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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The goal of this project is to investigate the geometry of subsets of the plane obtained by iterating contracting linear mappings defined by finite collections of disjoint triangles. The most famous construction of this type is the self-similar Sierpinski triangle (or Sierpinski gasket). See, for instance, http://www.math.uiuc.edu/~tyson/sg.html for pictures of the Sierpinski gasket and other generalized fractal triangles. In this project, we will be particularly interested in the case when the linear transformations are not similarity mappings. Various concepts of dimension (Hausdorff dimension, box-counting dimension) have been developed to describe the complexity of nonsmooth objects. We will compute or estimate dimensions for such generalized fractal triangles, and study how such objects and their dimensions behave under deformations.