A finitely generated group G is called residually finite if for every nontrivial element g of G there exists a subgroup H of finite index in G such that H does not contain g. Residual finiteness plays a fundamental role in group theory and its various applications, such as geometric topology, hyperbolic geometry, representation theory, logic, analysis, and so on. The project aims to study the quantitative aspects of residual finiteness of the free group F(a,b). This group is known to be residually finite and thus for every nontrivial element g of F(a,b) one can defined d(g) as the smallest index of a subgroup of F(a,b) that does not contain g. Then for n>0 one puts f(n) to be the maximum of d(g) taken over all nontrivial elements g that are represented by freely reduced words over a,b of length at most n. It is known that f(n) is bounded above by n, but the best known lower bounds are roughly logarithmic in n. We will try to understand the true asymptotics of f(n) via computer experimentation. For a specific element g of F(a,b) the number d(g) can be computed precisely in two different ways: One is graph-theoretic, based on the use of "Stallings subgroup graphs". The other method is algebraic and involves working with homomorphisms from F(a,b) to finite symmetric groups. Time permitting, we will look at related problems for free groups and for the fundamental groups of hyperbolic surfaces, particularly the problem of "untangling" closed a closed geodesic for such a surface in some finite cover of the surface.
Here’s the problem: I have a database of jointly authored papers by members of a mathematical network (called GEAR). These collaborations can be visualized as a graph with a vertex for each author and with edges indicating joint authorship. The challenge is to display this graph in a way that indicates clearly patterns of interaction among the network members. This is precisely what cluster analysis is designed to do! In this project we will apply algorithms based on cluster analysis to the GEAR database and create attractive, informative graphical displays illustrating the results.
Long range first passage percolation (LRFPP) on the d-dimensional square lattice can be thought of as an epidemics model with no recovery when the weights are exponential. At time zero only the origin is infected and every other vertex in the lattice is susceptible. Every vertex has an independent Poisson clock of intensity one, that starts as soon as the vertex gets infected. Whenever the clock rings for an infected vertex, it will choose one of its neighbor at relative position $x \in Z^d$ with probability $c ||x||^{-a}$ for some fixed $a>d$ fixed and infect that chosen vertex, if it is not already so. The infected region $B_t$ grows with time $t$. It is known that the infected set $B_t$ grows very fast, at an stretched exponential rate in $t$ when $a\in (d,2d)$ whereas it grows polynomially in $t$ when $a>2d$. The goal of this project is to understand the growth rate in terms of the diameter and number of components for different values of $a\in (d,2d)$ and find the boundary fluctuations by analyzing the optimal path to infect a far away vertex starting from the origin using computer simulations. One important problem is to find a scaling limit for the growth process which is intimately connected with the optimal path structure. Possible extensions of this project are understanding the polynomial growth region for $a>2d$ and inclusion of positive recovery rate.
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. In past semesters, we focused on random walks associated with quadratic residues and with the Moebius functions, a well-known number-theoretic functions with values +1, -1, and 0, and a random-like behavior that is closely connected to the Prime Number Theorem and the Riemann Hypothesis. In the course of these investigations we came across another, broader class of number-theoretic random walks with mysterious fractal-like patterns. These random walks will be main focus of the current project. For further details and reports on past projects, see http://www.math.illinois.edu/~hildebr/ugresearch/ Interested students should contact Professor Hildebrand at ajh@illinois.edu before applying.
This project is part of an ongoing program to seek out and explore interesting problems in n-dimensional calculus and geometry 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. In past projects we considered generalizations of the Broken Stick Problem (If a stick is broken up randomly into 3 pieces, what is the probability that the pieces can form a triangle?) and the Random Triangle Problem (If a triangle is chosen at random, what is the probability that it is acute?). For further details and reports on past projects, see http://www.math.illinois.edu/~hildebr/ugresearch/ Interested students should contact Professor Hildebrand at ajh@illinois.edu before applying.
This is a somewhat open-ended project on Hamiltonian dynamics in dimension two and related dynamics in dimension three. We will start out by visualizing (images and animations) interesting types of examples (on a disk, sphere, and torus) and compute certain numerical invariants. Based on prior knowledge of the group and progress during the project, we may look at how these invariants behave under limits and conjugation; the latter is intended to be done with numerical computations that may support positive or negative answers to specific open problems in two-dimensional and three-dimensional topological dynamics (more specifically: area preserving dynamics, and the average asymptotic linking number of a divergence-free vector field in hydrodynamics, respectively).
The famous Thomson problem is to find the equilibrium position for n electrons on a sphere interacting under Coulomb's law (as they physically must). We study variations of this (various potentials) and also some particle interactions that correspond to a discrete version of the important "curve shortening flow".
Asymptotics are a fundamental notion which pervade nearly every field of mathematics and have very serious applications in computer science and physics. We learn in Calculus that $\exp(x)$ is in some sense "larger" than $x^n$ for all $n>0$ (which is in turn larger than $\log(x)$, etc.). We also learn that $\int_1^{\infty}\frac{dx}{x}$ diverges whereas $\int_1^{\infty}\frac{dx}{x^{1+\epsilon}}$ will converge. These types of functions (the ones built from $\exp$ and $\log$) can be compared on a certain large scale. Depending on where a particular function lies on this scale will determine many of its properties. The longterm goal of the project is to develop an interactive visual tool which will illustrate this scale and these properties. As a short-term goal, and confidence-building measure, we will develop a backend (based on the so-called asymptotic couples of Rosenlicht) suitable for the symbolic computations needed in this area.
The goal of this project is to visualize 3-dimensional hyperbolic space using the Oculus Rift, allowing the user to look around and walk through hyperbolic space as if they lived in it. One visualization of hyperbolic geometry “from the inside” was created in the 90’s at the Geometry Center at the University of Minnesota as part of the video Not Knot (which can be found on YouTube). The aim of this project is to create a visualization similar to the one in this video, with two major differences: it will be a real-time animation controlled by the user, and it will use the Oculus Rift instead of an ordinary display. This semester we will build upon the progress made during Fall 2014.
Given a fixed length of string, the simple closed curve made with the string that encompasses the most area is the circle. In a similar vein, the (non-closed) curve which best reflects (parallel) light rays through some fixed point is the parabola. If one starts with an arbitrary curve, which is gradually yet randomly perturbed with a bias toward some optimal trait, such as encompassing the most area or reflecting light rays in a specific way, then we should expect the curve to deform into an optimal one. This "action principle" is the foundation upon which mechanics, and other areas of physics, are based on. This project is designed to apply tools from multivariable calculus to illustrate the above principle with programmed animations created in Mathematica. The goal will be to complete this for the two scenarios described above, along with many others based on them, including 3-dimensional versions. If time permits, we may work with additional scenario types.