# Summer 2015 Research

## Danielle Champney with Chad Eckman, Alvaro Matias and Alex Cheng

Mathematics students Alvaro Matias, Chad Eckman and Alex Cheng worked with Danielle Champney on developing interdisciplinary science, technology, engineering and math (STEM) curriculum that could be adapted for multiple grades and age groups. The group focused on topics in mathematics, statistics, physics and engineering and developed several modules for implementation in middle school, high school and college courses.

Additionally, Eckman adapted one of these modules, which focuses on students' data-based decision making, and implemented it in a fifth grade classroom as part of his senior project in fall and winter. He and Champney are currently writing up the findings. Liberal studies student Colin Schaefer is building on these findings and implementing similar research strategies in another fifth grade classroom to further study how young students make data-driven decisions.

## Paul Choboter with Caleb Miller, Skyer Young and Tuyen Pham

Modern weather prediction relies on the careful incorporation of observational data into numerical simulations to improve accuracy. The process of merging data with a simulation is called data assimilation. The mathematics of data assimilation is well-developed and draws from variational calculus, optimization, control theory and statistics.

Data assimilation is used in simulations of ocean circulation as well, but ocean data is difficult to collect in large quantities, especially near the coast. This project seeks to answer the question: with a limited amount of data to assimilate in a coastal ocean model, where are the optimal locations to collect that data? Realistic simulations were run in several configurations, with synthetic data measured from one simulation and assimilated into a second simulation. The Regional Ocean Modeling System was used to perform the simulations. Preliminary results were reported at the Mathematical Association of America fall 2015 meeting, and the research is ongoing.

## Caixing Gu with Heidi Keas and Robert Lee

Caixing Gu and students Heidi Keas and Robert Lee worked on a project titled "The *n*-inverses of a matrix," which resulted in the submission of a manuscript with the same title in December 2015. The concept of a left *n-*inverse of a bounded linear operator on a complex Banach space was introduced recently. Previously, there have been results on products and tensor products of left *n*-inverses, and the representation of left *n*-inverses as the sum of left inverses and nilpotent operators was being discussed. In this paper, the group gave a spectral characterization of the left *n*-inverses of a finite (square) matrix. They also showed that a left *n*-inverse of a matrix *T* is the sum of the inverse of *T* and two nilpotent matrices.

## Tony Mendes with Thomas Taylor, Brian Jones and Shelby Burnett

Tony Mendes worked with students Shelby Burnett, Brian Jones and Thomas Taylor to study the following problem posed by the late Herb Wilf. Let *u _{k}*(

*n*) be the number of permutations of 1, 2,…,

*n*with no increasing subsequence of length

*k*+1 and let

*y*(

_{k}*n*) be the number of standard Young tableaux with

*n*cells with bottom row at most

*k*cells.

The problem was to find a sign reversing involution proof of the identity

when *k* is even.

Using the Robinson-Schenstead correspondence, we interpreted the problem as one involving red and blue paths drawn inside symmetric permutation matrices. (See attached figure for one of these matrices when *n* = 10.) The group hoped that this new interpretation could lead to new insights into the combinatorics of permutations. We were able to solve the problem in certain special cases, but a general solution to the problem remains elusive.

## Erin Pearse with Jon Lindren and Zach Zhang

Erin Pearse worked with mathematics student Jonathan Lindgren and engineering student Zach Zhang on the problems of reconstruction of missing data and denoising data. The team is developing an algorithm that converts a dataset (for example, a collection of images) into a network of connected points and then exploits the intrinsic geometry of the network to “fix” damaged points using linear algebra.

## Jonathan Shapiro with Buddy Galletti, Adam Mair and Christopher Hurley

Jonathan Shapiro worked with students Christopher Hurley, Adam Mair and Buddy Galletti. The group studied the numerical ranges of composition operators on the Hardy space. They examined the numerical ranges for composition operators whose symbols are automorphisms of the disk, paying particular attention to those whose symbols are elliptical automorphisms. They made several conjectures involving the continuity of the numerical range and the numerical radius. Some continuity results which showed that the numerical range of certain composition operators with elliptical automorphism as their symbols are not disks.

Other research groups were **Dave Camp with Darren Marotta and Ben Brown **and** Stepan Paul with Michael Blakeman, Matthew Varble and Madeleine Jacques.**