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

Thursday, July 30, 1:30 p.m. - 4:00 p.m., Philadelphia Marriott Downtown, Grand Ballroom C

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.

Matthew Wright, St. Olaf College

Patterns in Generalized Permutations

1:30 p.m. - 1:50 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.


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.


Bringing Intuition from Euclidean Geometry to Finite Metric Spaces

3:00 p.m. - 3:20 p.m.
Don Sheehy, North Carolina State University


In this talk, I will discuss some interesting challenges that arise when attempting to bring ideas from Euclidean Geometry into the realm of finite metric spaces. More and more often, data sets have a notion of distance that satisfies the triangle inequality, but doesn't necessarily correspond to a Euclidean distance. I will discuss some interesting algorithmic and mathematical approaches to dealing with such data (metrics) and some open source tools for viewing and searching them.


An Undergraduate Course in Computational Mathematics

3:30 p.m. - 3:50 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.