Undergraduate Research Showcase

The Truncated Octahedral Conjecture 

Lark Song

Abstract: 

The Truncated Octahedral Conjecture (Bezdek, 2005) states that in Euclidean space, any parallelohedron of volume 1 has a surface area at least as large as the surface area of the truncated octahedral Voronoi cell of the body-centered cubic lattice of volume 1. This brief talk will describe the isoperimetric problem and report on our research progress. We have established local minima for the first two types among Fedorov’s (1885) five parallelohedra and are currently attacking the most sophisticated type.

Revised Proof Of The $a=2$ Case of the Chung-Diaconis-Graham Random Process 

Jen Wang

Abstract:

Define a simple random walk $(X_n)$ on $\mathbb{Z}/p\mathbb{Z}$ by $ X_{n+1}=aX_n+b_n \text{(mod $p$)}$, where $p$ is a fixed integer and $a_n$ and $b_n$ are independent random variables chosen by a generator. This construction of the recurrences $(X_n)$ is often used by computers to generate pseudorandom sequences. Chung, Diaconis, and Graham proved in 1987 that for the case when $a=2$ with common distribution $\mu$, it takes $N\geq c\log p\log\log p$ steps for $(X_n)$ to become random, i.e. its distribution converges to uniform [Chung et al., 1987]. In this paper, we revise the proof to offer explanations of certain setups and arithmetic that were omitted in the original work. More specifically, we provide combinatorial explanations behind the motivation of different parts of the proof along with a matrix visualization of the binary expansion technique.

Analysis and Simulation of Spiral Propagating Waves in Neuronal Networks

Neil Maclachlan

Abstract:

Our work is motivated by the discovery of spiral waves in fMRI data from the human cortex, which vary in distribution and chirality based on cognitive task. We propose a biophysically motivated firing rate model of excitatory and inhibitory neurons on a 2D lattice with isotropic distance dependent connectivity. This approach is novel since phase models, previously used to study spirals, do not include inhibition. Mean field theory yields a planar synaptic activity model, allowing bifurcation analysis. Simulations show spirals forming from random initial conditions. Local excitatory connectivity yields smaller, denser spirals, while long-range excitatory connections produce a few large spirals. Long range inhibitory connectivity causes spirals to be sparse as the surrounding medium goes to synchrony, aligning with the hypothesis that such connections disrupt spiral formation. Spirals persist through the Hopf bifurcation, supporting analysis in the oscillatory regime. We derive a principled phase model using weak coupling theory, and approximate it via its Fourier transform. We discover a connection between spiral activity and specific Fourier coefficients in a simple phase model and study how these coefficients can be altered in the full model.

Poincaré Disk Image Tiling

Sam Brunacini

Abstract:

We discuss the development and implementation of a method for generating “Circle Limit”-style (p, q) tilings in the Poincaré disk model of hyperbolic geometry from an arbitrary starting image. We explore the benefits and drawbacks of several methods of mapping a square pixel array to the unique rotationally symmetric p-gon with interior angle 2pi/q. These include radial projection, concentric mapping, and the Schwarz-Christoffel mapping. We discuss our algorithms, obstacles to implementation, design choices, and how this work can be extended.