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Invited Paper Session Abstracts - Open and Accessible Areas in Computational Mathematics

Please note: all sessions are listed in Mountain Daylight Time (MDT = UTC-6:00)

Wednesday, August 4, 1:00 p.m. - 3:30 p.m.

Mathematics research employ modern computational tools (such as computer algebra systems and programming environments) to investigate mathematical concepts, formulate questions, perform mathematical experiments, gather numerical evidence, and test conjectures. Computational tools can help make certain areas of mathematics research accessible to students, providing points of entry where students can formulate and explore questions in number theory, algebra, topology, and more.

This session will highlight areas of mathematics where computational tools allow students to grapple with open questions. Talks will be aimed at a broad, non-expert audience. The use of computation for investigating mathematical topics, rather than computation employed for statistical analysis, is preferred. Discussion of connections between computational investigation and proof is encouraged.

Patterns in Generalized Permutations

1:00 p.m. - 1:20 p.m.
Lara Pudwell, Valparaiso University


A permutation is an arrangement of the numbers 1, 2,..., n. Permutation p is said to contain pattern q if p has a subsequence whose digits appear in the same relative order as q. Permutations that avoid certain patterns have been well studied over the past 30 years, but many enumeration and characterization questions remain open, especially when we consider patterns in combinatorial objects that generalize permutations. In this talk, we'll consider several variations inspired by undergraduate research projects.


An Undergraduate Course in Computational Mathematics

1:30 p.m. - 1:50 p.m.
Matthew Richey, St. Olaf College


In order to better prepare the next generation of mathematicians to use computational methods, we need to consider changes to the traditional mathematical curriculum. Currently, there are few undergraduate courses that introduce students to the ideas and methods of advanced computing techniques as a means of exploring interesting mathematical ideas. At St. Olaf College, for the last decade we have been teaching a course entitled "Modern Computational Mathematics" in which (mostly sophomore and junior) students use computing environments such as Mathematica, Python, and R to investigate topics such as distributions of primes, RSA, number theory, and combinatorics. In this talk, I will describe examples of topics and methods covered in the course along with a discussion of how this sort of course fits into a traditional mathematics major curriculum.


How Neuroscience Provides an Accessible Context for Undergraduate Research in Mathematics

2:00 p.m. - 2:20 p.m.
Victor Barranca, Swarthmore College


The study of computation in the brain provides a fertile ground for research questions employing ideas from diverse areas of mathematics. Depending on the scale and level of abstraction of the research problem, numerous types of mathematical models, differential equations, graph structures, mappings, activity patterns, and probabilistic processes may arise. In each case, the resultant mathematical questions will typically not furnish fully analytical answers and will therefore require some level of modern computation. This talk will highlight accessible mathematics often arising in neuroscience applications and recent undergraduate research in the field employing computational mathematics in concert with mathematical analysis.


Computing Hyperelliptic Invariants from Period Matrices

2:30 p.m. - 2:50 p.m.
Christelle Vincent, University of Vermont


In this talk we present an obstacle to computing invariants of curves whose Jacobian has CM (complex multiplication), when the genus of the curve is greater than 1. The problem is essentially that while the Jacobian has everywhere potential good reduction, the curve does not. We show the connection between this obstacle and a certain embedding problem which we define in the talk, and present our progress on analyzing the embedding problem. This is joint work with Ionica, Kilicer, Lauter, Lorenzo Garcia, Massierer and Manzateanu.


Using Simulation to Investigate Distributions of Piercing Numbers

3:00 p.m. - 3:20 p.m.
Tia Sondjaja, New York University


The piercing number of a collection of sets is the size of the smallest set of points such that each set contains at least one point. We consider piercing numbers of random collections of arcs on a circle. We desire to answer the question: If n congruent arcs are positioned randomly on a circle, what is the probability distribution of the piercing number of the n arcs? Probabilistic simulation allows us to estimate such distributions, which depend on the length of the arcs. In certain cases we can prove exact formulas. This work is an example of a large class of problems in geometric probability that students can investigate with introductory probability theory and computational skill.