Dr. Lidia Mrad: Torus Formation in Chromonic Liquid Crystals
Liquid crystals are classical examples of phases intermediate in character between isotropic liquids and crystalline solids. In addition to being partially ordered, some liquid crystals manifest sensitivity to changes in concentration when added to a solution. Chromonic liquid crystals fall under this category. Chromonics are composed of disc-like molecules that tend to form rings when their concentration in a solution reaches a threshold value. These rings then aggregate into interesting geometrical shapes. An important question in this setup is how the dominant mechanism - shape formation in this case - is affected by specific system parameters. We formulate this question as an energy minimization problem which allows us to use several variational tools. Taking into account key parameters, we prove the existence of a torus solution to the ensuing equation and compute its dimensions. As confined deoxyribonucleic acid (DNA) forms chromonic liquid crystal phases, these results can shed light on the packing mechanism of a viral DNA in a capsid.
Dr. Anna Aboud: Lost in Translation?
When digital signals are sent, they are broken down into small components which must be reconstructed upon receipt. One well-established reconstruction method is the Kaczmarz algorithm, which uses iterative projections to construct a sequence of approximations converging to the signal in question. In this talk, we will introduce the Kaczmarz algorithm, discuss its applications in signal processing and data science, and conclude by presenting a more flexible alternative.
Dr. Elizabeth Drellich: The Abstract Algebra behind Animation
How do animators make sure that complicated surfaces appear smooth on screen? How do you program a 3D printer to get the sleek curves on your model of your favorite car? And what do these questions have to do with algebra? The surface of your model car or your favorite Pixar character's face is made up of a collection of polynomials that fit together smoothly called a spline. These splines are models, but they are also algebraic objects that can be added and multiplied. Some splines contain deep information about the structure of groups and algebraic varieties. This talk will introduce you to splines and their traditional uses, and show you how a concept that started centuries ago with shipbuilders can transform into a powerful algebraic tool.
Dr. Eva Goedhart, Lebanon Valley College: Pat-A-Cake, Pat-A-Cake Baker’s Man, Bake Me a Logarithm as Fast as You Can
After a quick run through a little history of the mathematician Diophantus, Dr. Goedhart presented a friendly version of Baker's method of linear forms in logarithms to solve Diophantine equations.
Dr. Michael Shearer, North Carolina State University: Nonlinear Waves
As an introduction to the broad subject of nonlinear waves, I’ll show a variety of wave equations that are fundamental to the study of shock waves and dispersive waves. G.B. Whitham formulated a system of partial differential equations that describe the evolution of the essential features of oscillatory waves. Through examples, asymptotics and numerical experiments, I’ll discuss phenomena related to properties of the Whitham modulation equations.
Ben Orlin, author of Math with Bad Drawings: The Unlikely Friendship of Math and Science
Math and science are often lumped together - in universities, on bookshelves, and in acronyms. But what, exactly, do the hard-nosed pragmatist science and soft-spoken dreamer mathematics have in common? On the one hand, mathematics is a toolkit, crafted to serve real-world purposes. On the other hand, math is a realm of pure logic, ideas pursued for their own sake. My question: How is it that, from knot theory to meta-logic to higher-dimensional geometry, the math that sounds the most fanciful turns out to be the most useful?
Dr. Robert Grizzard, Lafayette College: Counting Polynomials
An active area of research in number theory is to take an infinite set of interest, and find an asymptotic formula for the number of them less than a given ``height.'' A classical example is the set of prime numbers, where the notion of ‘’height'' is just absolute value, and the asymptotic formula is given by the prime number theorem. In this talk, we’ll discuss some questions about counting polynomials of a fixed degree with integer coefficients. There are two notions of height here -- the ``naive height," which is just the largest absolute value of a coefficient, and the Mahler measure, which is a more subtle height function determined by the roots of the polynomial. The question of how many polynomials there are of bounded naive height is trivial, but the corresponding question for Mahler measure leads to a very interesting problem of counting lattice points inside certain ``star bodies'' in Euclidean space (there will be pictures!). For the naive height, a more interesting question is to give an upper bound on the number of reducible polynomials of bounded height.
Dr. Michael Allocca, Muhlenberg University: The Not-So-Trivial Problem of Sorting Circular Permutations
Permutations are a common tool used in mathematics, computer science, and the natural sciences in general. Circular permutations, the ordered rearrangement of enumerated objects in circular fashion, introduce a surprising degree of complexity. We will discuss the mathematics behind sorting circular permutations and connect it to a strategy for modeling evolutionary distance among organisms with circular chromosomes.
Sahana Balasubramanya, Vanderbilt University: Groups as Geometric Objects
Dr. Balasubramanya introduced the notion of a group, which is an algebraic structure on a set of elements with a binary operation. She then introduced the notion of generating sets and Cayley graphs of a group, which allow us to view the group as a geometric object - a graph. This duality provides us with new methods to study groups and is particularly useful when the Cayley graph is hyperbolic; a notion she also explained. The talk ended with a discussion on "Bridson's Universe" to provide an overview of the universe of groups.
Dr. Pedro Ontaneda, Binghamton University: Constructing Negative Curvature
Euclidean geometry is characterized ty the fact that the sum of the angles of triangles is 180 degrees. We say that Euclidean objects are flat, or that they have zero curvature. In negatively curved geometries the sum of the angles of (small) triangles is less than 180 degrees. Negative curvature is everywhere in mathematics: negatively curved spaces are fundamental objects in geometry, topology, analysis, algebra and dynamical systems. I will give some ideas of how to construct such objects.
Dr. Daniel Droz, Susquehanna University: Latin Squares and Why We Care
Latin squares have always been intriguing mathematical playthings – even today Sudoku puzzles show our fascination with these patterns. They are also the subject of serious mathematical study as combinatorial designs. These squares also have applications in efficient experimental design and ensuring messages are transmitted clearly over noisy channels. We give an introduction to the classical theory of how these objects are systematically constructed and the contexts in which they are applied.
Senior Mathematics Capstone Presentations
- Ruiwen Fu: Different Definitions of Chaos
- Yiran Mao: A Quantitative Measurement of Chaos: the Topological Entropy
- Blain Schiff: Markov Chains and Oscar the Grouchy Gambler
- Adrian Navarro: Traffic Flow: a New Cellular Automaton Model for Racing Cars
- Stephanie Leonardo: Exploring Sharkovsky's Theorem
- Sarah Walsh: Image Inpainting with PDEs
- Yulin Zhu: Mathematical Modeling of Atmospheric Dispersion
- Elizabeth Fox: Outsmarting the Burglar: A Mathematical Model to Beat Burglary