characterize the numerical range of a composition C_{A} acting on the Hardy space where A is an n by n matrix that is a self-map of the unit ball in C^n. In the case of n=1, the result is known and interesting. We will work through the n=1 case carefully as we learn about the numerical range, composition operators, and the Hardy space. For the case n>1 the results are not known. Our first goal will be to characterize the numerical range of the composition operator C_{A} when A is 2 by 2. We will then investigate what results for n=2 generalize for all n>1. Once we are done with the n=2 we will have many interesting questions to explore.
Selected References:
Shapiro, J.H., Notes on the numerical range.
Psarrakos, P.J., and Tsatsameros, M.J., Numerical range: (in) a matrix nutshell.
Properties of Moving Mesh Systems (Advisor: Joan Remski):
Students: Marianne DeBrito (Lawrance Tech. University), Annaliese Keiser (Bowling Green State University), Taima Younes (University of Michigan-Dearborn)
Moving mesh methods are adaptive numerical techniques for solving partial differential equations (PDEs) that model physical phenomena in engineering and the sciences. The key idea is that mesh points move in the domain toward places where the solution of the physical PDE has particular characteristics, like steep gradients or large curvature. This movement depends on the choice of a function called the monitor function. Because more mesh points are concentrated where the solution changes most, the solutions can be computed with greater accuracy and efficiency than with a fixed mesh. It has been shown that the moving mesh system nicely balances properties between the mesh equation and the physical PDE for one particular class of monitor functions. In this project, we will see whether other types of monitor functions exhibit a similar balance of desirable properties. Students will study how the moving mesh equation works in 1D with a fixed function acting as the PDE solution and then later study a moving mesh system based on an appropriate application of their choice. The essential prerequisites are linear algebra and differential equations. Some programming experience may be helpful but is not necessary.
Selected References:
Weizhang Huang and Robert D. Russell. “Adaptive Moving Mesh Methods,” Springer, 2011
J. Remski, J. Zhang, and Q. Du, “On Balanced Moving Mesh Methods,” Journal of Computational and Applied Mathematics, vol 265, Current Trends and Progresses in Scientific Computation — Dedicated to Professor Ben-yu Guo on His 70th Birthday, August, 2014. pp. 255-263.
If you have any questions about REU at the University of Michigan-Dearborn, please contact us at um-dearborn-math-reu@umich.edu.