# Illinois Geometry Lab

## IGL Projects, Fall 2017

Applications

To apply for one of these projects, use one of the following forms. Application deadline: Monday August 21, 2017.

• #### Complex dynamics and zeros of derivatives

##### Faculty Mentor: Aimo Hinkkanen

This is a continuation of the IGL project of the same name from Fall 2016. 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.

Weekly
• ##### Project Difficulty
Basic
Completion of Calculus 3
• #### Stability of Quasicrystal Frameworks in 3D

##### Faculty Mentor: George Francis

This project concerns the stability of a rod-and-pinion framework under unrestricted deformations. Such a framework is the 1-skeleton a Penrose cluster of 3D quasicrystals. We develop real-time interactive computer animations for experiments to generate conjectures, verify them experimentally, and then develop proofs of the theorem if possible.

Weekly
• ##### Project Difficulty
Intermediate
Some linear algebra, 3D vector geometry and programming experience are needed for the advanced component. But even if you have never programmed an animation you are welcome to join the novice subgroup. We will be using the (free) online textbook on computer graphics by David Eck at Hobart College.
• #### Visual Cliffs, Virtual Reality, and Movement Disorders

##### Faculty Mentors: Richard Sowers (Math, ISE) and Manuel Hernandez (Kinesiology)

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 cofaculty mentor for this project. We intend to connect this project to the UIUC Health Care Engineering Systems Center.

• ##### Team Meetings
Every other week
• ##### Project Difficulty
Solid coding ability is required.
• #### Video as a Sensor

##### Faculty Mentors: Richard Sowers (Math, ISE) and Daniel Work (CEE)

This project is a continuation of the Spring 2017 "Geometry of Autonomous Vehicles" project. The near future promises a rise in sensor-equipped vehicles.. While machine learning is a core enabling technology, we believe that some models and context may provide additional benefits. 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.

• ##### Team Meetings
Every other week
• ##### Project Difficulty
Solid coding ability is required.
• #### Surfaces moving under Hamiltonian flows

##### Faculty Mentors: Ely Kerman

In the study of Hamiltonian flows (classical mechanics) the motion of a point (particle) is determined by a single function (the energy). In this project we will study a new evolution equation for star-shaped surfaces that is defined in terms of Hamiltonian flows. To each starshaped surface one can associate a simple function which measures how far the surface is from a round sphere. This function defines a Hamiltonian flow which moves the surface, which changes the flow, which moves the surface, .... etc. The resulting evolution of the original surface is governed by a differential equation which in two dimensions equals the standard heat equation, and in higher dimensions resembles it. The hope is that, like solutions of the heat equation, the evolving surface quickly converges to an interesting equilibrium. The first goal of the project will be to model the relevant equation in order to study the convergence of the solutions and the special features of the equilibrium surfaces.

• ##### Team Meetings
once a week initially, biweekly eventually
• ##### Project Difficulty
Intermediate
Completion of Calculus 3 and familiarity with differential equations at the level of Math 285. Familiarity with software like Matlab and Python is also desired but not expected.
• #### Automata and Space-Filling Curves

##### Faculty Mentors: Philipp Hieronymi and Erik Walsberg

In the last few years novel connections between mathematical logic, automata theory and metric geometry have emerged. A question that often arises in this area is the following: let $r \in \mathbb{N}_{>1}$ and let $C \subseteq \mathbb{R}^n$ be a geometrically interesting set (often a fractal), can the set of all $r$-ary representations of elements of $C$ be recognized by a Büchi automaton? For example, let C be the usual middle-thirds Cantor set. Elements of C are precisely those real numbers in $[0,1]$ that have a ternary representation in which the digit 1 does not occur. Therefore it is not hard to see that the set of ternary representations of elements of $C$ can be recognized by such an automaton. The goal of this project to answer similar questions in the case that $C$ is the graph of a function. In particular, this project aims to determine whether graphs of space-filling curves can be recognized in this way.

• ##### Team Meetings
At least biweekly
• ##### Project Difficulty
Intermediate
Completion of MATH 347 and 400-level real analysis course are required. Completion of MATH 414 or CS 374 helpful, but knowledge of logic or automata theory is not a prerequisite.
• #### Poncelet theorem and complex analysis

##### Faculty Mentors: Jeremy Tyson

The goal of this project is to investigate a surprising connection between the geometry of conic sections and finite Blaschke products. Poncelet’s theorem states that if a polygon is inscribed in a conic section and circumscribes another conic section, then each point of the outer conic is a vertex of a polygon with the same property. Daepp, Gorkin and Mortini (2002) identified a surprising connection between Poncelet’s theorem and the geometry of finite Blaschke products. A finite Blaschke product is the product of finitely many automorphisms of the unit disc. Let B(z) be a Blaschke product of degree three with one zero at the origin and two other nonzero and distinct zeros. Each point on the unit circle has three preimages. The Daepp-Gorkin-Mortini result states that the (Euclidean) triangle formed by these preimages is of Poncelet type. In fact, the triangles formed by the three preimages of any point of the unit circle all circumscribe a fixed ellipse, whose foci lie at the remaining two zeros of B. In this project we’ll investigate various related questions. For instance, what happens if we replace “Euclidean triangle” by “hyperbolic triangle”? Alternatively, what is the locus spanned by all such triangles under other normalization for degree three Blaschke products? No background is needed in complex analysis; this project is accessible to students who have completed Math 241.

• ##### Team Meetings
weekly at first, then periodically
• ##### Project Difficulty
Basic
Completion of Calculus 3
• #### Calculus, Geometry, and Probability in n Dimensions

##### Faculty Mentors: A.J. Hildebrand

This project is part of an ongoing program that began in Fall 2012 and that is now in its tenth semester. The general goal of this program is to seek out and explore interesting problems in n-dimensional calculus and geometry that are accessible at the calculus level, motivated by applications in probability, statistics, economics, and other areas, and that 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. During the past two semesters we focused on mathematical questions arising in game theory, and in particular on mathematical models of poker. Depending on the interests and backgrounds of the participants, we may continue with the "Mathematics of Poker" theme in the coming semester, or explore other problems of similar flavor at the interface of mathematics and economics. For further details, and reports on past projects, visit http://www.math.illinois.edu/~ajh/ugresearch/.

• ##### Team Meetings
Twice a week for two hours each time
• ##### 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. Programming experience with Mathematica and/or Python, would be a plus, but is 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.
• #### Chaotic maps, exotic number systems, and arithmetic coding schemes

##### Faculty Mentors: A.J. Hildebrand and Ken Stolarsky

The goal of this project is to explore surprising connections between two seemingly unrelated problems, one belonging to analysis, and the other belonging to number theory: The behavior of chaotic maps on the one hand, and the representation of numbers in "non-standard", or "exotic", number systems on the other hand. A chaotic map is, roughly speaking, a function that when iterated a large number of times behaves in unpredictable ways. We will focus on a particular class of such maps, namely piecewise linear functions from [0,1] to [0,1]. Examples include the "tent map", which consists of an up-slope followed by a down-slope, and maps of the form f(x)= {ax+b}, where the braces denote the fractional part. These maps lead in a natural way to representations, or "encodings", of real numbers with respect to generalized, or "exotic", number systems, a connection that has been used in recent years to design encryption and compression schemes. Despite the apparent simplicity of these maps, their behavior under iteration is still not well understood. In this project we will investigate these maps - from both the "chaotic map" and the "generalized number systems" angle - using computer experimentation and visualization, and hopefully also gain new theoretical insights.

• ##### Team Meetings
Twice a week
• ##### Project Difficulty
Intermediate