Imagine filling a box with glass and metal balls placed randomly. A natural question is: does the entire box act like a conductor or an insulator? In other words, is there a spanning cluster, a group of metal balls touching each other that reaches from one side of the box to the other? Imagine pouring a liquid on top of a block of porous material and letting the liquid trickle down. Will the liquid be able to travel through an open path from top to bottom? The first experiment is an example of site percolation and the second an example of bond percolation. Both can be modeled using a network with a vertex set of "sites" and an edge set of "bonds"; In the first example, consider a random network where vertices are metal balls with probability p and glass balls with probability 1-p. Edges in the network correspond to where balls touch each other. The second example is modeled by a random network in which an edge is open (and therefore lets liquid through) with probability p or closed (and therefore blocking liquid) with probability 1-p. The pertinent question is: is there an open path from top to bottom? At a critical threshold or percolation threshold p = p_c large clusters appear. For various networks, one can calculate the site percolation and bond percolation thresholds; however, the exact thresholds are only known for some 2-dimensional lattices. Another interesting type of percolation is bootstrap percolation; vertices are successively removed from the network if it does not have at least k neighbors. Imagine that a number of computers in a computer network have become infected and that each healthy computer with at least k infected neighbors also becomes infected. We continue this process continues until no more computers can become infected. The main goal of this project is to analyze and visualize different types of percolation on a number of lattices. Depending on interest, some other directions include studying percolation on toroidal lattices, quasi-lattices, random graphs, and Ramanujan graphs (sparse networks with high connectivity).
The partition function p(n) counts the number of ways to write the integer n as the sum of positive integers (where order does not matter). An old unsolved conjecture says that the values of this function are equally often even and odd (asymptotically). The even and odd values are thought to be distributed "randomly." We will research different definitions and measures of randomness for an infinite sequence of zeros and ones. Then we will apply these measures to the sequence given by p(n) in an attempt to determine just how random this sequence is. Our investigations may lead to other natural questions (related to other functions, or to primes other than 2).
Consider a particle bouncing between two stationary walls. The collision of the particle wit one wall is elastic and with the other one is given by the following rule: the velocity of the particle increases by one if $t\in [2n, 2n+1]$ and decreases by one if $t\in [2n+1, 2n+2]$, where $n$ is a natural number and $t$ is the moment of time at which the collision occurs. The goal of the project is to understand the velocity growth rate in this problem. This is a model problem in the stability of switching systems. The first step would be to carry out numerical experiments and conjecture the growth rate.
If you stand at the origin and there is a tree at each integer point in the plane, then only a fraction of the trees of this infinite forest will be visible. Surprisingly enough this fraction is $6/\pi^2$. If you can only see trees within a certain distance, then you can study their gap distribution. That is, you can order the visible trees counterclockwise and look at the angle between two consecutive ones. The statistics of these gaps is very interesting and has some unexpected features. For example, gaps cannot be "too small". The current project aims to generalize this to the case when only a subset of the integers points is occupied. What happens, for example, if at each lattice point there is a tree with probability $1/2$ ? Or what if we only consider subsets of lattice points of number-theoretical interest? Will the gap distribution change and, if so, how?
The aim is to investigate numerically, for a few examples of interest, the spacing statistics of angles between geodesic arcs connecting a fixed point z in the upper half-plane ${\mathbb H}^2$ with points from the orbit $\Gamma z=\left\{ \gamma z= \frac{az+b}{cz+d} : \gamma= \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \right\}$, where $\Gamma$ is a lattice (discrete subgroup of finite covolume) in the group $PSL(2,{\mathbb R})$. The more difficult situation where ${\mathbb H}^2$ is replaced by the 3-hyperbolic space ${\mathbb H}^3$ and $\Gamma$ by a lattice in the group $PSL(2,{\mathbb C})$ will also be considered.
We seek geometric interpretations and number-theoretic properties of various "CR" mappings between spheres in different dimensions. It is easy to write Mathematica Code to find many, many examples. Formulas for these mappings exhibit striking combinatorial and number-theoretic properties, many of which are accessible to a calculus student. These mappings are fun, delightful, but not fully understood!
Many properties of the natural numbers can be encoded as sequences of 1's and -1's. On the surface, such sequences often show no obvious pattern and indeed seem to behave much like sequences generated by true random experiments such as coin tosses. In this project we seek to obtain a deeper understanding of the behavior of such sequences via certain "random walks" in the plane formed with these sequences. These random walks provide a natural way to visualize the degree of randomness inherent in a sequence and to detect, and possibly explain, hidden patterns, but they can also open up new mysteries that defy explanation.
Investigate numerical solutions of differential equations associated to problems in structural design of Lithium-Ion batteries. These equations model charging and discharging processes of batteries as we change the arrangements of silicon or tin particles in the anode of the battery.
An important curvature condition in Lorentz spaces is determined by the behavior of geodesics. The project is to visualize this geodesic behavior. We study geodesics in Lorentz convex polyhedral surfaces in Minkowski space-time. (1) For convex polyhedral surfaces in Euclidean 3-space, geodesics close to a vertex can be studies by "development". i.e. If you cut the surface and roll it out to a sector in the plane, the geodesics are the curves that roll out to straight lines. (2) The same idea works in Minkowski 3-space, only now you roll out using Lorentz motions instead of Euclidean motions. This is no problem for a computer, but the result will be surprising and informative for our Euclidean brains. (3) For 3-dimensional convex polyhedra in Minkowski 4-space, the same method works. Now the surface has two spacelike axes and one timelike axis at a vertex, and so the geodesics near a vertex will have behavior that combines types (1) and (2). By making this behavior clear graphically, we will illustrate how curvature works in Lorentz spaces.
This research project will focus on the first steps in human vision processing. Much is known about the network of neurons in the retina and first layer of the visual cortex, and several models of the network's signal propagation have been suggested by the mathematical community. Together with neurovision researchers at UCSB, we will work on testing the validity of these models. Our end of the project will consist of three goals: 1) Study the biological data driving the models, 2) Understand the sub-Riemannian geometry used by the models, 3) Assemble software implementing the models.
Although this project relates to the quasicrystal project Spring 2013, we take a step back and start at a more elementary level. We will study a monograph of the Mathematical Association of America:
Jack Graver "Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures", MAA "Dolciani Math. Expos., No 25".
It's a beautifully written, short book on the bracing problem in architecture, beginning with counting degrees of freedom, some graph theory, 2D rigidity, and sliding into 3D (where there are plenty of unsolved, if elementary problems to keep a beginning graduate mentor intrigued.) But it is the final historical chapter, beginning with Euler, through Peaucellier linkages, ending with Buckminster Fuller's tensegrity, that caught my eye. A field trip to the Monticello RR Museum to study the linkage on a live steam locomotive, maybe a trip to the Science and Industry together with a viewing of Helmut Jahn's exposed truss architecture, are not out of the question.
This project is part of a broader program to seek out and explore interesting problems in n-dimensional space that are accessible at the calculus level, but rarely covered in standard calculus courses. These problems typically arise as natural generalizations of familiar problems in 2 or 3 dimensions, and they are often motivated by applications to probability and statistics or other areas. Last year we considered two such problems, the volume of the region of intersection of cylinders in n dimensions, and the "Broken Stick Problem", which concerns the distribution of the pieces created when a stick is broken up randomly into n pieces. Depending on the interest and background of the team members, we may continue some of these investigations and/or explore other problems of similar flavor.