# Group Members

Kyle Evans

## Overview

As today’s world becomes increasingly globalized, there exists a greater need to develop intercultural competence (ICC) in children through education. Three interdisciplinary units were designed by graduate students at UConn in the past year to develop ICC in sixth grade students, but lack assessment tools that track this development over the course of a school year. With a focus on Michael Byram’s model of ICC, we created two assessment tools – a survey and a rubric. The survey contains Likert scale items measuring four dimensions – attitudes towards cultures, knowledge of cultures, mathematics learning, and interdisciplinary learning – along with open-ended questions to add a qualitative component for each dimension. The rubric, designed for teachers, provides a resource for evaluating ICC in students’ interactions and written reflections. The tools have the potential to form a new baseline for assessment of ICC in children and can be refined and adapted for use across school districts and grade levels. In addition, we created three examples of lesson plans – one for third grade, one for sixth grade, and one for ninth grade – that demonstrate how intercultural competence and global issues can be incorporated across different levels of student development while also adhering to the Common Core State Standards for Mathematics.

ICC Presentation

Poster

# Spectrum of the Magnetic Laplacian on the Diamond Fractal

## Overview

The diamond fractal is a fractal that is obtained in the following manner.  Start with a graph with two vertices and an edge and replace the edge with two new vertices connected to our original vertices to obtain a diamond shaped graph.   The diamond fractal is defined to be the limiting object after continuing with the edge replacement indefinitely.  In the project, the spectrum of magnetic Laplacian operators on graph approximations to the diamond fractal was computed.

Given a level n approximation to the fractal with known magnetic field strengths through cells and holes, it is possible to determine the net magnetic field through the cells and holes of the preceding graph approximations.  The spectral similarity relation between the operators on successive graph approximations was worked out, with the corresponding spectral decimation polynomial depending on the magnetic field strengths.  A poster and talk on this work was presented at the REU Mini-Symposium at UConn.

# arXiv:1704.01609

## Presentation

Magnetic Spectral Decimation

Poster

# Minimal Length Maximal Green Sequences and Triangulations of Polygons

## Supervisors

Khrystyna Serhiyenko

## Overview

Maximal green sequences (MGS’s) are combinatorial objects that involve local transformations of directed graphs, also called quivers.  We studied minimal length MGS’s for quivers of type A.   It is know that such quivers are in bijection with triangulations of polygons.  Moreover, these local transformations of quivers behave well under this correspondence, and there is a related notion of mutation of triangulations.  This enabled us to study MGS’s both in terms of quivers and in terms of triangulations. We showed that any minimal length MGS has length n+t, where n is the number of vertices in the quiver and t is the number of 3-cycles.  We also developed an algorithm that yields such sequences of minimal length.

Journal of Algebraic Combinatorics volume 44, pages 905–930 (2016)

arxiv 1508.02954

## Presentation

Minimal Length Maximal Green Sequences

Poster

# Power Dissipation in Fractal AC Circuits

## Supervisors

Alexander TeplyaevLuke Rogers, Antoni Brzoska, Ulysses Andrews

## Overview

In this project, Feynman’s analysis of an infinite ladder circuit is extended to fractal circuits, in particular, a Sierpinski Ladder circuit and two variants of a Hanoi circuit.   These circuits are constructed by taking the limit of graph approximations with inductors and capacitors placed along edges in a well-defined manner.  Inductors, capacitors and resistors all contribute to the impedance within a circuit; but whereas a resistor imparts a real impedance, inductors and capacitors impart a purely imaginary impedance.

For each circuit, the following was accomplished.  First, the net impedance between certain boundary points of the circuit was computed.  Second, the filter conditions for each circuit were found.  A circuit becomes a filter when the net impedance between boundary vertices has a positive real part!  Third, it was proved that these impedances can be obtained by placing a small positive resistance epsilon on each edge of the graph approximations, finding the limiting impedance between the boundary vertices, and then taking epsilon to zero.  Finally, the construction of harmonic functions on these circuits was outlined.

# arXiv:1605.03890

## Presentation

Fractal AC Circuits

Poster

# Stochastic Stability of Planar Flows

## Publication: arXiv:1510.09221

 Journal reference: Stochastics and Dynamics, Vol. 17, No. 6 (2017) 1750046 DOI: 10.1142/S0219493717500460

## Overview

We investigated systems of complex-valued ordinary differential equations (ODEs) that blows up in finite time, which we refer to as explosive systems. The goal is to understand for what initial conditions does the system explode and will the addition of noise stabilize it; that is, if we were to perturb the system with an additive Brownian motion, will the system of stochastic differential equation (SDE) still be explosive? In fact, we were able to prove a toy model of the stochastic Burgers’ equation to be ergodic; that is, the SDE is nonexplosive and it has a unique limiting distribution.

