A mathematical billiard consists of particles moving freely inside a planar domain, bouncing preserving the angle at the boundary. A braid is a collection of trajectory of $n$ points in the plane, distinct at all times, such that all points end at their starting positions. From $n$ generic trajectories of a billiard running for some long time $T$ one can construct a braid with $n$ strands, by moving the particles back to their starting positions along straight lines. Understanding the topological characteristics of the resulting braids (such as the winding numbers, counting how often one strand wraps around the other) are the main goal of this project. We expect very different asymptotics for chaotic and integrable dynamics.
The famous Riemann hypothesis ($1 million problem) states that all the non-trivial (i.e. critical) zeros of the Riemann zeta-function are located on the critical line Re(s) = 1/2 of the complex plane. Recent developments have increased the previous record of 40% of zeros on the line to 41%. New analytical techniques are now available (due to the author and other members of the faculty) that should allow us to increase this to perhaps 42%. Help is needed in coding the analytical aspects of the project. Generous credit will be given.
Feynman's interpretation for quantum mechanics using path integrals has a toy version in combinatorics. The objective of this project is to continue the numerical and graphical analysis of the exponential Laplacian graph (IGL Project Fall 2016) as the combinatorial candidate for the partition function, and to interpret the path integral in terms of paths in the graph. 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, and it carries applications in information theory and materials science.
weekly, two hours per week
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.
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). Professor Dan Work of Civil and Environmental Engineering is a co-faculty mentor for this project. Solid coding ability is required. See http://www.math.illinois.edu/~r-sowers/ResearchGroup/MTP.html for some prior IGL work.
The near future promises a rise in autonomous vehicles. While machine learning is a core enabling technology, we believe that some models and context may provide additional benefits. We will be working with an enhanced driving video dataset from comma.ai. Our goal is to compute some of the geometry of traffic surrounding the driver and understand how this affects driving decisions. Hopefully you will gain some expertise which is valuable in this exciting area (think Uber and Tesla). Solid coding ability is required. Professor Dan Work of Civil and Environmental Engineering is a co-faculty mentor for this project.
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 analog 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 of the form 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.
Much remains to be learned about orbits of elements under the present [0,1] -> [0,1] golden ratio iteration (tent map iteration), including detailed distribution of period lengths, patterns in the purely periodic behavior of maximal elements, and interactions between the two unstable fixed points. Further examples of iterations with similar properties that emerged during the Fall semester are as yet barely explored.
Starting with four mutually tangent circles in the plane, one can iteratively inscribe circles into triangular interstices, and obtain a fractal set called an Apollonian circle packing. See https://en.wikipedia.org/wiki/Apollonian_gasket for pictures of Apollonian circle packings. The purpose of this project is to study the fine structure of Apollonian circle packings. We will compute several spatial statistics of the circles from an Apollonian circle packings, like electrostatic energy and minimal distance distribution, and study the limiting behavior of these statistics.
Virtual reality (VR) is entering into the public consciousness. One area where it may have significant impact is the medical industry. In this project, we seek to create a VR component of a testbed for understanding responses to visual stimuli and its relation to Parkinson's disease and other movement disorders. We will likely develop on a Hololens (computer languages are likely to be C# and Unity) or a similar VR platform. Hopefully you will gain some expertise which is valuable in this exciting area. Solid coding ability is required. Professor Manuel Hernandez of the Dept. of Kinesiology & Community Health is a co-faculty mentor for this project. We intend to connect this project to the UIUC Health Care Engineering Systems Center.
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 in probability, statistics, economics, 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 we focused on mathematical questions arising in voting theory and game theory, and in particular on mathematical models of poker. We plan to continue with this theme in Spring 2017. For further details, and reports on past projects, visit http://www.math.illinois.edu/~ajh/ugresearch
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 are comparing two geometric realizations of the space of all phylogenetic trees to discover which is better suited for performing biological data analysis via various methods. We are using a combination of custom-built and open-source software.
In this project, we will investigate the representation of integers from orbits of subgroups of SL(2, Z). There is a notion 'critical exponent'. It has been known that when the critical exponent of a group is large enough, almost every integer passing some mild algebraic condition will appear in an orbit of this group. First we will use an algorithm from Curtis McMullen to compute the critical exponents for several groups, then we study what happens for the integer orbits when the critical exponents of the corresponding groups are not large.
Weekly initially, then depends on progress.
This project will investigate new notions of curvature attached to curves and surfaces in a non-Riemannian 3-dimensional manifold, the sub-Riemannian Heisenberg group. The goal is to develop geometric intuition for these new notions via the computation and visualization of examples. We will start by studying curvature of curves and surfaces in flat (Euclidean) 3-space as well as in Riemannian 3-dimensional manifolds. One possible goal of the project is to investigate analogs of the Fenchel and Fary-Milnor theorems on the total curvature of closed curves, possibly knotted curves. Students should be familiar with basic differential geometry (at the level of Math 423 or Math 481).
Initially twice per week, then once per week.
"The goal of this project is to better understand symplectic circle actions with isolated fixed points. In practice, students will study these by looking for sets of integers which satisfy certain equations. For more details, see full project description.
Initially once per week, then every other week.
Surfaces can be made by connecting together sides of polygons. This idea of interlocking polygons is key to constructing jigsaws however, usually, these only use small pieces to build a larger polygon. By adding even more connections to obtain a surface, we should be able to construct jigsaw puzzles with many different solutions. Each solution coming from a symmetry of the underlying surface. This project will focus on researching how to find these symmetries (the Veech group), but additionally we will try to manufacture some of these puzzles at the IGL or FabLab.
Initially weekly, then biweekly
The study of self-intersection numbers of closed geodesics in finite volume hyperbolic spaces has received heavy attention and witnessed significant progress in the last decade. In particular, Mirzakhani proved an asymptotic polynomial growth for the number of geodesics with no self-intersection. But a typical geodesic can have many self intersections, and after proper normalization, the self intersection numbers are distributed like Gaussian, as shown by Chas and Lalley. In this project, we will numerically reexamine these results in the infinite volume setting.
once a week initially, then depending on progress
This is an elementary project to introduce beginners to programming a mathematical real-time interactive computer animation. Specifically, we address the problem of modelling linkages that have limited motion, and need to stay together at the joints, such as flexing chemical structures, and quasicrystal frameworks. See bottom of the http://new.math.uiuc.edu webpage. We'll use VPython for this. Participants will learn Python, and basic unix communication tools, HTML, and making animated GIFs of their creations.
once a week (and more online)
Weekly, and more online