project

Determining the Spectrum of the Laplacian on 3N-Gaskets

Posted on by
Jason Marsh, Nikhaar Gupta, Max Margenot, William Oakley

Group Members

Jason Marsh, Nikhaar Gupta, Max Margenot, and Will Oakley

Overview

The Laplacian is the central object of analysis on fractals. While most work on the Laplacian has been focused on computing the Laplacian spectrum on specific fractals, this group instead found the spectrum on an entire class of fractals—the 3N-Gaskets. This is the class of finitely ramified fractal 3N-Gons that are the attractors of iterated function systems containing 3N contraction mappings. For example, the 3-Gasket is the Sierpinski triangle.

The Laplacian on the fractal, and thus its eigenvalues, must be studied by examining the graph Laplacian on approximating graphs. The group found a function R(z) relating the Laplacian eigenvalues on consecutive graph approximations. They used R(z) along with the eigenvalues of the Laplacian on graph approximations to derive part of the spectrum on the next level. The rest of the spectrum was found using an “exceptional set”— a set of values which the group provided for arbitrary N. By repeating this infinitely many times, the spectrum of the Laplacian on the fractal could be found.

Presentation

Determining the Spectrum of Laplacian on 3N-Gaskets
Poster

Publication

arXiv.org

Frobenius Splitting of Projective Toric Varieties

Posted on by
Jed Chou, Ben Whitney

Group Members

Jed Chou and Ben Whitney

Overview

A toric variety is an algebraic variety containing the algebraic torus (C*)n as an open dense subset such that the action of the torus extends to the whole variety. Every n-dimensional toric variety can be associated to a fan, which can be given as a set of primitive vectors in an n-dimensional lattice N. Because of this association, many properties of toric varieties can be studied using combinatorial methods. This group was interested in determining which projective toric varieties are Frobenius split. If a projective toric variety is Frobenius split, it has many nice properties. For example, it can be given as the solution set to homogeneous degree two polynomials.

Sam Payne proved in 2008 that a toric variety is Frobenius split if and only if an associated polytope called the splitting polytope contains representatives of every residue class of (1/q)M/M where M is the dual lattice to N. The group’s goal was to use this characterization of Frobenius splitting to classify the Frobenius split projective toric varieties in n dimensions. Given a trivalent tree where edges are labeled with variables and integers, it’s possible to construct the fan of a toric variety. They determined which toric varieties arising in this way are Frobenius split for certain classes of edge labelings.

Lyapunov Exponents of Multiplicative Stochastic Processes

Posted on by
David Wierschen, Becky Simonsen

Group Members

David Wierschen and Becky Simonsen

Overview

This group considered the stability of matrix Lie group valued stochastic differential equations, dXt = AXtdt + BXtdt. Random dynamical systems such as this arise in many applications (e.g., oceanic turbulence, helicopter blade motion, light in random channels, wireless networks) in which stability is of practical and theoretical concern. The stability of the zero solution, Xt = 0, is determined by the top Lyapunov exponent. But in practice, analytic calculations of the Lyapunov exponent are often impossible, so time discrete approximations and simulations are necessary. Oceledet’s famous multiplicative ergodic theorem ensures that the Lyapunov exponent of Xt is almost surely constant. But the Lyapunov exponent of a time discrete approximation is itself a random variable. The mean of this random variable has been studied. The group provided estimates on the variance, distribution and rate of convergence in certain numerical approximation methods. In addition, they expanded on and provided simulations for recent results regarding the top Lyapunov exponent of certain Lie group valued SDEs.

Exploring Learning Difficulties in Multivariable Calculus

Posted on by
Fabiana Cardetti, Cathy Matta, Gabe Feinberg

Contributors

Cathy Matta, Gabriel Feinberg, and Fabiana Cardetti

Overview

This pilot study used student perceptions about their understanding of mathematics to guide the development of learning aids for multivariable calculus classes. Studies on the use of computer technology in advanced mathematics classrooms have shown that technology can help with the understanding of abstract concepts (Godaszi, Elahe Aminifar, & Bakhshalizadeh, 2009; Verner, Aroshas, & Berman, 2008). In addition, other researchers have found that using real-world applications and Inquiry Based Learning (IBL) projects can also help students not only with their learning but also with their enjoyment of mathematics (Hassi & Laursen, 2009; Spronken-Smith, Walker, Batchelor, O’Steen, & Angelo, 2012; Stillman, Galbraith, Brown, Edwards, 2007). In this study, these approaches were used in conjunction with students’ perceptions (Pierce, Stacey, & Barkatsas, 2007; Schoenfeld, 1989; Szydlik, 2000) to develop learning aids for multivariable calculus.

