Title: Invisible Lattice Points
Speaker: Pam Harris (Williams College)
Abstract: This talk is about the invisibility of points on the integer lattice ℤ ✕ ℤ, where we think of these points as (infinitely thin) trees. Standing at the origin one may notice that the tree at the integer lattice point (1, 1) blocks from view the trees at (2, 2), (3, 3), and, more generally, at (n, n) for any n ∈ ℤ≥0. In fact any tree at (ℓ, m) will be invisible from the origin whenever ? and m share any divisor d, since the tree at (ℓ/D, m/D), where D = gcd(ℓ, m) blocks (ℓ, m) from view. With this fact at hand, we will investigate the following questions. If the lines of sight are straight lines through the origin, then what is the probability that the tree at (ℓ, m) is visible? Meaning, that the tree (ℓ, m) is not blocked from view by a tree in front of it. Is possible for us to find forests of trees (rectangular regions of adjacent lattice points) in which all trees are invisible? If it is possible to find such forests, how large can those forests be? What happens if the lines of sight are no longer straight lines through the origin, i.e. functions of the form f(x) = ax with , but instead are functions of the form f(x) = axb with b a positive integer and a ∈ ℚ? Along this mathematical journey, I will also discuss invisibility as it deals with the underrepresentation of women and minorities in the mathematical sciences and I will share the work I have done to help bring more visibility to the mathematical contributions of Latinx and Hispanic Mathematicians.
Math work is joint with Bethany Kubik, Edray Goins, and Aba Mbirika. Diversity work with Alexander Diaz-Lopez, Alicia Prieto Langarica, and Gabriel Sosa.
Biography: Pamela E. Harris is a Mexican-American Assistant Professor in the department of Mathematics and Statistics at Williams College. She received her B.S. from Marquette University, and M.S. and Ph.D. in mathematics from the University of Wisconsin-Milwaukee. Her research interests are in algebra and combinatorics, particularly as these subjects relate to the representation theory of Lie algebras. Her recent research on vector partition functions and projects in graph theory has been supported through awards from the National Science Foundation and the Center for Undergraduate Research in Mathematics. Harris co-organizes research symposia and professional development sessions for the national conference of the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science, was a Mathematical Association of America’s Project NExT (New Experiences in Teaching) Fellow from 2012-2013, and is an editor of the e-Mentoring Network blog of the American Mathematical Society. In 2016, she co-founded www.Lathisms.org an online platform that features prominently the extent of the research and mentoring contributions of Latin@s and Hispanics in the Mathematical Sciences. She is also the lead editor for the Special Issue on Motherhood and Mathematics of the Journal of Humanistic Mathematics.