Poster

# Group Members

## Overview

We consider the wave equation on the unit interval with fractal measure, and use two numerical models to study wave speed and propagation distance.  The first approach uses a Fourier series of eigenfunctions of the fractal Laplacian, while the second uses a Markov chain to model the transmission and reflection of classical waves on an approximation of the fractal. These models have complementary advantages and limitations, and we conjecture that they approximate the same fractal wave.

Poster

# Spectrum of the Magnetic Laplacian on the Sierpinski Gasket

## Overview

One project investigated magnetic gauge fields on the Sierpinski Gasket. After numerical experimentation, using Mathematica and MatLab, this team determined that specific portions of the spectrum of the Laplacian are unaffected by a perturbating magnetic field given by a harmonic 1-form of finite topological type and continued on to prove that this. In a specific case they also gave a description of the spectrum via a covering space and symmetry argument. They presented a poster and a talk on this work at the REU mini-conference at the University of Massachusetts, Amherst and gave a talk at the Young Mathematician’s Conference at Ohio State University.

## Presentation

Spectrum of the Magnetic Laplacian

arXiv.org

# Measurable Riemannian Structure on Higher Dimensional Sierpinski Gaskets

## Overview

This team worked on the existence of measurable Riemannian structures in the sense of Kigami on higher dimensional Sierpinski-type gaskets. Some time ago, Kusuoka proved existence of a measure, metric (in the Riemannian sense) and gradient operator on a class of fractals that includes these gaskets, such that these objects bear the same relation to the Dirichlet form as do the Riemannian volume, metric and gradient on Euclidean space. Kigami later completed this picture in the case of the usual 3-vertex Sierpinski gasket by constructing a geodesic length that is the analogue of that occuring in the Riemannian case and proving Gaussian heat kernel estimates, and Kajino has subsequently proved very re ned estimates for the heat kernel in this setting. It was believed that although Kigami’s approach relied on certain 2-dimensional techniques the results would also be valid on Sierpinski-type gaskets with more vertices, and that is what our research team have proved. They presented a poster on this work at the REU mini-conference at the University of Massachusetts, Amherst, and are writing the results up for publication.

# arXiv:1703.03380

## Presentation

Geodesics and a Riemannian Metric on Harmonic Sierpinski Gaskets

# Resources to Aid the Transition into an IBL Mathematics Course

## Supervisors

Gabriel Feinberg and Fabiana Cardetti

## Overview

This group conducted a research study to create resources to support both instructors and students transition into an Inquiry-based learning (IBL) course. The IBL approach has been shown to deepen student conceptual learning and increase student engagement and motivation in a subject without taking away from procedural understanding. The experiences in an IBL class are significantly different from traditional courses; however, there are few research-based resources to aid instructors and college students adapt to this approach. For this project, the creation of such resources was guided by extensive review of the literature as well as informed by experienced instructors teaching undergraduate mathematics courses and students who had positive and negative experiences in IBL courses. These methods along with the group’s experiences and expertise as instructors and undergraduate students of mathematics helped determine specific aspects that would be most challenging for an instructor, as well as the difficulties students would face, in the transition to an IBL course. The results of this study include a teacher’s and a student’s guide that address those difficulties, provide guidance for each audience, and contribute suggestions to achieve the desired learning outcomes of an IBL course. Lily An presented results at the Women in Mathematics in New England Conference (WIMIN13) on September 21 at Smith College.

## Presentation

Resources for Teachers and Students Transitioning Into an IBL Mathematics Course

# An Inquiry-Based Approach to Teaching Parameterization

## Supervisors

Gabriel Feinberg and Fabiana Cardetti

## Overview

In the summer of 2012, the UConn Math Education REU team identified parameterization of curves as a challenging topic for students in multivariable calculus courses. Encouraged by the positive research results of inquiry-based learning (IBL) on student performance and attitudes, the research focus for the group in 2013 was to develop IBL curricular materials aimed at supporting student’s understanding of this topic. The group conducted an extensive literature review, studied popular multivariable calculus textbooks, and consulted with experienced instructors to create an original IBL module. The module engages students in collaborative discovery to gain a deep conceptual understanding of parameterization in addition to providing opportunities for procedural practice. In addition, the group developed a detailed guide to support instructors in the effective classroom implementation of the module.

## Presentation

An Inquiry-Based Approach to Teaching Parameterization