Research by Undergraduates in Mathematics Boston University Symposium 2007

Faculty Sponsor: Steven J. Miller

Title: The Sato-Tate Conjecture 'on average'

ABSTRACT: The Sato-Tate conjecture is a beautiful and deceptively simple statement on the arithmetic of elliptic curves over finite fields, with surprising links to many other deep results in number theory. Recently the Sato-Tate conjecture was proven for a wide class of elliptic curves. I will explicate this conjecture (with numerical examples and motivation), and explain why it is easy to prove 'on average' in certain situations.

Presenter: Marc Kelechava, CAS 2008, Boston University

Faculty Sponsor: Emmma Previato

Title: On Lattice Path Enumeration and Epidemiology

ABSTRACT: We explore an application of lattice path enumeration to a problem in epidemiology, the study of the spread of diseases within human populations. An epidemiologist, Dr. D. Ozonoff, discovered equivalence between this problem and a problem in lattice path enumeration, a branch of enumerative combinatorics. A geometric interpretation allows us to compute the number of underdiagonal paths that touch the diagonal line a fixed number of times. We derive a formula encapsulating information about the sub-lattices, and then we deduce the probability that a path is underdiagonal. We have found one-to-one correspondences with well-known number sequences and the number of underdiagonal paths that touch the diagonal once.

ABSTRACT [by Emily Berman, SMG 2009, Campus Campaign Coordinator]: As math concentrators/majors, let me throw some numbers at you: 13 million, 50%, 1 in 10. Of the 13 million children growing up in poverty in the United States, only 50% on average will graduate from high school. If they graduate, they do so at an 8th grade reading and math skill set. Only 1 in 10 of these 13 millionwill graduate from a four-year college. That's only 1.3 million. (Source: National Assessment of Educational Progress, 2005, cited on www.teachforamerica.org) Teach For America, a nonprofit organization committed to closing the achievement gap, recruits seniors to teach in low-income schools across the nation for a two year period. It is a full salary job in which you can impact the life of low-income students, giving them educational opportunities that they do not have based upon their socioeconomic background. Registering on the website gives you access to look at the application at no consequences.

Glenn H. Stevens (Department of Mathematics and Statistics, Boston University, Program Director, Focus on Mathematics)

ABSTRACT: Do you like doing challenging mathematics? Do you enjoy sharing ideas with others? If so, then Boston University's Noyce Scholars Program may be the perfect opportunity for you. With funding from the National Science Foundation, Boston University offers full scholarships for outstanding graduating math majors to our MAT program in Mathematics Education. The Noyce Program features a number of unusual elements -- focusing on significant mathematics and developing applications to classrooms in high need school settings. This presentation will describe the Noyce Program in detail and will offer sample mathematics activities for everyone to enjoy.