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Abstracts for the MAA Undergraduate Poster Session Baltimore, MD January 18, 2019 Abstracts for the MAA Undergraduate Poster Session Baltimore, MD January 18, 2019 MATHEMATICAL ASSOCIATION OF AMERICA Abstracts for the MAA Undergraduate Poster Session Baltimore, MD January 18, 2019 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Dear Students, Advisors, Judges and Colleagues, If you look around today you will see approximately 350 posters and more than 550 student presenters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and universities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new PhDs, and administrators. Many of the judges signed up when they registered for the conference, but there are also a number of judges here today who volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by Tudor Investments and Two Sigma. We are also helped by the members of the MAA Committee on Undergraduate Students (CUS) in some way or other. They are: Emily Cilli- Turner, James Collins, Janine Janoski, Darci Kracht; Emille Davie Lawrence; Aihua Li; Sara Louise Malec; Rhonda McKee; Stacy Ann Muir; Andy Niedermaier; Pam Richardson; Peri Shereen; Hortensia Soto; and Violeta Vasilevska. There are many details of the poster session that begin with putting out the advertisement in FOCUS, making sure the online submission system works properly, and organizing poster boards and tables in the room we are in today that are attributed to Hortensia Soto (MAA Associate Secretary), Kenyatta Malloy (MAA), and Donna Salter (AMS). Our online submission system and technical support is key to managing the ever-growing number of poster entries we receive. Thanks to MAA staff, especially Kenyatta Malloy, for her work managing the system this year. Preparation of the abstract book is a time-consuming task. Thanks to Beverly Ruedi for doing the final production work on the abstract book. We would also like to thank James Collins (University of Mary Washington), Emille Davie Lawrence (University of San Francisco), Pam Richardson (Westminster College), Peri Shereen (California State Uni- versity Monterey Bay), and Doug Ensley (MAA) for their help in authoring the new judging form, as well as Aihua Li (Montclair State University) and Peri Shereen for organizing an orientation for the judges. Thanks to all the students, judges, volunteers, and sponsors. We hope you have a wonderful experience at this years poster session! Eric Ruggieri College of the Holy Cross Chasen Smith Georgia Southern University 2 Organized by the MAA Committee on Undergraduate Students The MAA gratefully acknowledges the support of Tudor Investment Corporation and Two Sigma. Their generosity on behalf of the 2019 Undergraduate Student Poster Session enables students to interact with peers and role models in the mathematical sciences during the largest mathematics meeting in the world. Titles, Authors, Advisors and Abstracts 1. Noetherian Rings with Unusual Prime Ideal Structures. Anya Michaelsen Williams College Advisor(s): Susan Loepp, Williams College For a ring R, the set of prime ideals of R, called the spectrum of R, is a partially ordered set with respect to inclusion. Given a partially ordered set X, M. Hochster showed exactly when X can be realized as the spectrum of a commutative ring. It is unknown, however, when a partially ordered set can be realized as the spectrum of a commutative Noetherian ring. In 2016, C. Colbert showed that there exists an uncountable Noetherian commutative ring with Krull dimension at least 2 and a countable spectrum. We extend this result in two ways. First, we consider a spectrum with a countable and uncountable branch and discuss progress toward constructing a Noetherian ring with this spectrum. Second, we construct a 2-dimensional uncountable excellent ring with a countable spectrum. We will outline both constructions as well as future work to extend these results. This research was done as part of the 2018 SMALL REU program at Williams College. 2. Transition Matrices for Young’s Representations of Sn Sam Armon Macalester College Advisor(s): Tom Halverson, Macalester College The irreducible representations of the symmetric group Sn are indexed by integer partitions n. The corresponding ` simple modules are denoted Sn n , and the dimension of Sn equals the number f of standard Young tableaux f j ` g of shape . In the 1920s, A. Young defined two bases of Sn — the natural and seminormal bases — by describing the action of Sn on vectors indexed by standard Young tableaux of shape . We give a formula for the entries in the transition2 matrix between the seminormal and natural bases, answering an open question in the representation theory of the symmetric group. Our method is to