Projects

Geodesic Interpolation on the Sierpinski Gasket

July 9, 2018

Group Members

Cory McCartanLaura LeGareCaitlin Davis.

Supervisors

Luke RogersSweta Pandey.

Overview

Geodesics (shortest paths) on manifolds such as planes and spheres are well understood.  Geodesics on fractal sets such as the Sierpinski Triangle are much more complicated.  We begin by constructing algorithms for building shortest paths and provide explicit formulas for computing their lengths.  We then turn to the question of interpolation along geodesics—given two subsets of the Sierpinski Triangle, we “slide” points in one set along geodesics to the other set.  We construct a measure along the interpolated sets which formalizes a notion of the interpolation of a distribution of mass, and we prove interesting self-similarity relations about this measure.

Publication

Presentation

Poster

 

Computations on the Koch Snowflake with Boundary and Interior Energies

Group Members

 

Carlos Lima,  Malcolm Gabbard.

Supervisors

Gamal Mograby,  Luke Rogers,  Sasha Teplyaev.

Overview

At the end of the 20th century studies had been conducted on the Koch Snowflake which had been motivated through work done by physicists on “fractal drum” experiments. These investigations focused on the eigenfunctions of the negative Dirichlet Lapcian generated on a planar domain with a fractal boundary, particularly with the condition that the boundary be set to zero. Here we study the eigenfunctions on the Koch Snowflake with a non-zero boundary condition and we consider a Laplacian defined on the boundary. To generate an n-level fractal and apply the Laplacian matrix, Python programming was implemented for both the n-level fractal and Laplacian matrix. This then gives us insight into the eigenvalues and visualization of the corresponding eigenfunctions on the Koch Snowflake. Initial observations indicate a kind of localization of the eigenfunctions on the fractal boundary. 

Publication

Presentation

Poster

 

Lipschitz Continuity of Laplacian Eigenfunctions on a class of postcritically finite (PCF) self-similar sets

Group Members

Benjamin YorkAnchala Krishnan

Supervisors

Gamal MograbyChristopher Hayes, Luke Rogers.

Overview

We prove a general result for Lipschitz and Hölder continuity of functions defined on a class of postcritically finite (PCF) self-similar sets. Intuition for this theorem comes from formulating arguments on the unit interval, which is well understood in a classical setting. We generalize these results for many kinds of self-similar sets endowed with different measures and metrics. As a corollary to this general result, we prove that eigenfunctions of the Laplacian on the harmonic Sierpinski Gasket are Lipschitz continuous.

Publication

Presentation

Poster

 

 

Probability, Analysis and Mathematical Physics on Fractals 2018

February 23, 2018

Each year we are looking for a group of undergraduate students to work on Probability, Analysis and Mathematical Physics on Fractals. The aim of the projects will be exploration of differential equations and various operators on fractal domains. Students in the project are supposed to have the usual background in linear algebra and differential equations. Knowledge of Matlab, Mathematica, other computer algebra systems, or programming, as well as proof writing, mathematical analysis, and probability may be helpful but is not required. Previous undergraduate work includes published papers on the eigenmodes (vibration modes) of the Laplacian (2nd derivative) of functions that live on Sierpinski gasket type fractals, and the electrical resistance of fractal networks, as well as work on Laplacians on projective limit spaces. The exact choice of the topics to study will depend on the students’ background and interests. Besides being interesting, taking part in a research project like this may be very useful in the future (for instance, when applying to graduate schools).

Luke Rogers, Gamal Mograby, Sasha Teplyaev, Patricia Alonso-Ruiz

Financial Math: Portfolio Optimization and Dynamic Programming

December 20, 2017

Group Members

Ayelet Amiran, Skylyn Brock, Ryan Craver, Ugonna Ezeaka, Mary Wishart

Supervisors

Fabrice Baudoin, Berend Coster, Phanuel Mariano

 

Overview

Financial markets obviously have asymmetry of information. That is, there are different type of traders whose behavior is induced by different types of information that they possess. Let us consider a “small” investor who trades in a arbitrage free financial market so as to maximize the expected utility of her wealth at a given time horizon. We assume that she possesses extra information about the future price of a stock. Our basic question is: What is the value of this information ?

To answer the question, we will consider some standard basic discrete models of financial markets, like binomial trees and solve the portfolio optimization problem with asymmetry of information by developing  dynamic programming tools.

Publication

arXiv:1808.03186

Presentation

tba

Poster

tba

Gradients on Higher Dimensional Sierpinski Gaskets

July 28, 2017

Group Members

Luke Brown,  Giovanni E Ferrer SuarezKaruna Sangam.

Supervisors

Gamal MograbyDan KelleherLuke RogersSasha Teplyaev.

Overview

Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.

Publication

Presentation

Poster

Spectral Analysis on Graphs Related to the Basilica Julia Set

Group Members

Courtney GeorgeSamantha Jarvis.

Supervisors

Dan KelleherLuke RogersSasha Teplyaev.

Overview

We analyze the spectra of a sequence of graphs constructed from the Schreier graphs of the Basilica group.  Our analysis differs from earlier work of Grigorchuk and Zuk in that it is based on a macroscopic decomposition of the graphs. This method gives precise information about the multiplicities of eigenvalues and, consequently, good information about the spectral measures of large graphs. It also permits a proof of the existence of gaps in the spectrum of limiting graphs.

Publication

Presentation

Poster

Multiplicative LLN and CLT and their Applications

July 25, 2017

Group Members

Lowen PengAnthony SistiRajeshwari Majumdar

Supervisors

Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo

Overview

We study the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturb the random matrices with a parameter and we look to find the dependence of the the Lyapunov exponent on this parameter. We also study the variance related to the multiplicative CLT. We prove and conjecture asymptotics of various parameter dependent plots.

Publications — tba

Presentations:

Raji Majumdar and Anthony Sisti, will present posters Applications of Multiplicative LLN and CLT for Random Matrices and Black Scholes using the Central Limit Theorem on Friday, January 12 at the MAA Student Poster Session, and give talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.

 

Investigating the Interplay of Argumentation and Mathematics in Classroom Tasks

June 2, 2016

Group Members

Megan Brown, Grace WrightMath Ed

Supervisors

Steven Lemay

Overview

This project was motivated by the current emphasis on argumentation in the Common Core State Standards for Mathematics. In particular, we focused on tasks that address the third standard for Mathematical Practice that highlights argumentation as an expertise all students should develop. Our data consists of 157 tasks from 40 elementary and high school teachers participating in a professional development program focused on mathematical argumentation. We analyzed how argumentation affects the cognitive demand levels of the tasks, as well as how argumentation detracts or contributes to the mathematical content in the tasks. Our findings show a change in cognitive demand levels when tasks were analyzed with and without the argumentation component. In addition, five different themes emerged from our analysis with regards to the interplay of argumentation and mathematical concepts, some of which were common across elementary and high school tasks.

Presentation

Poster

Mathematics and Intercultural Competence in the Middle School

Group Members

Chris Bennett, Megan BrunnerDiscussion

Supervisors

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.

Presentation

ICC Presentation

Poster