Illinois Geometry Lab

IGL Projects, Fall 2016

• Traffic Patterns in Manhattan

Faculty Mentor: Rich Sowers

We have some data for traffic in Manhattan and have processed it to find some reduced-dimension characterizations. This research is ongoing, but a part yet to be fully developed is visualization. This will be fairly challenging and involve various software packages. Hopefully you will gain some expertise which is valuable in the developing field of traffic (think Uber, Lyft, Apple, Google Maps).

weekly
• Project Difficulty
Contact the instructor.
• Iterated Functions and the Golden Ratio

Faculty Mentor: Kenneth B. Stolarsky

What happens when a function that maps a set into itself is iterated? Instances of this lead to both chaos theory and notable fractals. We consider a case that leads to a number system based on the golden ratio, and use it to illuminate a number of problems involving topics connected with the golden ratio, including possibly Fibonacci numbers, continued fractions and paper folding. Of particular interest is the analogue of a basic concept in base 10 arithmetic. Here a number is rational if and only if its decimal expansion is periodic. In the new system it is periodic if and only if it is r+s*g where g is the golden ratio and r and s are rational. This raises some new questions, both in number theory and in the construction of exploratory computer algorithms.

• Team Meetings
mostly biweekly
• Project Difficulty
Intermediate
Completion of Calculus 3. It would be very desirable to have students who have completed subjects at the level of elementary number theory or elementary complex variables (i.e. that level of sophistication independent of subject material). Also students reasonably proficient with some computer algebra system such as mathematica.
• Quantum mechanics for graphs and CW-complexes

Faculty Mentor: Ivan Contreras

The beautiful Feynman approach for quantum mechanics using path integrals has a toy version in combinatorics. The objective of this project is to study numerically and graphically the exponential Laplacian graph as the combinatorial candidate for the partition function, and to interpret the path integral in terms of paths in the graph. In a same fashion, there is a combinatorial Laplacian in the case of CW-complexes and its exponential is related to the Euler characteristic. Graphical simulations for the solutions of the combinatorial version of the heat and Klein-Gordon equations will be useful for a better understanding of the problem.

• Team Meetings
weekly, 1-2 hours
• Project Difficulty
Intermediate
Completion of Calculus 3, Familiarity with ODEs at the level of Math 285. Some knowledge in basic topology is desired but not expected. Familiarity with software like Matlab and Python is ideal.
• Complex dynamics and zeros of derivatives

Faculty Mentor: Aimo Hinkkanen

This is a continuation of the IGL project "Graphics in Complex Dynamics" from Fall 2015. The team will produce graphics of Julia sets of suitable rational functions (these are fractals arising in complex dynamics) to illustrate the behavior of the zeros of the derivatives of a given function. The zeros of the first derivative of a function are poles of the associated Newton's method function, and the zeros of the second derivative of a function are among the zeros of the derivative of the Newton's method function, which, in turn, are often connected to certain components of the Fatou set (the complement of the Julia set) of the Newton's method function. For this reason, complex dynamics has been used to study the location of the zeros of the derivatives of a given function. Graphics will be produced to illustrate situations that can occur.

• Team Meetings
once a week
• Project Difficulty
Basic
Completion of Calculus 3.
• Calculus, Geometry, and Probability in n Dimensions

Faculty Mentor: A J Hildebrand

This project is part of an ongoing program, begun in Fall 2012, aimed at seeking out and exploring interesting problems in n-dimensional calculus and geometry that are accessible at the calculus level, but rarely covered in standard calculus courses. These problems are often motivated by applications to probability, statistics and other areas, they tend to have a broad appeal and are well-suited for creating interactive visualizations for presentation at outreach events. and for publication at the Wolfram Demonstrations website. Last year (2015/2016), we focused on the "geometry of voting", an intriguing geometric approach to voting theory in which voter preferences and election outcomes are represented by points in the plane or in a higher-dimensional space. This approach allows one to visualize different voting methods, and it can help explain voting paradoxes. Depending on the interests and background of the team members, we may continue with the voting theory theme or explore other directions - for example, game theory. For further details and reports on past projects, visit Professor Hildebrand's webpage.