use a graph , which has vertices labeled by the standard tableaux of shape and colored edges corresponding to adjacent transpositions in Sn. This graph is the Hasse diagram of weak Bruhat order on standard tableaux. The entries in the transition matrix are calculated using weights on walks on . We generalize our method to the wreath product group Sn Zr and the Iwahori-Hecke algebra Hn.q/ of Sn. o 3. Putting the “k” in Curvature: k-Plane Constant Curvature Conditions Maxine Calle Reed College Advisor(s): Corey Dunn, California State University, San Bernardino Differential geometry studies properties of manifolds using the tools of calculus and linear algebra, and we can char- acterize the local behavior of manifolds by studying representative model spaces. Studying curvature in this con- text allows us to generate representative numbers for model spaces, particularly by taking measurements that are basis-independent. This research generalizes two such curvature invariants known as constant sectional curvature and constant vector curvature. While both conditions are well understood considering 2-plane sectional curvature in 3- dimensional model spaces, little is known with regard to higher dimensions. We generalize these curvature conditions using k-plane scalar curvature in Riemannian model spaces of arbitrary finite dimension, and many results are gener- alizations of known aspects of 2-plane constant curvature conditions. By studying these k-plane curvature invariants, we can further characterize model spaces by generating representative numbers for various subspaces. This research was conducted as a part of the 2018 REU program at California State University, San Bernardino. 4. The Hilbert Series of the Polynomial Representation of the Rational Cherednik Algebra of type A in Characteristic p for p n1 n1 j Merrick Cai Kings Park High School Advisor(s): Daniil Kalinov, MIT We study the polynomial representation Lt;c./ of the rational Cherednik algebra Ht;c.Sn; h/ of type An 1 in an algebraically closed field of positive characteristic p where p n 1. In the t 0 case, for all c 0 we give a complete description of the polynomials in the maximal proper gradedj submodule kerD B, the kernel of¤ the contravariant form B, and subsequently find the Hilbert series of the irreducible quotient L0;c./. In the t 1 case, we give a complete D 5 6 Titles, Authors, Advisors and Abstracts description of the polynomials in ker B when the characteristic p 2 and c is transcendental over F2, and compute D the Hilbert series of the irreducible quotient L1;c./. 5. A Generalization of the Indefinite Spectral Theorem Andrew Lavengood-Ryan California State University, San Bernardino Advisor(s): Corey Dunn, California State University, San Bernardino One of the most profound and important theorems in Linear Algebra is the Spectral Theorem. This theorem is well- known for positive definite inner product spaces, but is less developed in indefinite inner product spaces.We aim to develop a generalization of the Indefinite Spectral Theorem by examining Jordan Form matrices and their associated metric. We start with an operator that is self-adjoint with respect to some inner product on a real vector space. We then complexify and extend the inner product to be Hermitian; from this, we determine the ”best basis” for this matrix, forcing the metric to be a direct sum of Standard Involuntary Permutation matrices. The theorem we’re after is simply a corollary of this result. This research has had applications in Differential Geometry, but would undoubtedly have further applications in Linear Algebra, as well. 6. On Characteristics of Hyperfields Obtained as Quotients of Finite Fields Hahn Lheem PROMYS Dylan Liu PROMYS Advisor(s): Matthew Baker, Georgia Tech Let p be an odd prime. We show that there exists an explicit form for the characteristic of the hyperfield Fp=G where G 1; 2; 3; 4. Finally, we prove a general form of the characteristic for hyperfields where G is prime. j j D j j 7. The voiced Tonnetz and the J -group, with illustrations in Schubert’s B[ major Sonata Patrick DeBonis University of New Mexico Advisor(s): Thomas Fiore, University of Michigan-Dearborn Motivated by Schubert’s Piano Sonata in B[ Major, D. 960, we expand knowledge of the J group developed by Fiore and Noll. In the spirit of David Lewin, we use the PLR-group to make both global and local maps of the sonata, following Richard Cohn. We use the Structure Theorem of Fiore-Noll to find J group operations that realize some of these musical motions while preserving voice ordering. As an enrichment of the neo-Riemannian Tonnetz we develop a voice leading Tonnetz for the J group as a simplicial set, rather than simplicial complex.
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