Presentation

Exploring Learning Difficulties in Multivariable Calculus

Analyzing Properties of the C. Elegans Neural Network: Mathematically Modeling a Biological System

Posted on by
Tyler Reese, Dylan Yott, Antoni Brzoska

Contributors

Tyler Reese, Dylan Yott, Antoni Brzoska, and Daniel Kelleher

Overview

The brain is one of the most studied and highly complex systems in the biological world. This group analyzed the brain of the nematode Caenorhabditis elegans. They used eigenvalues and eigenvectors of the Laplacian matrix — a matrix representation — of the neural network of the nematode brain with an eye on indicators of self-similarity.

The following is a list of the programs this group used in their research. Most of them were written in Matlab.

Publications

arXiv.org
PLoS ONE

The Strichartz Hexacarpet and Higher Dimensional Analogues

Posted on by
Diwakar Raisingh, Gabe Khan, Matt Begue

Contributors

Matt Begue, Dan Kelleher, Gabe Khan, and Diwakar Raisingh

Overview

The Strichartz hexacarpet is a fractal which can be obtained by repeated barycentric subdivisions of a triangle. This group examined properties of graph approximations and looked at analogous fractals obtained by subdivisions of higher dimensional triangles — n-simplexes. They studied properties of random walks on these fractals, obtaining heat kernel estimates and resistance factors. See also Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet Experiment. Math. Volume 21, Issue 4 (2012), 402-417.

Presentation

Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet

Tangent Space Visualization

Posted on by

Overview

Even in low dimensional examples, basic object in differential geometry, such has tangent spaces and bundles, can only be perceived in at least 4-dimensions. This makes intuitive understanding of these objects difficult. This group created some tools for understanding these objects a little better.

In these “tangent space visualisers,” we explore the concept of tangent vectors and tangent spaces. Below we have two examples, the figure-8 space (left) and the “theta” space (right). Click on the images to download the applets. The arrows indicate the tangent vectors which span the tangent space at any given point, click on any point in the space to move the arrow.

Figure-8 Tangent Space Theta Tangent Space

Quantum Mechanics on Laakso Spaces

Posted on by

Contributors

Christopher Kauffman, Robert Kesler, Amanda Parshall, Evelyn Stamey, and Benjamin Steinhurst

Overview

This group first reviewed the spectrum of the Laplacian operator on a general Laakso space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, they computed the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculated the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the resulting zeta function. Lastly, they derived a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for some special cases.
See also B. Steinhurst, Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension arXiv:0811.1378

Publications

arXiv.org
AIP | Journal of Mathematical Physics

From Self-Similar Structures to Self-Similar Groups

Posted on by

Contributors

Daniel Kelleher, Benjamin Steinhurst, and Chuen-Ming Wong

Overview

This group explored the relationship between limit spaces of contracting self-similar groups and self-similar structures. They gave the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is p.c.f. They then gave the necessary and sufficient condition on a p.c.f. self-similar structure such that there exists a contracting group whose limit space has an isomorphic self-similar structure; in this case, they provided a construction that produces such a contracting group. Finally, they illustrated their results with several examples.

Publications

arXiv.org
International Journal of Algebra and Computation (IJAC)

Spectrum and Heat Kernel Asymptotics on General Laakso Spaces

Posted on by

Contributors

Matthew Begue, Levi deValve, David Miller, and Benjamin Steinhurst

Overview

This group introduced a method of constructing a general Laakso space while calculating the spectrum and multiplicities of the Laplacian operator on it. Using this information, they found the leading term of the trace of the heat kernel of a Laakso space and its spectral dimension.

Publications

arXiv.org
Fractals