Graphics of fractals arising in complex dynamics, such as Julia sets of complex analytic functions, will be constructed. Two basic types of algorithms, based on forward and on backward iteration, will be used, and the effectiveness of the algorithms compared.
Given a smooth oval, how does one place n points on its boundary so that the product of all "n choose two" Euclidean distances they determine is maximal(or nearly so)? This is the "transfinite diameter" problem. We shall study it(at the very least numerically) in the context of interacting particles. Two of many directions: What if removal of the points results in n arcs of equal length? What if the Euclidean distance between successive points is constant?
We want the students to combine geometric insight and Mathematica computations to study new notions of complexity of polynomials in two real variables. We hope to make connections with CR Geometry.
Completion of Calculus 3. Some experience with Mathematica.
Random-like behavior is ubiquitous in number theory. For example, the primes, the digits of pi and other famous constants, and the Moebius function and other number-theoretic functions, all appear to behave much like appropriately defined "true" random sequences. In this ongoing project we seek to explore such random features experimentally - via large scale computations and geometric visualizations as random walks - and possibly also theoretically. The specific focus varies from semester to semester, and will be decided at the beginning of the semester. For examples of past projects see http://www.math.illinois.edu/~hildebr/ugresearch/. Interested students should contact Professor Hildebrand at firstname.lastname@example.org 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. The specific focus varies from semester to semester and will be decided at the beginning of the semester. For examples of past projects see http://www.math.illinois.edu/~hildebr/ugresearch/. Interested students should contact Professor Hildebrand at email@example.com before applying.
The most basic principle of learning and pattern formation in neuroscience is the Hebbian paradigm, which can be stated simply as "Cells that fire together, wire together". More explicitly, when cells have similar firing patterns, the synaptic coupling between them grows stronger; when different, weaker. A simple model of this type of system is one in which we model each of the cells as an oscillator and the synapse by a nonlinear coupling. Thus, a mathematical description of this model is an ODE defined on a graph where the edge weights are allowed to evolve. Said yet another way, we want to understand the behaviour of nonlinearly coupled oscillators when the connection strengths between these oscillators are allowed to vary. These types of systems aren't very well-studied, so even some systematic numerical simulations of these could be interesting. Proving something analytic about them would be even more so.
Free groups are fundamental objects in algebra and topology. Free groups are known to have many nice "residual" properties related to finite quotients and finite index subgroups of free groups. In particular, a finitely generated free group F is "residually finite", meaning that for every nontrivial element g of the group there exists a finite K quotient of F where the image of this element is still nontrivial; equivalently, for every nontrivial element g of F the there exists a finite index subgroup H of F which does not contain this element. This project, which will be mostly computational in nature, aims to study quantitative aspects of these kind of residual properties; e.g. trying to bound the smallest size of K or the smallest index of H in F in the above context in terms of the length of the element g. We will also look at related questions for the "primitivity index function" in a free group F, where, given a nontrivial element g in F, one tries to bound, in terms of the length of g, the smallest index of a finite index subgroup of F that contains g as a "primitive" or "basis' element. This question is closely related to geometric problem about "untangling" closed geodesics on hyperbolic surfaces. The main tool used to study these question involves labeled directed graphs and certain types of "folding" and "collapsing" operations on them.
The goal of the project is to identify and to study applications of Linear Algebra in engineering and science. Among these applications are Google's famous PageRank algorithm, image compression schemes and face recognition software. The products of this project should include, but are not limited to implementations of these important algorithms and the creation of software and other material which illuminates the use of Linear Algebra in engineering applications.
It is my hope that products of this project can be used in future installments of the Department's Linear Algebra courses such as Math 415. Such material might help to bridge the gap between the lecture course and the state of the art applications in engineering.