• Team Meetings
Twice a week for around two hours each, with additional meetings during peak periods.
• Project Difficulty
Basic
Students should have taken Math 241, preferably in the honors version, with a grade of A- or better, or be concurrently enrolled in an honors section of Math 241. Beyond strong calculus skills, there are no formal prerequisites, and the project can accommodate a broad range of interests, backgrounds, and majors. Experience with Mathematica visualizations is helpful, but not required. More important than any formal prerequisites are an enthusiasm about the subject and the project, an open mind, a willingness to work with like-minded students in a team, and a willingness and to make the necessary time commitment for this project to succeed.
• Randomness in Number Theory

Faculty Mentor: A J Hildebrand

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 project we seek to explore such random features experimentally - via large scale computations and geometric visualizations as random walks - and, if possible, also theoretically. In 2015/2016, we focused on random and non-random features in the leading digits of arithmetic sequences such as the Fibonacci numbers. Depending on the interests and background of the team members, we may continue this line of investigation in Fall 2016, or explore other types of randomness in number theory - for example, random walks defined in terms of digital expansions of numbers. For further details and reports on past projects, visit Prof. Hildebrand's webpage.

• Team Meetings
Twice a week for around two hours each, with additional meetings during peak periods.
• Project Difficulty

All team members should have a strong general mathematical background, and have taken Math 453 with a grade of A- or better. At least two team members should also have strong coding skills in C/C++, and experience working in a Unix/Linux command-line environment (we will be using the Campus Computing Cluster for large scale computations). Experience with visualizations in Mathematica and/or Python would be plus, but is not required.

• Discrete Morse Theory, Vector Fields, and Materials Science

Faculty Mentor: Dr. Ruth Davidson and Dr. Rosemary Guzman

This is a continuation of a project from Spring 2016. The goal of this project is to compare structural properties of various materials using techniques of topological data analysis, derived from Discrete Morse Theory, in the hopes of developing an alternative to laboratory-based testing. We will perform these comparisons by using command-line open source software developed by a team of Australian scientists for image analysis. Adapting their software allows us to look for key structural features of substances that correlate to properties such as the relative conductivities of metals and the relative durabilities of biological substances such as tooth enamel.

Weekly
• Project Difficulty
Intermediate

Completion of Calculus 3, willingness to install and use open-source command-line software, interest in applied mathematics and interdisciplinary research. Experience with a programming language such as python, Java, C, or C++ is helpful, but not required. The applications of the mathematics and software in this project include disciplines such as bioengineering, aerospace, electrical engineering, and physics.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• Polyhedral Geometry for Analyzing Phylogenetic Methods

Faculty Mentor: Dr. Ruth Davidson and Dr. Rosemary Guzman

Polyhedra are generalizations of polygons in arbitrary dimension. Phylogenies are mathematical models of the common evolutionary history of a group of species. Polyhedra have historically connected many fields of applied mathematics to pure mathematics via optimization and computer science. Recently, polyhedra have been used to evaluate the accuracy and biases of methods for using biological data to construct phylogenies from a geometric point of view, leading to an explosion of important open problems and a shortage of prepared scientists to work on them. In this project we will learn about a variety of open problems in this area, decide which problem is most interesting to the group, and study it using open-source software.

Weekly
• Project Difficulty
Intermediate

Completion of Calculus 3. Willingness to install and use open-source command-line software and learn basic scripting skills, interest in applied mathematics and interdisciplinary research. The biology behind this project is important but can be learned by the team during the project. Experience with a programming language such as python, Java, C, or C++ is helpful, but not required.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• Connecting Algebraic Geometry to Phylogenies via Singular Value Decomposition

Faculty Mentor: Dr. Ruth Davidson

Algebraic geometry is one of the oldest, broadest, and famously difficult fields in mathematics. Phylogenies are mathematical models of the common evolutionary history of a group of species. The method SVDquartets of Chifman and Kubatko (2014) uses algebraic geometry in an accessible way, via singular value decomposition, to build phylogenies, and bypasses a known, nontrivial source of error in phylogeny estimation. To remedy the fact that this method has only been tested on simulated and biological data in a few studies with mixed results, this project will continue this line of inquiry with further testing on biological and simulated data and exploration of strategies for boosting its performance.

Weekly
• Project Difficulty
Intermediate

Completion of Calculus 3. Willingness to install and use open-source command-line software and learn basic scripting skills, interest in applied mathematics and interdisciplinary research. The biology behind this project is important but can be learned by the team during the project. Experience with a programming language such as python, Java, C, or C++ is helpful, but not required.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• Optimization of Digital Business through Analytics

Faculty Mentor: Shu Li and Nick Baier

Running a start-up means successfully allocating resources where they have the highest potential for a positive return; an optimization problem with a combination of hard data and soft assumptions. A small tech start-up in Chicago is looking for math majors to help analyze current product and digital marketing performance in light of individual and industry trends. The end result will be a professional presentation along with a recommendation directly to the company. There will be additional aspects of this opportunity that allow students to translate mathematical acumen to business situations. This project entails working closely with a University of Illinois math alumni in a professional setting.

Weekly
• Project Difficulty
Easy to Intermediate